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Research article

Robust passivity analysis of mixed delayed neural networks with interval nondifferentiable time-varying delay based on multiple integral approach

  • New results on robust passivity analysis of neural networks with interval nondifferentiable and distributed time-varying delays are investigated. It is assumed that the parameter uncertainties are norm-bounded. By construction an appropriate Lyapunov-Krasovskii containing single, double, triple and quadruple integrals, which fully utilize information of the neuron activation function and use refined Jensen's inequality for checking the passivity of the addressed neural networks are established in linear matrix inequalities (LMIs). This result is less conservative than the existing results in literature. It can be checked numerically using the effective LMI toolbox in MATLAB. Three numerical examples are provided to demonstrate the effectiveness and the merits of the proposed methods.

    Citation: Thongchai Botmart, Sorphorn Noun, Kanit Mukdasai, Wajaree Weera, Narongsak Yotha. Robust passivity analysis of mixed delayed neural networks with interval nondifferentiable time-varying delay based on multiple integral approach[J]. AIMS Mathematics, 2021, 6(3): 2778-2795. doi: 10.3934/math.2021170

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  • New results on robust passivity analysis of neural networks with interval nondifferentiable and distributed time-varying delays are investigated. It is assumed that the parameter uncertainties are norm-bounded. By construction an appropriate Lyapunov-Krasovskii containing single, double, triple and quadruple integrals, which fully utilize information of the neuron activation function and use refined Jensen's inequality for checking the passivity of the addressed neural networks are established in linear matrix inequalities (LMIs). This result is less conservative than the existing results in literature. It can be checked numerically using the effective LMI toolbox in MATLAB. Three numerical examples are provided to demonstrate the effectiveness and the merits of the proposed methods.



    We investigate equations of the form

    utuxx=fT(t,u),    tR, xR, (1)

    where fT:R×RR is of the type

    fT(t,u)=g(u)mT(t)u, (2)

    and T is a positive parameter. We suppose that g is a KPP (for Kolmogorov, Petrovsky and Piskunov) function of class C1(R+) with R+=[0,+). More precisely, we have

    g>0 on (0,1),  g(0)=g(1)=0,  g(0)>0,  g(1)<0, (3)

    and

    ug(u)u decreasing on (0,+). (4)

    The previous hypotheses imply in particular that

    g(u)g(0)u,  u[0,+), (5)

    and that

    g<0 on (1,+). (6)

    In Sections 2 and 4, the function mT is T-periodic, nonnegative and of class C1(R). In this case, the function fT is a T-periodic in time function of class C1(R×R+) such that fT(,0)=0 on R. Furthermore, according to (6) and the nonnegativity of mT, we have

    fT(t,u)<0,  (t,u)R×(1,+). (7)

    In Section 3, the function mT is asymptotically periodic in time. We give more details about this notion later in this introduction.

    Equations of the type

    utuxx=g(u)mT(t)u,    tR, xR, (8)

    are proposed to model the spatial evolution over time of a cancerous tumor in the presence of chemotherapy. The quantity u(t,x) represents the density of cancer cells in the tumor at the position x and at the time t. We begin by considering, for T>1, a particular case of periodic function mT:R+R of class C1(R+) for which there exists a nontrivial function φ:[0,1][0,+) with φ(0)=φ(1)=0 such that

    {mT=φ on [0,1),mT=0 on [1,T). (9)

    In the absence of treatment, cancer cells reproduce and spread in space. This reproduction is modeled by the reaction term of KPP type g(u), which takes into account the fact that the resources of the environment of the tumor are not infinite and so, that there is a maximal size beyond which the tumor cannot grow anymore. To treat the patient, cycles of chemotherapy are given. Every cycle lasts a lapse of time T and is composed of two subcycles. The duration of the first one is equal to 1. During this time, the drug acts on the tumor. At every moment of the first subcycle, the death rate of the cancer cells due to the drug is equal to φ(t). In this case, the total reaction term is g(u)φ(t)u. There is a competition between the reproduction term and the death term. The chemotherapy has a toxic effect on the body because it destroys white blood cells. It is thus essential to take a break in the administration of the treatment. This break is the second subcycle of the cycle of chemotherapy. It lasts during a time equal to T1. In this case, the reaction term is just g(u), and thus, the tumor starts to grow again. To summarize, the term mT(t) defined in (9) represents the concentration of drug in the body of the patient at time t, and the integral T0mT(s)ds=10φ(t)dt represents the total quantity of drug in the patient during a cycle of chemotherapy. Finally, we impose for this type of functions mT that

    g(0)10φ(t)dt<0. (10)

    This inequality is not really restricting. Indeed, we shall see after that this hypothesis is in fact a condition so that the patient is cured in the case where there is no rest period in the cycles of chemotherapy (that is T=1).

    We now refine the previous modelling. In fact, the concentration of drug in the patient's body is not a datum. We only know the concentration of drug injected to the patient. We denote DT(t) this concentration at time t, and we assume that the function DT:R+R+ is T-periodic and satisfies

    DT(t)={1,  t[0,1],0,  t(1,T). (11)

    The concentration of drug m is then the Lipschitz-continuous and piecewise C1 solution m:R+R of a Cauchy problem of the type

    {m(t)=DT(t)m(t)τ,  tR+,m(0)=m00. (12)

    The real number τ>0 is called clearance. It characterizes the ability of the patient's body to eliminate the drug. It is also possible to take into account that the patient does not necessarily take the treatment in an optimal way. It may happen to him/her, for example, to forget his/her medicine, or being forced to move a chemotherapy session if it is programmed on a holiday. So, we add to the nonlinearity a perturbative term of the type εp(t,u), where ε0 and p:R+×RR. It corresponds to study equations of the type

    utuxx=g(u)m(t)u+εp(t,u),    tR, xR,

    where m solves (12).

    The mathematical study of reaction-diffusion equations began in the 1930's. Fisher [12] and Kolmogorov, Petrovsky and Piskunov [17] were interested in wave propagation in population genetics modeled by the homogeneous equation

    utuxx=f(u),    tR, xR. (13)

    In the 1970's, their results were generalized by Aronson and Weinberger [1] and Fife and McLeod [11]. In particular, if f is a KPP nonlinearity (that is, f satisfies (3) and (5)), there exists a unique (up to translation) planar fronts Uc of speed c, for any speed cc:=2f(0), that is, for any cc, there exists a function uc satisfying (13) and which can be written uc(t,x)=Uc(xct),  with 0<Uc<1, Uc()=1 and Uc(+)=0. Furthermore, if c<c, there is no such front connecting 0 and 1. Another property for this type of nonlinearities is that if we start from a nonnegative compactly supported initial datum u0 such that u00, then the solution u of (13) satisfies u(t,x)1 as t+. Aronson and Weinberger name this phenomenon the "hair trigger effect". Moreover the set where u(t,x) is close to 1 expands at the speed c.

    Freidlin and Gärtner in [13] were the first to study heterogeneous equations. More precisely, they generalized spreading properties for KPP type equations with periodic in space coefficients. Since this work, numerous papers have been devoted to the study of heterogeneous equations with KPP or other reaction terms. We can cite e.g. [2,3,4,5,6,8,10,16,19,27,28,29] in the case of periodic in space environment, [14,18,19,24,25] in the case of periodic in time environment and [21,22,23] in the case of periodic in time and in space environment. The works of Nadin [21,22] and Liang and Zhao [19] are the closest of our paper. We will compare later the contributions of our work with these references. We now give the main results of the paper.

    When the nonlinearity is not homogeneous, there are no planar front solutions of (8) anymore. For equations with coefficients depending periodically on the space variable, Shigesada, Kawasaki and Teramoto [26] defined in 1986 a notion more general than the planar fronts, namely the pulsating fronts. This notion can be extended for time dependent periodic equations as follows.

    Definition 1.1. For equation (1), assume that fT is T-periodic and that (1) has a T-periodic solution θ:R(0,+), tθ(t). A pulsating front connecting 0 and θ(t) for equation (1) is a solution u:R×RR+ such that there exists a real number c and a function U:R×RR+ verifying

    {u(t,x)=U(t,xct),    tR,  xR,U(,)=θ,  U(,+)=0,    uniformly on R,U(t+T,x)=U(t,x),    tR,  xR.

    So, a pulsating front connecting 0 and θ for equation (1) is a couple (c,U(t,ξ)) solving the problem

    {UtcUξUξξfT(t,U)=0,    (t,ξ)R×R,U(,)=θ,  U(,+)=0,    uniformly on R,U(t+T,ξ)=U(t,ξ),   (t,ξ)R×R.

    In this definition, by standard parabolic estimates, the limiting state θ=U(,) solves the system

    {y=fT(t,y)  on R,y(0)=y(T), (14)

    whose solutions are called equilibrium states of the equation (1).

    If θ:RR is a solution of (14), let us now define λθ,fT and Φθ,fT:RR as the unique real number and the unique function (up to multiplication by a constant) which satisfy

    {(Φθ,fT)=(fTu(t,θ)+λθ,fT)Φθ,fT  on R,Φθ,fT>0  on R,Φθ,fT is Tperiodic. (15)

    These quantities are called respectively principal eigenvalue and principal eigenfunction associated with fT and the equilibrium state θ. Furthermore, if we divide the previous equation by Φθ,fT, and if we integrate over (0,T), we obtain an explicit formulation of the principal eigenvalue, namely

    λθ,fT=1TT0fTu(s,θ(s))ds.

    We now recall the definition of the Poincaré map PT associated with fT. For any α0, let yα:R+R+ be the solution of the Cauchy problem

    {y=fT(t,y) on R,y(0)=α. (16)

    Definition 1.2. The Poincaré map associated with fT is the function PT:R+R+ defined by

    PT(α)=yα(T).

    We conclude, with the fact that each nonnegative solution of (14) is associated with a fixed point of PT, and conversely. Furthermore, if αT0 is a fixed point of PT we have the following equality

    (PT)(αT)=eTλyαT,fT. (17)

    We can find these results concerning the notions of principal eigenvalue and Poincaré map in [7], [9], [15] and [20].

    Let T>0. In Section 2, we study (1) and (2) with functions mT which are T-periodic in time. For these functions we assume there exists T>0 such that

    λ0,fT{>0 if T<T,<0 if T>T,=0 if T=T. (18)

    This is indeed the case if mT is of the type (9) because

    λ0,fT=g(0)+1TT0mT(s)ds=g(0)+1T10φ(s)ds.

    Furthermore, for this type of functions, hypothesis (10) implies that λ0,fT=1>0. Hence, in this case T>1. The existence and uniqueness of positive solutions of (14) is summarized in the following result.

    Proposition 1. We consider the real number T defined in (18).

    (Ⅰ) If TT, there is no positive solution of (14).

    (Ⅱ) If T>T, there is a unique positive solution wT of (14). Furthermore,

    (i) For any tR we have wT(t)(0,1], and

    1TT0fTu(s,wT(s))ds0.

    (ii) If TmT is continuous in Lloc(R), then the function T(T,+)wT(0) is continuous and, if mT is of type (9) with assumption (10), it is increasing.

    (iii) If TmT is continuous in Lloc(R), then the function wT converges uniformly to 0 on R as T(T)+.

    (iv) If mT is of type (9) with assumption (10), then wT converges on average to 1 as T tends to +:

    limT+1TT0wT(t)dt=1.

    The same result of existence and uniqueness (result of the type (II)) was proved for KPP nonlinearities depending periodically on space by Berestycki, Hamel and Roques in [5] and for KPP nonlinearities depending periodically on space and time by Nadin in [22]. We give here a proof using the Poincaré map associated with fT. The last two points of the proposition are quite intuitive. Indeed, the limit as T(T)+ is explained by the fact that for TT, the only nonnegative equilibrium state is zero. The limit as T+ is explained by the fact that in this case, the nonlinearity fT is "almost" the KPP function g since the function mT has an average close to 0 when T is large.

    Let us now summarize a result in [22], which deals with the evolution of u(t,x) as t+.

    Proposition 2. [22] Let u0:RR be a bounded and continuous function on R such that u00 and u00. Under assumption (18), we consider the function u:R+×RR satisfying

    {utuxx=fT(t,u)  on (0,+)×R,u(0,)=u0  on R. (19)

    If T<T, then there exists M>0 depending only on u0 and Φ0,fT such that

    0u(t,x)MΦ0,fT(t)eλ0,fTt,  (t,x)R+×R. (20)

    If T=T, then

    supxR|u(t,x)|t+0.

    If T>T, then for every compact set KR, we have

    supxK|u(t,x)wT(t)|t+0.

    A similar result was proved for KPP nonlinearities depending periodically on space by Berestycki, Hamel and Roques in [5].

    In the biological context with mT satisfying (9), the treatment is effective (in the sense that u(t,x)0 uniformly on R as t+) if and only if the duration of cycles of chemotherapy is equal or less than T. In particular, since hypothesis (10) implies that T>1, the treatment is effective if there is no rest period between two injections of drug, that is as T=1. The result is interesting because it implies that T1 is the longest rest period for which the patient recovers. Inequality (20) refines the criterion of cure of the patient because according to the fact that the function Tλ0,fT is decreasing and positive on (0,T), the convergence rate of the density u(t,x) to 0 as t+ is all the faster as T is small. In other words, in the case of effective treatment, shorter the period between two injections, more quickly the patient will be cured. If the treatment is not effective, the equilibrium state wT invades the whole space as t+. In particular, the tumor can not grow indefinitely. Finally, Proposition 2 also allows to clarify the result (ii) of Proposition 1. The fact that TwT(0) is increasing on (T,+) implies that in the case where the treatment is not effective (that is wT>0 invades the whole space as t+), the longer the rest period between two injections, the denser the equilibrium state of the tumor.

    We now study in more detail the case where the treatment is not effective, that is, the case where T>T. We know that then, the equilibrium state wT invades the whole space as t+. The purpose of this part is to give the invasion rate of the zero state by wT. To answer this question, we quote two results. The first one is about the existence of pulsating fronts connecting 0 and wT, in the sense of Definition 1.1, and the second one concerns spreading properties. They are proved in [18] and in [21].

    Theorem 1.3. [18], [21] Let T>T, where T is given in (18). (Ⅰ) There exists a positive real number cT such that pulsating fronts with speed c connecting 0 and wT exist if and only if ccT.

    (Ⅱ) We denote u:R+×RR the solution of the Cauchy problem

    {utuxx=fT(t,u)  on (0,+)×R,u(0,)=u0  on R.

    If u0 is a bounded continuous function such that u00 and u00, then

    c(0,cT),    limt+sup|x|<ct|u(t,x)wT(t)|=0.

    If u0 is a continuous compactly supported function such that u00, then

    c>cT,    limt+sup|x|>ctu(t,x)=0.

    In his paper [21], Nadin considers in the first assertion of the spreading properties in Theorem 1.3 initial conditions which are more general. He assumes that u0 is not necessarily compactly supported but that u0 is of the form O(eβ|x|) as |x|+, where β>0. The previous theorem completes Proposition 2. Indeed, we know that in the case where the treatment is not effective, the equilibrium state wT invades the whole space as t+. Theorem 1.3 states that this invasion takes place at the speed cT.

    We can now characterize the critical speed cT with the principal eigenvalue λ0,fT. More precisely:

    Proposition 3. For every T>T, the critical speed cT is given by

    cT=2λ0,fT. (21)

    Hence, if TT0mT(s)ds is continuous, then the function T(T,+)cT is continuous and, if T0mT(s)ds does not depend on T, it is increasing. Furthermore, we have the two following limit cases:

    limT(T)+cT=0,

    and, if 1TT0mT(s)dsT+0, then

    limT+cT=2g(0).

    In the case where the treatment is not effective, the invasion of space by the equilibrium state wT is all the faster as the rest time between injections is long. The two limits cases T(T)+ and T+ are explained in the same manner as in Proposition 1. Let us note that in the case where mT is of the type (9), then the previous properties concerning T0mT(s)ds are satisfied.

    We end this section by stating the existence of pulsating fronts in the case of nonlinearities which are not of KPP type (that is hypotheses (4) and (5) are not necessarily verified, but we still assume (3), (6) and (18)). For these nonlinearities, there is still a positive solution to problem (14), but it may not be unique. According to Cauchy-Lipschitz theorem, solutions of (14) are ordered on [0,T]. For T>T, we can thus define yT:RR as the infimum of all positive solutions of (14). After showing that yT>0, we will prove there exists a critical speed cT>0 such that there is a pulsating front connecting 0 and yT for speed ccT and there is no pulsating front connecting 0 and yT for c<cT. In this case, cT is not necessarily equal to 2λ0,fT. For this type of nonlinearity, Nadin shows in [21] that there exist two critical speeds c and c for which there is a pulsating front for cc and there is no pulsating front for cc. Nevertheless the case c(c,c) is not treated in [21]. In [18], Liang and Zhao prove the result using a semiflow method. We give here an alternative proof. We begin by proving the existence of pulsating front U(t,ξ) for domains of the type R×[a,a] which are bounded in ξ, then we pass in the limit as a+. We state the result.

    Proposition 4. Let fT satisfy assumptions (2), (3), (6) and (18), and T>T. There exists a positive real number cT such that pulsating fronts U(t,ξ) monotone in ξ connecting 0 and yT exist if and only if ccT.

    We are interested in the case of nonlinearities which are no more periodic in time, but which are the sum of a function which converges as t+ to a time periodic nonlinearity and of a small perturbation. More precisely, for ε0, we consider equations of the type

    utuxx=g(u)m(t)u+εp(t,u),    tR, xR, (22)

    where m solves (12) with T>1 and DT defined in (11). We assume that p:R+×RR is a function of class C1 for which there exists C>0 such that

    |p(t,u)u|C,  (t,u)R+×(0,+). (23)

    The function m is not periodic, but it is asymptotically T-periodic in time. More precisely, there exists a T-periodic positive function mT:R(0,+) such that

    limt+|m(t)mT(t)|=0. (24)

    Indeed, an elementary calculation implies that for any nN, we have

    m(t)={τ[1+((e1τ1)(enTτ1)eTτ1+m0TenTτ)etτ],   t[nT,nT+1),τ[(e1τ1)(e(n+1)Tτ1)eTτ1+m0T]etτ,   t[nT+1,(n+1)T).

    Consequently, if we define the positive T-periodic function mT:R(0,+) by

    mT(t)={τ[1+(e1τ1eTτ11)etτ],    t[0,1],τe1τ1eTτ1eTtτ,    t[1,T),

    then the convergence result (24) holds. Furthermore, we have T0mT(t)dt=τ. Consequently the function fT:R+×R+R defined by fT(t,u)=g(u)mT(t)u satisfies (18) because λ0,fT=g(0)+τ/T. We assume that τ>g(0). We notice that mT is independent of m0. It was predictable because mT is the unique positive T-periodic solution of m=DTm/τ on R. We define the nonlinearities f:R+×R+R and fε:R+×R+R by

    f(t,u)=g(u)m(t)u,  and  fε(t,u)=f(t,u)+εp(t,u).

    According to (24), we have

    supu(0,+)|f(t,u)fT(t,u)u|t+0. (25)

    The function fT is T-periodic and satisfies the general assumptions given in Section 1.3. We still denote T the critical time (notice that T>1 because τ>g(0)), wT the unique positive equilibrium state for T>T and cT the critical speed associated with fT for T>T.

    The aim of this section is to show that Proposition 2 and the spreading results of Theorem 1.3 hold true when we replace fT by fε in the statements, for ε small enough. It is reasonable to hope so. Indeed, on the one hand, if ε is small, then the term εp is negligible compared to f, and on the other hand, these results deal with the large time behavior of the solutions, and precisely, hypothesis (25) implies that f "looks like" fT as t+. The first result is the generalization of Proposition 2.

    Theorem 1.4. Let u0:RR be a bounded and continuous function such that u00 and u00. For all ε0, we consider the function uε:R+×RR satisfying

    {utuxx=fε(t,u)  on (0,+)×R,u(0,)=u0  on R. (26)

    If T<T, there exists εT>0 such that for all ε(0,εT) we have

    limt+supxR|uε(t,x)|=0.

    If T>T and if λwT,fT>0, then there exist ˜εT>0 and MT>0 such that for all ε(0,˜εT) and for all compact KR, we have

    lim supt+ supxK|uε(t,x)wT(t)|MTε.

    We saw in Proposition 1 that λwT,fT0. In the previous theorem, in case T>T, we impose that λwT,fT>0. This property is not necessarily satisfied. Indeed, if we consider the function h:R+R defined by h(u)=u(1u)3, then we have h(0)=h(1)=0, h>0 on (0,1), h<0 sur (1,+), h(u)/u decreasing on (0,+) and h(1)=0. In the case where the function fT(t,) is concave for all tR+, the property λwT,fT>0 is verified for any T>T. Indeed, if we define F:[0,1]R by

    F(x)=1TT0fT(s,xwT(s))wT(s)ds,

    then we have F(0)=F(1)=0 and F is convex on [0,1]. Consequently, if F(1)=0, that is, if λwT,fT=0, then we have F=0 on [0,1]. It is a contradiction because F(0)=λ0,fT<0.

    Let us give a sketch of the proof. For T>0 and ε>0, we will frame fε by two T-periodic functions fTε and fTε for which the results of Proposition 2 will apply. In the case where T<T, if fTε is the upper bound function, we will show that for ε>0 small enough, we have λ0,fTε>0. Hence, the solution of (26) with fTε as nonlinearity is a supersolution of problem (26) and, according to Proposition 2, it converges to 0 as t+. In the case where T>T, we will prove that for ε>0 small enough, we have λ0,fTε<0 and λ0,fTε<0. Consequently, there is a unique positive solution wTε (resp. wTε) of system (14) with fTε (resp. fTε) as nonlinearity (owing to Proposition 1). The solution of (26) with fTε as nonlinearity is a supersolution of (26) and, according to Proposition 2, it converges to wTε as t+. In the same way, the solution of (26) with fTε as nonlinearity is a subsolution of (26), and it converges to wTε as t+. We will conclude using the fact that wTε and wTε are close to wT as ε is small enough.

    Note that the case T=T is not treated in Theorem 1.4. If ε=0, the solution of the Cauchy problem (26) converges uniformly to 0 as t+, whereas if ε>0, the convergence to 0 may not hold. We summarize these results in the following proposition.

    Proposition 5. Let T=T and ε0. We consider the function uε:R+×RR satisfying the Cauchy problem (26).

    (Ⅰ) If ε=0, then uε converges uniformly to 0 as t+.

    (Ⅱ) If ε>0, we can conclude in two cases.

    (i) If f(t,u)=fT(t,u) and p(t,u)=u, then, for ε small enough, uε converges to a positive solution of (14) with fε as nonlinearity as t+.

    (ii) If p(t,u)0, then, uε converges uniformly to 0 as t+.

    Concerning the spreading results of Theorem 1.3, they remain true if we replace fT by fε in the statement.

    Theorem 1.5. Let T>T. For any ε0, we consider uε:R+×RR satisfying

    {utuxx=fε(t,u)  on (0,+)×R,u(0,)=u0  on R.

    If u0 is a continuous bounded function such that u00 and u00, and if λwT,fT>0, then for all c(0,cT), there exists ˆεc,T>0 such that for all ε(0,ˆεc,T) we have

    lim supt+sup|x|<ct|uε(t,x)wT(t)|MTε,

    where MT is defined in Theorem 1.4.

    If u0 is a continuous compactly supported function such that u00, then, for all c>cT, there exists ˉεc,T>0 such that for all ε(0,ˉεc,T) we have

    limt+sup|x|>ctuε(t,x)=0.

    The proof of this theorem uses the same ideas as the proof of Theorem 1.4. For T>T and ε>0, we will frame fε by two T-periodic functions fTε and fTε for which the results of Theorem 1.3 will apply. An important point of the demonstration will be to notice that for ε small enough, the critical speeds cT,ε and cT,ε associated respectively with fTε and fTε are close to the critical speed cT associated with fT.

    As in Section 1.1, we consider a C1 and T-periodic function mT (with T1) of the type

    {mT=φ on [0,1),mT=0 on [1,T),

    where φ:[0,1][0,+) satisfies φ(0)=φ(1)=0. In this part, we are interested in equations of the type

    utuxx=g(u)mTτ(t)u,    tR, xR, (27)

    where 0<τT. The function g satisfies hypotheses (3), (4) and (6). The function mTτ:R+R+ is T-periodic and defined by

    {mTτ(t)=1Tφ(tT),  t[0,τ),mTτ(t)=0,       t[τ,T),

    where the function φ is the same as in mT. In these equations, the duration of the treatment is equal to τ. Furthermore, we have

    T0mTτ(t) dt=1Tτ0φ(tT)dt=10φ(t)dt. (28)

    So, it is clear that the quantity of drug administered during a cycle of chemotherapy is independent of the treatment duration τ. We will study the influence of the parameter τ with respect to the results of previous sections. We define the functions fTτ:R+×R+R and fTτ:R+×R+R by

    fT(t,u)=g(u)mT(t)u  and  fTτ(t,u)=g(u)mTτ(t)u.

    The first proposition deals with the principal eigenvalue associated with fTτ and the equilibrium state 0.

    Proposition 6. Let T>0 and τ(0,T]. The real number λ0,fTτ is independent of τ. Actually, we have

    λ0,fTτ=λ0,fT=g(0)+10φ(s)dsT.

    Consequently, if T>0 denotes the critical time for the function fT, then, for any τ(0,T), fTτ satisfies (18) for T[τ,+), and the critical time T associated with fTτ is the same as the one associated with fT. We are interested here in the solutions of the system

    {y=fTτ(t,y)  on R,y(0)=y(T). (29)

    The same proof as in Proposition 1 implies that for any τ(0,T) and T[τ,T], there is no positive solution of (29), while for any T>T and τ(0,T], there is a unique positive solution wTτ:R(0,1] of (29). Furthermore, the same proof as in Proposition 2 implies that if τ(0,T) and T[τ,T], then the treatment is efficient, and if T>T and τ(0,T], then the equilibrium state wTτ invades the whole space as t+. More precisely, Proposition 2 remains true by replacing fT by fTτ and wT by wTτ. To summarize, the optimal duration of a chemotherapy cycle for which the treatment is efficient does not depend on how the drug is injected.

    Let us now study the case where the treatment is not efficient, that is, T>T and τ(0,T]. Theorem 1.3 remains valid if we replace fT by fTτ and wT by wTτ, but with a critical speed cT,τ depending a priori on τ. Nevertheless Propositions 3 and 6 imply that cT,τ=2λ0,fTτ=2λ0,fT=cT, where cT is the critical speed associated with fT. Consequently, the invasion rate does not depend on how the drug is administered.

    Finally, we are interested in the influence of the parameter τ on the equilibrium state wTτ.

    Proposition 7. Let T>T. The function

    {(0,T)(0,+)τwTτ(0)

    is continuous and decreasing.

    Consequently, in the case where the treatment is not efficient, the shorter the duration of the chemotherapy cycle, the larger the value of the equilibrium state wTτ(0). This means that it is better to administer the treatment over long periods.

    Outline. Section 2 is devoted to the proof of Propositions 1, 3 and 4. Section 3 gathers the proof of Theorem 1.4, Proposition 5 and Theorem 1.5. Finally, we prove in Section 4 Propositions 6 and 7.

    We first investigate solutions of (14), showing Proposition 1. We begin with the case where TT. We argue by way of contradiction, supposing there is a positive solution w of (14). Then

    (w)(t)w(t)=g(w(t))w(t)mT(t),  t[0,T].

    We integrate this equation between 0 and T. We obtain

    T0(g(w(s))w(s)mT(s))ds=0. (30)

    Yet, as w>0 on [0,T] and according to (4) and (18), we have

    1TT0(g(w(s))w(s)mT(s))ds<λ0,fT0,

    which contradicts (30).

    We now consider the case where T>T. To prove the existence of a positive solution of (14), we give two lemmas demonstrating the existence of a positive fixed point of the Poincaré map PT defined in Definition 1.2.

    Lemma 2.1. There exists α0>0 such that for all α(0,α0] we have PT(α)>α.

    Proof. Indeed, according to the fact that fT(,0)=0, we have PT(0)=0, and owing to (17) and the fact that λ0,fT<0 we have (PT)(0)>1.

    Lemma 2.2. For all α>1, we have PT(α)<α.

    Proof. Let α>1. We consider yα solution of (16). Two cases can occur.

    1st case. If yα(t)>1 for all t0, then, according to (7), we have yα(t)=fT(t,yα(t))<0 for all t0. Consequently yα(T)<yα(0), that is PT(α)<α.

    2nd case. If there exists t00 such that yα(t0)1, then, owing to (7), we have yα(t)1 for all tt0. In particular, for n0N such that n0Tt0, we have yα(n0T)1<yα(0). Yet, the sequence (yα(nT))n is constant or strictly monotone. So it is decreasing. Consequently we have yα(T)<yα(0), that is PT(α)<α.

    Lemma 2.1 and Lemma 2.2 imply that there exists α(α0,1] such that PT(α)=α. Consequently, the solution of (16) with α=α is a positive solution of (14). We prove now the uniqueness of such a solution. Let w1:RR and w2:RR two positive solutions of (14). There exists ρ>1 such that w1ρw2 on [0,T]. We can define

    ρ=inf{ρ1 | w1(t)ρw2(t),  t[0,T]}.

    We have

    w1(t)ρw2(t),  t[0,T]. (31)

    Moreover there exists t[0,T] such that

    w1(t)=ρw2(t). (32)

    We are going to show that ρ=1. We argue by way of contradiction supposing that ρ>1. So

    w1(t)=fT(t,w1(t)),  t[0,T]. (33)

    Furthermore

    (ρw2)(t)>fT(t,ρw2(t)),  t[0,T]. (34)

    Indeed, for all t[0,T],

    (ρw2)(t)=ρw2(t)=ρw2(t)(g(w2(t))w2(t)mT(t))>ρw2(t)(g(ρw2(t))ρw2(t)mT(t)) (according to (4) since ρ>1)=fT(t,ρw2(t)).

    According to (31), (32), (33), (34) and the T-periodicity of w1 and w2, we have

    w1(t)=ρw2(t),  t[0,T].

    It is a contradiction because w1 is a solution of y=fT(t,y) whereas ρw2 is a strict supersolution. So ρ=1. Consequently, by the symmetry of the roles played by w1 and w2, we have w1w2 on [0,T], and then on R by periodicity.

    We denote wT the positive solution of (14). We now show the properties of wT. The previous proof implies that (PT)(wT(0))1. Hence, according to (17), it follows that λwT,fT0. We also saw that wT(0)(0,1]. Consequently, owing to (7) and the fact that fT(,0)=0 on R, we have wT(t)(0,1] for any tR.

    We now study the function T(T,+)wT(0). We show the monotonicity of TwT(0) if mT is of type (9), with assumption (10) (in this case T>1). We consider two real numbers T1 and T2 such that T<T1<T2. For i{1,2}, the Poincaré map PTi associated with fTi is defined on R+ by

    PTi(α)=yTiα(Ti),  α0,

    where yTiα is the solution of the Cauchy problem

    {y=fTi(t,y)  on R,y(0)=α. (35)

    We saw in () that the function PTi has a unique positive fixed point αTi. Furthermore αTi(0,1]. The unique equilibrium state wTi:R(0,1] associated with fTi is the solution of the Cauchy problem (35) with α=αTi. Consequently, if we prove that PT1<PT2 on (0,1], then we will deduce that αT1<αT2, that is wT1(0)<wT2(0). Let α(0,1]. The functions yT1α and yT2α are solutions on [0,T1] of the equation

    y=fT1(t,y).

    Consequently, since yT1α(0)=yT2α(0)=α, we have

    yT1αyT2α  on [0,T1].

    Furthermore, from (3), (7) and the fact that φ in (9) is nonnegative and nontrivial, there holds

    0<yT1α(T1)=yT2α(T1)<1.

    On [T1,T2], yT2α is a solution of y=g(y). Consequently, according to (3), we have yT2α(T1)<yT2α(T2). Finally, it follows that

    yT1α(T1)=yT2α(T1)<yT2α(T2).

    In other terms

    PT1(α)<PT2(α).

    Finally, we have necessarily αT1<αT2, that is wT1(0)<wT2(0).

    We show now the continuity property. Let ˜T(T,+) and (Tn)n be a sequence of (T,+) such that Tnn+˜T. We fixe T(T,˜T). There exists nN and T+>T such that

    T<T<Tn<T+,  nn. (36)

    We will demonstrate that wTn(0)n+w˜T(0). Since 0<wTn1 and TmT is continuous in Lloc(R), the sequence (wTn)n converges up to extraction of a subsequence to a function ˜w in C0,δ([0,T+]) for any δ(0,1). The equilibrium state wTn satisfies

    {wTn(t)=wTn(0)+t0fTn(s,wTn(s))ds,  t[0,T+],wTn(0)=wTn(Tn).

    Passing to the limit as n+, we obtain

    {˜w(t)=˜w(0)+t0f˜T(s,˜w(s))ds,  t[0,˜T][0,T+],˜w(0)=˜w(˜T).

    The function tt0f˜T(s,˜w(s))ds is of class C1([0,˜T]). Consequently ˜w is of class C1([0,˜T]) and it satisfies

    {˜w=f˜T(t,˜w)  on [0,˜T],˜w(0)=˜w(˜T),

    and 0˜w1 in [0,˜T]. Owing to (II), it follows that ˜w0, or ˜ww˜T. If ˜w=0, then wTn0 as n+ uniformly on [0,T+]. For any nN, we have

    (wTn)(t)wTn(t)=fTn(t,wTn(t))wTn(t),  t[0,Tn].

    We integrate the previous equation over [0,Tn], then we pass to the limit as n+. We obtain ˜Tλ0,f˜T=0. It is a contradiction because λ0,f˜T<0, as ˜T>T. Hence, we have necessarily ˜ww˜T. The uniqueness of the accumulation point of (wTn)n implies that the convergence holds for the whole sequence. In particular, wTn(0)n+w˜T(0), and consequently, the function TwT(0) is continuous on (T,+).

    We study now the behavior of the equilibrium state wT for the limit cases where T(T)+ and T+. We begin by showing that the function wT converges uniformly to 0 on R as T(T)+. Let (Tn)n be a sequence such that Tnn+T and Tn>T for any nN. Since (Tn)n is bounded, there exists T+>T such that for any nN we have Tn(T,T+). Up to extraction of a subsequence, (wTn)n converges to a function w in C0,δ([0,T+]) for any δ(0,1). The equilibrium state wTn satisfies

    {wTn(t)=wτn(0)+t0fTn(s,wTn(s))ds,  t[0,T+],wTn(0)=wTn(Tn).

    Passing to the limit as n+, we obtain

    {w(t)=w(0)+t0fT(s,w(s))ds,  t[0,T][0,T+],w(0)=w(T).

    The function tt0fT(s,w(s))ds is of class C1([0,T]). Consequently w is of class C1([0,T]) and it satisfies

    {(w)=fT(t,w)  on [0,T],w(0)=w(T),

    and 0w1 on [0,T]. According to (II), w0. The uniqueness of accumulation point of (wTn)n implies that the convergence holds for the whole sequence. Furthermore since [0,Tn][0,T+] for any nN, by Tn-periodicity of wTn, it occurs that

    supR|wTn|=sup[0,Tn]|wTn|sup[0,T+]|wTn|n+0,

    which completes the proof of this point.

    We study now the case where T+ under assumptions (9) and (10). The function wT converges on average to 1 as T tends to +. We give a technical lemma.

    Lemma 2.3. Under assumptions (9) and (10), the real number δ defined by

    δ:=inf{wT(1) | TT+1}

    is positive. Furthermore, δ<1.

    Proof. We argue by way of contradiction. Let us suppose there exists a sequence (Tn)n such that Tnn++ and wTn(1)n+0. We fix T+>T. There exists n+N such that for any nn+, we have Tn[T+,+). According to the monotonicity of TwT(0), it follows that

    0<wT+(0)<wTn(0),  nn+.

    Up to extraction of a subsequence, (wTn)n converges to a function w in C0,β([0,1]) for any β(0,1). Passing to the limit as n+ in the previous inequalities implies that

    0<wT+(0)w(0). (37)

    The same reasoning as previously implies that the function w is of class C1([0,1]) and satisfies the Cauchy problem

    {(w)=g(w)φ(t)w  on [0,1],w(1)=0.

    By uniqueness, we have necessarily w0, that is, wTn converges uniformly to 0 on [0,1], which contradicts (37). Lastly, each function wT ranges in (0,1], and due to (7) and the nontriviality of φ in (9), one has wT<1 on R. Hence, we have δ<1.

    We return to the proof of the last point of Proposition 1. We consider yδ the solution of the Cauchy problem

    {y=g(y)  on (1,+),y(1)=δ,

    where δ(0,1) is defined in Lemma 2.3. Let ε>0 be such that δ<1ε<1. Since yδ(t)t+1, there exists lε>1 such that yδ(lε)=1ε/2. We define Tε=4lε/ε (>lε), and we consider TTε. The function wT is a solution of

    {y=g(y)  on (1,T),y(1)=wT(1).

    Since wT(1)δ, we have wTyδ on [1,T). In particular wT(lε)1ε/2, and since wT is increasing on (lε,T), we have

    1ε2wT(t)<1,  t(lε,T).

    Furthermore

    |1TT0wT(t)dt1|1T(lε0|wT(t)1|dt+Tlε|wT(t)1|dt).

    Yet,

    1Tlε0|wT(t)1|dt2lεT2lεTε=ε2.

    and

    1TTlε|wT(t)1|dtTlεTε2ε2.

    So |1TT0wT(t)dt1|ε, and the proof of Proposition 1 is complete.

    We begin by showing the characterization of cT with the principal eigenvalue λ0,fT. Let μR. We denote λμ the principal eigenvalue and Φμ the principal eigenfunction associated with the operator Lμ:C1(R)C0(R) defined by LμΨ=Ψt(μ2+fTu(t,0))Ψ. Consequently, we have

    (Φμ)t=(μ2+fTu(t,0)+λμ)Φμ   on R.

    We divide the previous equation by Φμ, then we integrate between 0 and T. According to the fact that Φμ is T-periodic, we obtain λμ=μ2+λ0,fT. In [21], Nadin gives the following characterization of the critical speed cT:

    cT=inf{cR | there exists μ>0 such that λμ+μc=0}.

    Consequently, we have

    cT=inf{cR | there exists μ>0 such that μ2μcλ0,fT=0}.

    We thus look for the smallest real number c for which the equation μ2μcλ0,fT=0 of the variable μ admits a positive solution. An elementary calculation leads to cT=2λ0,fT. Consequently, we have

    cT=2g(0)1TT0mT(t)dt.

    Hence the function T(T,+)cT is continuous, increasing if T0mT(t)dt does not depend on T, and we have the two limits cases

    limT+cT=2g(0)  if 1TT0mT(t)dtT+0,  and  limT(T)+cT=0,

    which concludes the proof of Proposition 3.

    Let α[0,1]. We recall that if yα:RR is the solution of the Cauchy problem

    {y=fT(t,y) on R,y(0)=α,

    then we denote PT:α[0,1]PT(α)=yα(T) the Poincaré map associated to the function fT. According to the proof of Proposition 1, there exists a fixed point of PT in (0,1]. Nevertheless, since hypothesis (4) is not satisfied here, this fixed point is not necessarily unique. We define

    α0=inf{α(0,1] | PT(α)=α}.

    To simplify the notations, we denote yT:RR the function yT=yα0. We begin by proving that this infimum is not equal to zero.

    Lemma 2.4. We have α0>0.

    Proof. We assume that α0=0. So, there exists a sequence (αn)n(0,1]N such that PT(αn)=αn and αnn+0. We divide the equation yαn=fT(t,yαn) by yαn, then we intregrate between 0 and T. We obtain

    T0fT(s,yαn(s))fT(s,0)yαn(s)ds=0.

    Passing to the limit as n+, since yαn0 uniformly on [0,T] as n+ by Cauchy-Lipschitz theorem, we have

    T0fTu(s,0)ds=0,

    which contradicts the fact that λ0,fT<0. Consequently α0>0. Notice also that, by continuity of PT, there holds PT(α0)=α0, and yT=yα0 solves (14). Furthermore 0<yT1 on R.

    Since fT is of class C1(R×[0,1],R) and T-periodic, there exists ε0(0,1) such that for all ε(0,ε0] and for all tR we have

    |fT(t,εΦ0,fT(t))εΦ0,fT(t)fTu(t,0)||λ0,fT|2εΦ0,fT(t), (38)

    where Φ0,fT is the principal eigenfunction associated with fT and 0. Since λ0,fT<0 and yT is the smallest positive solution of system (14), we can apply Theorem 2.3 of the Nadin's paper [21]. Consequently, there exists a couple (c0,U0), where U0:R×R[0,1], (t,ξ)U0(t,ξ) is of class C1,2(R2) and solves

    {(U0)t(U0)ξξc0(U0)ξ=fT(t,U0)  on R×R,U0(,)=U0(+T,)  on R×R,U0(,)=yT , U0(,+)=0  uniformly on R. (39)

    Necessarily c0>0 because Nadin shows in [21] that for c<2λ0,fT, which is a positive real number, there is no pulsating front of sped c connecting 0 and yT. Furthermore, we have

    ξU0(t,ξ)<0,  (t,ξ)R×R.

    Let c1>0 be a real number such that there exists a pulsating front U1 with speed c1 such that ξU1<0 on R×R, and let c2>c1. We are going to prove the existence of a pulsating front U2 such that (c2,U2) solves (39) and ξU2<0 on R2. Yet, by [21], the set

    C={cR | there exists a pulsating front U of speed c such that ξU<0 on R×R}

    is closed and included in [2λ0,fT,+). This will conclude the proof of Proposition 4 by denoting cT=infC.

    Given c1<c2 as above, let a>0 and rR. We define

    εa,r=min{min[0,T]×[a,a]U1(,+r)2Φ0,fT(),ε0,yT(0)Φ0,fT(0)}.

    We consider the problem

    {UtUξξc2Uξ=fT(t,U)  on (0,T)×(a,a),U(0,)=U(T,)  on [a,a].U(,a)=U1(,a+r) , U(,a)=εa,rΦ0,fT  on [0,T]. (40)

    We begin by showing that the previous problem has a solution.

    Proposition 8. There exists a solution to problem (40).

    Proof. We consider the problem

    {UtUξξc2Uξ=fT(t,U)  on (0,+)×(a,a),U(,a)=U1(,a+r) , U(,a)=εa,rΦ0,fT  on [0,+),U(0,)=ψ  on [a,a],

    where ψC0([a,a],[0,1]). This Cauchy problem admits a solution Uψ defined on R+×[a,a]. Furthermore, 0Uψ1 in R+×[a,a] from the maximum principle and the definition of εa,r. We define the closed convex set

    C={ψC0([a,a],[0,1]) | εa,rΦ0,fT(0)ψU1(0,+r)  on [a,a]}.

    Note that this set is not empty since Φ0,fT>0, U11 and εa,rΦ0,fT(0)U1(0,+r) on [a,a] according to the definition of εa,r. We start by proving that if ψC, then Uψ(T,)C using a comparison lemma.

    Lemma 2.5. Let ψC. Then we have

    εa,rΦ0,fT(t)<Uψ(t,ξ)<U1(t,ξ+r)   (t,ξ)(0,+)×(a,a). (41)

    Proof. Since ξU1<0 on R×R and c1<c2, the function U1(,+r) satisfies on [0,+)×(a,a),

    (U1(+r))t(U1(+r))ξξc2(U1(+r))ξfT(t,U1(+r))=(c1c2)(U1(+r))ξ>0.

    Moreover, since ψC, we have U1(0,+r)ψ on [a,a] and, according to the definition of εa,r and the T-periodicity of U1 and Φ0,fT, we have U1(,a+r)εa,rΦ0,fT on [0,+). Consequently, we can apply a comparison principle, and we obtain

    Uψ(t,ξ)U1(t,ξ+r)  (t,ξ)[0,+)×[a,a].

    In the same way, since εa,rε0, and according (38) and the negativity of λ0,fT, we have on [0,+)×(a,a)

    (εa,rΦ0,fT)t(εa,rΦ0,fT)ξξc2(εa,rΦ0,fT)ξfT(t,εa,rΦ0,fT)=εa,rΦ0,fT(λ0,fT+fTu(t,0))fT(t,εa,rΦ0,fT)=εa,rλ0,fTΦ0,fT(fT(t,εa,rΦ0,fT)εa,rΦ0,fTfTu(t,0))εa,rλ0,fTΦ0,fTεa,rλ0,fT2Φ0,fT<0.

    Furthermore since ψC, we have εa,rΦ0,fT(0)ψ on [a,a] and, according to the definition of εa,r and the T-periodicity of U1 and Φ0,fT, we have εa,rΦ0,fTU1(,a+r) on [0,+). Consequently, we can apply a comparison principle and we conclude that

    εa,rΦ0,fT(t)Uψ(t,ξ)   (t,ξ)[0,T]×[a,a],

    The fact that the inequalities in (41) are strict is a consequence of the strong maximum principle.

    We return to the proof of Proposition 8. We consider

    T:CCψUψ(T,)

    Owing to (41) and the T-periodicity of Φ0,fT and U1, T is well defined. We are going to demonstrate using the Schauder's fixed point theorem that the function T has a fixed point in the closed convex set C. We show now that T is continuous. In fact we show that T is a Lipschitz-continuous function. Let ψ and φ in C. We have on (0,T]×[a,a]

    (UψUφ)t(UψUφ)ξξc2(UψUφ)ξ=β(t,ξ)(UψUφ),

    where β:(0,T]×[a,a]R is defined by

    β(t,ξ)={fT(t,Uψ(t,ξ))fT(t,Uφ(t,ξ))Uψ(t,ξ)Uφ(t,ξ),  if Uψ(t,ξ)Uφ(t,ξ),fTu(t,Uψ(t,ξ)),if Uψ(t,ξ)=Uφ(t,ξ).

    Since |β|fTuL([0,T]×[0,1]) on (0,T]×[a,a], and UψUφ=0 on [0,T]×{a,a}, the maximum principle yields for any (t,ξ)[0,T]×[a,a]

    |Uψ(t,ξ)Uφ(t,ξ)|ψφL([a,a])efTuL([0,T]×[0,1])t.

    If we take t=T, we obtain

    Uψ(T,)Uφ(T,)L([a,a])efTuL([0,T]×[0,1])TψφL([a,a]).

    So T is a Lipschitz-continuous function.

    We prove now that T(C) is compact. Let (ψn)n be a sequence of C. By standard parabolic estimates, the sequence (Uψn(T,))n is bounded in C2,α([a,a],[0,1]) for any α(0,1). Since C2,α([a,a],[0,1]) embeds compactly into C0([a,a],[0,1]), (Uψn(T,))n converges up to extraction of a subsequence in C.

    So, according to Shauder's fixed point theorem, there exists ψa,rC([a,a],[0,1]) such that T(ψa,r)=ψa,r, that is Uψa,r(T,)=Uψa,r(0,). Actually, the function Uψa,r is solution of (40). By uniqueness and T-periodicity of fT, Uψa,r can be extended as a T-periodic solution of (40) in R×[a,a].

    To simplify the notations, we denote now Ua,r instead of Uψa,r. Owing to Lemma 2.5 and the T-periodicity of Ua,r, we have the following inequalities

    εa,rΦ0,fT(t)<Ua,r(t,ξ)<U1(t,ξ+r)   (t,ξ)[0,T]×(a,a). (42)

    We are now going to use a sliding method and we first give a comparison lemma.

    Lemma 2.6. Let U and V be two T-periodic functions solving problem (40). Let h[0,2a]. We define Vh(t,ξ)=V(t,ξ+h) for any (t,ξ)[0,T]×[a,ah]. Then, we have

    VhU  on [0,T]×[a,ah].

    Proof. We denote Ih=[a,ah]. For h=2a, we have Ih={a}. Since U(,a)=U1(,a+r), V2a(,a)=V(,a)=εa,rΦ0,fT and

    εa,rΦ0,fTU1(,a+r)2<U1(,a+r) on [0,T],

    it occurs that V2a<U on [0,T]×I2a. Furthermore, VhU on [0,T]×Ih for all h[0,2a] sufficiently close to 2a, by continuity of U and V. Consequently, we can define

    h=inf{h_0 | h[h_,2a], VhU on [0,T]×Ih}.

    We have 0h<2a. We are going to show by way of contradiction that h=0. Thus let us suppose that h>0. By continuity and T-periodicity of U and Vh, the definition of h implies that

    VhU on R×Ih. (43)

    Furthermore, if we define the bounded function η:R×IhR by

    η(t,ξ)={fT(t,U(t,ξ))fT(t,Vh(t,ξ))U(t,ξ)Vh(t,ξ),  if U(t,ξ)Vh(t,ξ),fTu(t,U(t,ξ)),if U(t,ξ)=Vh(t,ξ),

    then, we have on R×Ih

    (UVh)tc2(UVh)ξ(UVh)ξξ=η(t,ξ)(UVh). (44)

    Consequently, according to (43) and (44), if there exists (t,ξ)R×(a,ah) such that U(t,ξ)=Vh(t,ξ), then, by the strong maximum principle, the continuity and the T-periodicity of U and Vh, we have

    Vh=U on R×Ih. (45)

    Yet, according to (42) (which is automatically fulfilled from the arguments used in Lemma 2.5), and since ξU1<0 on R×R, we have for any tR,

    Vh(t,a)=V(t,a+h)<U1(t,a+h+r)<U1(t,a+r)=U(t,a).

    Consequently, Vh<U on R×[a,ah). Furthermore, according to (42), for any tR, we also have

    Vh(t,ah)=V(t,a)=εa,rΦ0,fT(t)<U(t,ah).

    So, it occurs that

    Vh<U on R×Ih.

    Since [0,T]×Ih is a compact set, and both U and V are continuous on [0,T]×[a,a], there exists h0(0,h) such that for any η(0,h0), we have Vhη<U on [0,T]×Ihη. This contradicts the definition of h. Consequently we have h=0 and the proof of Lemma 2.6 is complete.

    Corollary 1. There exists a unique function Ua,r solving (40).

    Proof. We apply the conclusion of Lemma 2.6 with h=0 and reverse the roles of U and V.

    Corollary 2. The function rRUa,rC0([0,T]×[a,a],[0,1]) is continuous.

    Proof. Let rR and (rn)n be a sequence of real numbers such that rnnr. According to standard parabolic estimates and the T-periodicity of each function Ua,rn, there exists U such that, up to extraction of a subsequence, Ua,rnnU in C1,α2 in t and in C2,α in ξ, for any α(0,1). Consequently,

    {(U)t(U)ξξc2(U)ξ=fT(t,U)  on R×(a,a),U(0,)=U(T,)  on [a,a],U(,a)=U1(,a+r) , U(,a)=εa,rΦ0,fT  on [0,T].

    The uniqueness of the solution of the previous problem (Corollary 1) implies that we have U=Ua,r, and that the whole sequence (Ua,rn) converges to U.

    Corollary 3. For any t[0,T] and ξ(a,a), we have

    ξUa,r(t,ξ)<0.

    Proof. We apply Lemma 2.6 with U=V=Ua,r. The strict inequality is a consequence of the maximum principle applied to ξUa,r.

    Proposition 9. There exist εa(0,ε0] and raR such that Ua,ra(0,0)=εaΦ0,fT(0)2.

    Proof. There exists (ta,r,ξa,r)[0,T]×[a,a] such that

    εa,r=min{U1(ta,r,ξa,r+r)2Φ0,fT(ta,r),ε0,yT(0)Φ0,fT(0)}

    Let (rn)n be a sequence of real numbers such that rnn+. There exists a function Ua, such that up to extraction of a subsequence, Ua,rnn+Ua, in C0,α([0,T]×[a,a]) for any α(0,1). Since (ta,rn)n is bounded, there exists ta[0,T] such that up to extraction of a subsequence, we have ta,rnn+ta. So, according to the fact that (ξa,rn) is also bounded (because a is fixed here), it follows that

    εa,rnn+εa:=min{yT(ta)2Φ0,fT(ta),ε0,,yT(0)Φ0,fT(0)}

    We thus have Ua,(,a)=εaΦ0,fT on [0,T]. Consequently, since ξUa,0 on [0,T]×[a,a], it occurs that Ua,(0,0)εaΦ0,fT(0). So there exists n0N such that rn0<0 and

    Ua,rn0(0,0)34εaΦ0,fT(0).

    Let now (˜rn)n be a sequence of real numbers such that ˜rnn++.There exists a function Ua,+ such that up to extraction of a subsequence Ua,˜rnn+Ua,+ in C0,α([0,T]×[a,a]) for any α(0,1). Furthermore, for any t[0,T], we have Ua,˜rn(t,a)=U1(t,a+˜rn)n+0. Consequently, since ξUa,+0 and Ua,+0 on [0,T]×[a,a], it occurs that Ua,+0. So, there exists n1N such that ˜rn1>0 and

    Ua,˜rn1(0,0)14εaΦ0,fT(0).

    According to Corollary 2, there exists ra(rn0,˜rn1) such that

    Ua,ra(0,0)=12εaΦ0,fT(0),

    which completes the proof.

    Proposition 10. There exists a sequence ann++ such that Uan,ran converges on any compact set in C1,α2 in t and in C2,α in ξ, for any α(0,1), to a function U2 solving (39) with c=c2, and such that (U2)ξ<0 on R2.

    Proof. Since ta is bounded, there exist t[0,T] and a sequence ann++ such that tann+t. Consequently,

    εann+ε:=min{yT(t)2Φ0,fT(t),ε0,,yT(0)Φ0,fT(0)}>0.

    According the standard parabolic estimates, up to extraction of a subsequence, Uan,ran converges on any compact set to a function U2 in C1,α2 in t and in C2,α in ξ, for any α(0,1). The function U2 satisfies

    {(U2)t(U2)ξξc2(U2)ξ=fT(t,U2)  on [0,T]×R,U2(0,)=U2(T,)  on R,U2(0,0)=12εΦ0,fT(0),(U2)ξ0 on [0,T]×R.

    Since (εΦ0,fT)fT(t,εΦ0,fT) and (yT)=fT(t,yT) on [0,T] and since we have εΦ0,fT(0)yT(0), it occurs that εΦ0,fTyT on [0,T]. Consequently

    U2(0,0)(0,yT(0)2].

    The functions U2(,) and U2(,+) solve the equation y=f(t,y) on [0,T]. Furthermore, U2(t,ξ)yT(t) for all t[0,T] and all ξR, since this inequality holds for U1 and since each function Ua,r satisfies (42). Consequently, since (U2)ξ 0 on [0,T]×R, we have necessarily U2(,)=yT and U2(,+)=0. Finally we apply the strong maximum principle to the equation satisfied by (U2)ξ and obtain (U2)ξ<0 on R2 (otherwise (U2)ξ would be identically equal to zero, which is impossible since U2(,)=yT and U2(,+)=0).

    Let T>0 with TT (that is λ0,fT0). We define

    εT=1C+1min{|λ0,fT|,g(2)2}>0,

    where C is defined in (23). Let ε(0,εT). According to (23) and (25), there exists nεN such that for all tnεT and for all u0 we have

    fT(t,u)(C+1)εufε(t,u)fT(t,u)+(C+1)εu. (46)

    We define the Tperiodic functions fTε:R×R+R and fTe:R×R+R by

    fTε(t,u)=fT(t,u)(C+1)εu,  and  fTe(t,u)=fT(t,u)+(C+1)εu. (47)

    According to (7), it occurs that

    {fTε(t,u)0,  (t,u)R×[2,+),fT(t,u)0,  (t,u)R×[2,+). (48)

    Furthermore, according to (4) and (6), for any u[2,+), we have g(u)/ug(2)/2<0. Consequently, since ε(0,1C+1g(2)2), the following inequality is true

    fTe(t,u)0,  (t,u)R×[2,+), (49)

    Concerning the principal eigenvalues associated with the equilibrium 0 and functions fT, fTε and fTe, the following relations hold

    {λ0,fTε=λ0,fT(C+1)ε,λ0,fTε=λ0,fT+(C+1)ε. (50)

    We begin by handling the case where T<T. Owing to (50), the fact that λ0,fT>0 and since ε(0,λ0,fTC+1), we have

    λ0,fTe>0. (51)

    We consider vε:R+×RR the solution of the Cauchy problem

    {(vε)t(vε)xx=fTε(t,vε)  on (0,+)×R,vε(0,)=uε(nεT,)  on R.

    Owing to (46) and the Tperiodicity of fTε, the function uε(+nεT,) satisfies on (0,+)×R

    (uε(+nεT,))t(uε(+nεT,))xx=fε(t+nεT,uε(+nεT,))fTε(t,uε(+nεT,)).

    So, applying a comparison principle, we obtain

    0uε(t+nεT,x)vε(t,x),  (t,x)R+×R. (52)

    According to (51), Proposition 2 applied with the T-periodic nonlinearity fTε implies that

    limt+supxRvε(t,x)=0.

    Hence, owing to (52),

    limt+supxRuε(t,x)=0,

    which concludes the proof of the first part of Theorem 1.4.

    We now consider the case where T>T. Since λwT,fT>0, there exists μT>0 such that for all μ(0,μT) and for all (t,u,v)R×[0,2]2, we have

    |uv|μ|fT(t,v)fT(t,u)fTu(t,u)(vu)|λwT,fT2|vu|. (53)

    We define the two positive real numbers ˜MT and ˜εT by

    ˜MT=8(C+1)λwT,fTsup[0,T]wTinf[0,T]ΦwT,fT>0,

    and

    ˜εT=min{εT,λwT,fT4(C+1),inf[0,T]wT2˜MTsup[0,T]ΦwT,fT,min{μT,1}˜MTsup[0,T]ΦwT,fT}>0, (54)

    where ΦwT,fT is the principal eigenfunction associated with fT and the equilibrium state wT. Let ε(0,˜εT). According to (50), the fact that λ0,fT<0 and since ε(0,λ0,fTC+1), we have

    λ0,fTε<0, λ0,fT<0, and λ0,fTe<0. (55)

    Owing to (48), (49) and (55), the same proof as in Proposition 1 implies that there exists a unique T-periodic positive equilibrium state wTε (resp. wTε) associated with fTε (resp. fTε). Furthermore, for any tR, we have wTε(t)(0,2] (resp. wTε(t)(0,2]).

    Lemma 3.1. There exists MT>0 independent of ε such that

    {supt[0,T]|wTε(t)wT(t)|MTε,supt[0,T]|wTε(t)wT(t)|MTε. (56)

    Proof. We begin by proving the first inequality. We define the function ˉvε:RR by

    ˉvε(t)=wT(t)+˜MTεΦwT,fT(t).

    We are interested in the problem

    {y=fTε(t,y) on R,y(0)=y(T). (57)

    We will show that ˉvε is a strict supersolution and wT is a strict subsolution of (57). Let tR. We have

    (ˉvε)(t)fT(t,ˉvε(t))(C+1)εˉvε(t)=fT(t,wT(t))+˜MTεΦwT,fT(t)fTu(t,wT(t))fT(t,ˉvε(t))+˜MTεΦwT,fT(t)λwT,fT(C+1)εˉvε(t).

    Since ε(0,μT˜MTsup[0,T]ΦwT,fT), we have |ˉvε(t)wT(t)|μT. Furthermore, wT(t)[0,1], and since ε(0,1˜MTsup[0,T]ΦwT,fT), the definition of ˉvε implies that ˉvε(t)[0,2]. Consequently, it follows from (53) that

    fT(t,wT(t))+˜MTεΦwT,fT(t)fTu(t,wT(t))fT(t,ˉvε(t))λwT,fT2˜MTεΦwT,fT(t).

    Consequently,

    (ˉvε)(t)fT(t,ˉvε(t))(C+1)εˉvε(t)λwT,fT2˜MTεΦwT,fT(t)(C+1)εˉvε(t)=˜MTεΦwT,fT(t)(λwT,fT2(C+1)ε)(C+1)εwT(t).

    Yet ε(0,λwT,fT4(C+1)). So

    λwT,fT2(C+1)ελwT,fT4.

    Hence

    (ˉvε)(t)fT(t,ˉvε(t))(C+1)εˉvε(t)˜MTεΦwT,fT(t)λwT,fT4(C+1)εwT(t)=ε(λwT,fT4˜MTΦwT,fT(t)(C+1)wT(t)).

    Consequently, according to the definition of ˜MT, it follows that

    λwT,fT4˜MTΦwT,fT(t)(C+1)wT(t)=(2ΦwT,fT(t)inf[0,T]ΦwT,fTsup[0,T]wTwT(t))(C+1)>0.

    Finally, ˉvε is a strict supersolution of (57).

    We now show that wT is a strict subsolution of this problem. Let tR. We have

    (wT)(t)fT(t,wT(t))(C+1)εwT(t)=(C+1)εwT(t)<0.

    According to Lemma 3.1 of [22], there exists a solution ˜wε of (57), and one has

    wT(t)<˜wε(t)<wT(t)+˜MTεΦwT,fT(t),  tR. (58)

    In particular, ˜wε is a positive solution of (57). So, by uniqueness, we have ˜wε=wTε. Finally, inequalities (58) rewrite

    supt[0,T]|wT(t)wTε(t)|εMT,

    where MT is defined by MT=˜MTsup[0,T]ΦwT,fT.

    We now give a sketch of the proof of the second inequality of Lemma 3.1. We define the function v_ε:RR by

    v_ε(t)=wT(t)˜MTεΦwT,fT.

    We are interested in the problem

    {y=fTε(t,y) on R,y(0)=y(T). (59)

    We can show in the same way as previously that v_ε is a strict subsolution and that wT is a strict supersolution of (59). According to Lemma 3.1 of [22], there exists a solution ˆwε of (59), and one has

    wT(t)˜MTεΦwT,fT(t)<ˆwε(t)<wT(t),  tR. (60)

    Yet ε(0,inf[0,T]wT2˜MTsup[0,T]ΦwT,fT). So for any tR

    wT(t)˜MTεΦwT,fT(t)wT(t)12ΦwT,fTsup[0,T]ΦwT,fTinf[0,T]wT>0.

    Consequently ˆwε is a positive solution of (59). So, by uniqueness, we have ˆwε=wTε. Finally, inequalities (60) rewrite

    supt[0,T]|wT(t)wTε(t)|εMT,

    which completes the proof of Lemma 3.1.

    Let us now complete the proof of Theorem Theorem 1.4. We recall that ε(0,˜εT), where ˜εT is defined in (54). Let KR be a compact set and let η>0. We consider ˜uε:R+×RR and ˜uε:R+×RR solving respectively

    {(˜uε)t(˜uε)xx=fTε(t,˜uε) on (0,+)×R,˜uε(0,)=uε(nεT,)  on R,

    and

    {(˜uε)t(˜uε)xx=fTε(t,˜uε) on (0,+)×R,˜uε(0,)=uε(nεT,)  on R,

    where nεN is such that (46) holds for all (t,u)[nεT,+)×R+, and uε solves (26). The function vε:R+×RR,(t,x)uε(t+nεT,x) satisfies

    {(vε)t(vε)xx=fε(t+nεT,vε) on R+×R,vε(0,)=uε(nεT,) on R.

    Owing to (46) and the T-periodicity of fTε, it occurs that on R+×R

    (vε)t(vε)xx=fε(t+nεT,vε)fTε(t+nεT,vε)=fTε(t,vε)

    Consequently, since vε(0,)=uε(nεT,)=˜uε(0,) on R, applying a comparison principle, we obtain

    vε(t,x)˜uε(t,x),  (t,x)R+×R.

    In other words

    uε(t+nεT,x)˜uε(t,x),  (t,x)R+×R.

    Actually, we can show in the same way that

    ˜uε(t,x)uε(t+nεT,x)˜uε(t,x),  (t,x)R+×R.

    According to the T-periodicity of wT, we have wT=wT(+nεT) on R. Hence, for any (t,x)R+×R

    ˜uε(t,x)wT(t)uε(t+nεT,x)wT(t+nεT)˜uε(t,x)wT(t). (61)

    Therefore, for any (t,x)R+×K,

    {˜uε(t,x)wT(t)supxK|˜uε(t,x)wTε(t)|supt[0,T]|wTε(t)wT(t)|,˜uε(t,x)wT(t)supxK|˜uε(t,x)wTε(t)|+supt[0,T]|wTε(t)wT(t)|.

    On the other hand, owing to Proposition 2, there exists tε,K,η>0 such that for any ttε,K,η

    supxK|˜uε(t,x)wTε(t)|+supxK|˜ue(t,x)wTe(t)|η. (62)

    According to Lemma 3.1, (61) and (62), we thus have, for any (t,x)[tε,K,η,+)×K

    |uε(t+neT,x)wT(t+neT)|η+MTε.

    In other words, for any ttε,K,η+neT we obtain

    supxK|uε(t,x)wT(t)|η+MTε,

    That is

    lim supt+supxK|uε(t,x)wT(t)|MTε,

    which completes the proof of Theorem 1.4.

    We begin by proving (I). According to (25), there exists t00 such that

    f(t,u)fT(t,u)g(2)2u,  t[t0,+),u[0,+), (63)

    where we recall that g(2)<0. According to (4) and (6), for any u[2,+), we have g(u)/ug(2)/2<0. Consequently, (63) implies that

    f(t,u)0,  t[t0,+),u[2,+), (64)

    We define

    M=max{2,supRu0}.

    The real number M is a supersolution of (26). Furthermore, 0 is solution of (26) and 0u(0,)M on R. Consequently, according to the maximum principle we have

    0u(t,x)M,  tR, xR. (65)

    We denote v:R+R the function satisfying

    {v=f(t,v) on R+,v(0)=M.

    Owing to (65), we have 0u(t0,)M on R. It follows from the comparison principle that

    0u(t+t0,x)v(t),  t0, xR.

    Furthemore, since 2M, it follows from (64) that

    v(t)M,  t0.

    To summarize

    0u(t,+t0,x)v(t)M,  t0, xR. (66)

    We will show that v(t)t+0. We argue by way of contradiction assuming there exists a real number δ0>0 and a sequence tnn++ such that

    v(tn)>δ0,  nN.

    For any nN, we write tn=˜tn+knT, where ˜tn[0,T) and knN, and we define the function vn:[knT,+)R by vn(t)=v(t+knT). The function vn satisfies

    {vn(t)=f(t+knT,vn(t))  t[knT,+),vn(˜tn)=v(tn)>δ0.

    Up to extraction of a subsequence, ˜tnn+t[0,T]. Consequently, according to (25) and the Arzela-Ascoli theorem, there exists v:RR such that vnn+v locally uniformly on R and which satisfies

    {(v)=fT(t,v)  on R,v(t)δ0. (67)

    Furthermore, owing to (66), we have

    0v(t)M,  tR. (68)

    We consider σ:R+R such that

    {σ=fT(t,σ)  on R+,σ(0)=M.

    Owing to (7) and the fact that M1, we have σ(0)σ(T). Consequently, the sequence (σ(nT))n is nonincreasing. Furthermore, it is bounded below by 0. Hence, it converges up to extraction of a subsequence to a real number l0. For any nN, we define the function σn:R+R by σn(t)=σ(t+nT). The sequence (σn)n converges up to extraction of a subsequence in C1([0,T]) to a function σ satisfying

    {(σ)=fT(t,σ)  on [0,T],σ(0)=σ(T)=l.

    According to Proposition 1, we have necessarily σ=0, and thus, the convergence holds for all the sequence. Owing to (68), for any nN, we have v(nT)M. Consequently, since fT is Tperiodic, we can apply a comparison principle and we obtain

    v(nT+t)σ(t),  tR+,  nN.

    In particular

    v(t)σn(t),  nN.

    Passing to the limit as n+, we obtain

    v(t)σ(t)=0,

    which is a contradiction with (67). Consequently v(t)t+0 and thus, we conclude the proof of () using (66).

    We now prove (). We begin by considering the case where f(t,u)=fT(t,u) and p(t,u)=u for any (t,u)R+×R+. In this case, we have

    fε(t,u)=fT(t,u)+εu,  (t,u)R×R+.

    Let ε(0,g(2)/2). The function fε is T-periodic, and we have

    fε(t,u)0,  tR, u[2,+).

    Furthermore λ0,fε=λ0,fTε=ε<0. Consequently, owing to Theorem 1, there exists wTε:R(0,+) solving (14) with fε as nonlinearity. According to Proposition 2, for all compact set KR, we have

    supxK|uε(t,x)wTε(t)|t+0.

    We now consider the case where p(t,u)0 for any (t,u)R×R+. In this case

    fε(t,u)fT(t,u),  (t,u)R+×R+.

    We denote u the solution of the Cauchy problem

    {utuxx=fT(t,u)  on (0,+)×R,u(0,)=u0  on R.

    From the comparison principle, it occurs that

    0uε(t,x)u(t,x)  (t,x)R+×R. (69)

    According to (), we have supxRu(t,x)=0. Consequently supxRuε(t,x)=0, which concludes the proof.

    Proof. Let T>T and c(0,cT), where cT is the critical speed associated with fT defined in Proposition 1.3. We recall that for ε(0,˜εT), where ˜εT is defined in (54), inequalities (46), (48), (49) and (55) are satisfied. Furthermore, the critical speeds associated with nonlinearities fTε and fTε are respectively defined by

    cT,ε=2|λ0,fTε|=2λ0,fT+(C+1)ε,

    and

    cT,ε=2|λ0,fTε|=2λ0,fT(C+1)ε.

    In particular, since cT=2|λ0,fT|=2λ0,fT, there exists εc,T>0 such that for all ε(0,εc,T) we have

    c(0,cT,ε)(0,cT,ε). (70)

    We define

    ˆεc,T=min{˜εT,εc,T}>0. (71)

    We consider ε(0,ˆεc,T). According to the strong maximum principle, we have uε(nεT,)>0 on R, where nεN is such that (46) holds for all (t,u)[nεT,+)×R+. Consequently, there exists a nonnegative and nontrivial compactly supported function ˜uε,0:RR such that

    uε(nεT,x)˜uε,0,    xR. (72)

    Let ˜uε:R+×RR be the solution of the Cauchy problem

    {(˜uε)t(˜uε)xx=fTε(t,˜uε)  on (0,+)×R,˜uε(0,)=˜u0,ε  on R.

    Owing to (46), (72) and the fact that fTε is T-periodic, we can apply a comparison principle and get that

    ˜uε(t,x)uε(t+nεT,x),    (t,x)R+×R. (73)

    According to (49), we have fε0 on R+×[2,+). Hence, since u0 is bounded, if we define

    ˜C=max{2,supRu0},

    then according to the maximum principle, we have uε˜C on R+×R. In particular

    uε(nεT,x)˜C,  xR. (74)

    Let vε:R+R be the solution of

    {(vε)t=fTε(t,vε)  on R+,vε(0)=˜C. (75)

    Owing to (46) and (74), we can still apply a comparison principle to get that

    uε(t+nεT,x)vε(t),    (t,x)R+×R. (76)

    According to (49) and the fact that ˜C2, it occurs that vε(T)vε(0). So the sequence (vε(nT))n is nonincreasing. Furthermore, this sequence is bounded below by 0. Consequently, it converges to a real number l0. For any nN, we define vε,n:R+R by vε,n(t)=vε(t+nT). The sequence (vε,n)n converges up to extraction of a subsequence to vε0 in C1([0,T]) satisfying

    {(vε)=fTε(t,vε)  on [0,T],vε(0)=vε(T)=l.

    So vε is equal to 0 or wTε. Yet, there exists κε>0 such that 0<κεΦ0,fTε(0)˜C and

    |fTε(t,κεΦ0,fTε(t))(fTε)u(t,0)κεΦ0,fTε(t)|λ0,fTε2κεΦ0,fTε(t),  t[0,T].

    Consequently, we have on R+

    (κεΦ0,fTε)fTε(t,κεΦ0,fTε)κεΦ0,fTε(λ0,fTε+(fTε)u(t,0))(κεΦ0,fTε(fTε)u(t,0)+λ0,fTε2κεΦ0,fTε)λ0,fTε2κεΦ0,fTε0.

    Hence, the function κεΦ0,fTε is a subsolution of the problem (75) on R+. Therefore

    0<κεΦ0,fTε(t)vε(t),  tR+.

    Using the T-periodicity of Φ0,fTε and passing to the limit as n+, we obtain

    0<κεΦ0,fTε(t)vε(t),  tR+.

    Consequently, we have necessarily vεwTε on [0,T]. In particular, the uniqueness of accumulation point of the sequence (vε,n)n implies that the convergence to wTε holds for the whole sequence. Let η>0. There exists nη,εN such that

    nnη,εsupt[0,T]|vε(t+nT)wTε(t)|η. (77)

    On the other hand, according to (70), the spreading properties in periodic case (Proposition 1.3) give the existence of tc,η,ε0 such that

    ttc,η,εsup|x|<ct|wTε(t)˜uε(t,x)|η. (78)

    Let (t,x)R+×R such that tmax{tc,η,ε,nη,εT} and |x|<ct. According to (73) and (76), it occurs that

    ˜uε(t,x)uε(t+nεT,x)vε(t)

    The fact that tnη,εT implies that we can write t=ntT+˜t, where ˜t[0,T) and ntN such that ntnη,ε. Consequently, as the function wT is T-periodic, we have

    ˜uε(t,x)wT(t)uε(t+nεT,x)wT(t+nεT)vε(ntT+˜t)wT(˜t)

    Hence, according to (77) and Lemma 3.1

    uε(t+nεT,x)wT(t+nεT)|vε(ntT+˜t)wTε(˜t)|+|wTε(˜t)wT(˜t)|η+MTε,

    and on the other hand, owing to (78) and Lemma 3.1, it occurs that

    uε(t+nεT,x)wT(t+nεT)sup|y|<ct|wTε(t)˜uε(t,y)|sup[0,T]|wTεwT|ηMTε.

    To conclude, for any tmax{tc,η,ε,nη,εT}+nεT, we have

    sup|x|<ct|uε(t,x)wT(t)|η+MTε,

    which concludes the proof of the first assertion of Theorem 1.5.

    We now show the second part of the theorem. We consider c>cT and c such that cT<c<c. There exists εc,T>0 such that for all ε(0,εc,T) we have

    c>min{cT,ε,cT,ε}. (79)

    Furthermore, according to (4), (23) and (25), there exists D>0 such that for all ε[0,1), we have

    fε(t,u)Du,  tR+, uR+. (80)

    We define ˉεc,T=min{1,ˆεc,T,εc,T}>0, where ˆεc,T is defined in (71). Let ε(0,ˉεc,T). We consider H:R+×RR solving the heat equation

    {HtHxx=0  on (0,+)×R,H(0,)=u0  on R.

    The function H is given by

    H(t,x)=12πtSupp(u0)e(xy)24tu0(y)dy,  t(0,+),xR, (81)

    where Supp(u0) is the support of u0, which is here assumed to be compact. We define the function HD:R+×RR by HD(t,x)=H(t,x)eDt. We have (HD)t(HD)xx=DHD on (0,+)×R. Furthermore, owing to (80), we have (ue)t(ue)xx=fε(t,ue)Duε on (0,+)×R. Consequently, since HD(0,)=uε(0,)=u0 on R, the comparison principle yields

    uε(t,x)H(t,x)eDt,  tR+,xR.

    In particular, owing to (81), it occurs that

    uε(nεT,x)eDnεT2πnεTSupp(u0)e(xy)24nεTu0(y)dy,  xR. (82)

    We define the real number

    γc,ε=c+(c)2+4λ0,fTe2.

    Let us note that (c)2+4λ0,fTε>0 because c>cT,ε=2λ0,fTε. According to (82), uε(nεT,) has a Gaussian decay as |x|+ and in particular, there exists a real number Mc,ε>0 such that

    uε(nεT,x)Mc,εΦ0,fTe(0)eγc,εx,  xR. (83)

    We also define the function vc,ε:R+×RR by

    vc,ε(t,x)=Mc,εΦ0,fTe(t)eγc,ε(xct),

    We have on R+×R

    (vc,ε)t(vc,ε)xx=(γ2c,ε+γc,εc+λ0,fTe)Mc,εΦ0,fTeeγc,ε(xct)+(fTe)u(t,0)vc,ε.

    Hence according to (5) and the fact that γ2c,ε+γc,εc+λ0,fTe=0, we obtain on R+×R

    (vc,ε)t(vc,ε)fTe(t,vc,ε)

    Furthermore, owing to (46), (47) and the T-periodicity of fTε, it occurs that on R+×R

    (uε)t(uε)xx=fε(t+nεT,uε)fTε(t+nεT,uε)=fTε(t,uε)

    Consequently, since (83) implies that uε(nεT,)vc,ε(0,) on R, the comparison principle implies that

    0uε(t+nεT,x)vc,ε(t,x),  (t,x)R+×R.

    For all t0, since vc,ε(t,) is decreasing on R, we have

    0supx>ctuε(t,x)supx>ctvc,ε(t,x)vc,ε(t,ct)=Mc,εΦ0,fTe(t)eγc,ε(cc)tt+0.

    In the same way, we can show that

    0supx<ctuε(t+nεT,x)t+0.

    To summarize

    limt+sup|x|>ctuε(t,x)=0,

    which concludes the proof of the second assertion of Theorem 1.5.

    We begin by proving Proposition 6.

    Proof. Owing to (28), the principal eigenvalue associated with 0 and fTτ is given by

    λ0,fTτ=g(0)+T0mTτ(t) dt=g(0)+10φ(t)dt=λ0,fT.

    We now demonstrate Proposition 7.

    Proof. Let T>T. We denote PTT the Poincaré map associated with fTT. We recall that PTT is defined on R+ by

    PTτ(α)=yτ,α(T),

    where yτ,α is the solution of the Cauchy problem

    {(yτ,α)=fTT(t,yτ,α)  on R+,yτ,α(0)=α. (84)

    In the same way as in the proof of Proposition 1, we show that the function PTτ has a unique positive fixed point αTτ. Furthermore αTτ(0,1]. Consequently there is a unique equilibrium state wTτ:R(0,1] associated with fTτ. It is the solution of the Cauchy problem (84) with α=αTτ.

    We begin by showing the continuity property. Let τ(0,T) and (τn)n be a sequence of (0,T) such that τnn+τ. We will demonstrate that wTτn(0)n+wTτ(0). The sequence (wTτn)n converges up to extraction of a subsequence to a function w in C0,δ([0,T]) for any δ(0,1). The equilibrium state wTτn satisfies

    {wTτn(t)=wTτn(0)+t0fTτn(s,wTτn(s))ds,  t[0,T],wTτn(0)=wTτn(T).

    Passing to the limit as n+, we obtain

    {w(t)=w(0)+t0fTτ(s,w(s))ds,  t[0,T],w(0)=w(T).

    The function tt0fTτ(s,w(s))ds is of class C1([0,T]). Consequently w is of class C1([0,T]) and it satisfies

    {(w)=fTτ(t,w)  on [0,T],w(0)=w(T).

    Owing to Proposition 1, it follows that w0, or wwTτ. If w=0, then wTτn0 as n+ uniformly on [0,T]. For any nN, we have

    (wTτn)(t)wTτn(t)=fT(t,wTτn(t))wTτn(t),  t[0,T].

    We integrate the previous equation over [0,T], then we pass to the limit as n+. We obtain Tλ0,fTτ=0. It is a contradiction because since T>T, we have λ0,fTτ=λ0,fT<0. Hence, we have necessarily wwTτ. So the function τwTτ(0) is continuous on (0,T).

    We now study the monotonicity of this function. We consider two real numbers τ1 and τ2 such that 0<τ1<τ2<T. The Poincaré map PTτi associated with fTτi is defined on R+ by

    PTτi(α)=yτi,α(T),

    where yτi,α is the solution of (84), with τ=τi. We recall that the equilibrium state wTτi is the solution on R+ of (84) with α=αTτi. Consequently, if we prove that PTτ1>PTτ2 on (0,+), then we will deduce that αTτ1>αTτ2, that is, wTτ1(0)>wTτ2(0). Fix α>0. We define the function zτi,α:R+R by

    zτi,α(t)=yτi,α(t)et0mTτi(s)ds. (85)

    This function solves on R+ the equation

    (zτi,α)=g(zτi,αet0mTτi(s)ds)et0mTτi(s)ds

    For any t[0,T], we have

    et0mTτ1(s)dset0mTτ2(s)ds. (86)

    According to (4) and the fact that zτ1,α>0, it follows that for any t[0,T]

    g(zτ1,αet0mTτ1(s)ds)zτ1,αet0mTτ1(s)dsg(zτ1,αet0mTτ2(s)ds)zτ1,αet0mTτ2(s)ds. (87)

    In other terms, zτ1,α is a subsolution of the equation satisfied by zτ2,α. Since zτ1,α(0)=zτ2,α(0)=α, we can apply a comparison principle and we obtain

    zτ1,α(t)zτ2,α(t),   t[0,T].

    Actually, the previous inequality is strict with t=T because (86) and (87) are strict on (0,τ2). Owing to (85), we have

    yτ1,α(T)eT0mTτ1(s)ds>yτ2,α(T)eT0mTτ2(s)ds

    According to (28), it occurs that

    T0mTτ1(s)ds=T0mTτ2(s)ds=10φ(s)ds.

    Consequently

    yτ1,α(T)>yτ2,α(T).

    In other words, PTτ1(α)>PTτ2(α), which concludes the proof.



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