In this paper, we consider a coupling non-linear system of two plate equations with logarithmic source terms. First, we study the local existence of solutions of the system using the Faedo-Galerkin method and Banach fixed point theorem. Second, we prove the global existence of solutions of the system by using the potential wells. Finally, using the multiplier method, we establish an exponential decay result for the energy of solutions of the system. Some conditions on the variable exponents that appear in the coupling functions and the involved constants that appear in the source terms are determined to ensure the existence and stability of solutions of the system. A series of lemmas and theorems have been proved and used to overcome the difficulties caused by the variable exponent and the logarithmic nonlinearities. Our result generalizes some earlier related results in the literature from the case of only constant exponent of the nonlinear internal forcing terms to the case of variable exponent and logarithmic source terms, which is more useful from the physical point of view and needed in several applications.
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Nasser-Eddine Tatar. On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results[J]. AIMS Mathematics, 2023, 8(9): 19971-19992. doi: 10.3934/math.20231018
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In this paper, we consider a coupling non-linear system of two plate equations with logarithmic source terms. First, we study the local existence of solutions of the system using the Faedo-Galerkin method and Banach fixed point theorem. Second, we prove the global existence of solutions of the system by using the potential wells. Finally, using the multiplier method, we establish an exponential decay result for the energy of solutions of the system. Some conditions on the variable exponents that appear in the coupling functions and the involved constants that appear in the source terms are determined to ensure the existence and stability of solutions of the system. A series of lemmas and theorems have been proved and used to overcome the difficulties caused by the variable exponent and the logarithmic nonlinearities. Our result generalizes some earlier related results in the literature from the case of only constant exponent of the nonlinear internal forcing terms to the case of variable exponent and logarithmic source terms, which is more useful from the physical point of view and needed in several applications.
In this paper, we deal with the existence and asymptotic behavior of solutions of the following system:
{ψtt+Δ2ψ+ψ+ψt+h1(ψ,φ)=αψln|ψ|,in Ω,t>0,φtt+Δ2φ+φ+φt+h2(ψ,φ)=αφln|φ|,in Ω,t>0,ψ(⋅,t)=φ(⋅,t)=∂ψ∂ν=∂φ∂ν=0,on∂Ω,t≥0,(ψ(0),φ(0))=(ψ0,φ0),(ψt(0),φt(0))=(ψ1,φ1),in Ω, | (P) |
where Ω is a bounded and regular domain of R2, with smooth boundary ∂Ω. The vector ν is the unit outer normal to ∂Ω and the constant α is a small positive real number satisfying some conditions. The coupling functions h1,h2 are of the form
h1(ψ,φ)=c1|ψ+φ|2(p(⋅)+1)(ψ+φ)+c2|ψ|p(⋅)ψ|φ|p(⋅)+2,h2(ψ,φ)=c1|ψ+φ|2(p(⋅)+1)(ψ+φ)+c2|ψ|p(⋅)+2|φ|p(⋅)φ, | (1.1) |
where c1,c2>0 are constants, and p is a continuous function on ¯Ω satisfying some conditions to be mentioned later.
We study the existence and asymptotic behavior of solutions for the nonlinear coupling system of two plate equations with logarithmic source terms (P). For this purpose, we use the well-known logarithmic Sobolev inequality and logarithmic Gronwall inequality to treat the terms involving logarithms. We also use the embedding properties to treat the terms involving variable exponents. In addition, the main methods used to achieve our results are the Faedo-Galerkin method, the Banach fixed point theorem and the multiplier method.
From both the theoretical point of view and the application point of view, it is of great importance to have an idea about the existence and asymptotic behavior of solutions for coupling systems with logarithmic source terms in which the coupling functions are nonlinear with variable exponents. The significance of studying our system (P) is important in many fields. For example, from the logarithmic point of view, the logarithmic nonlinearity appears naturally in inflation cosmology and supersymmetric field theories, quantum mechanics and many other branches of physics, such as nuclear physics, optics and geophysics [1,2,3] and [4]. These specific applications in physics and other fields attract a lot of mathematical scientists to work with these problems.
Regarding problems with logarithmic source terms, we refer to the works of [5,6,7,8,9,10,11]. From the variable exponent non-linearity point of view, there has been an increasing interest in treating equations with variable exponents of nonlinearity. This great interest is motivated by the applications to the mathematical modeling of non-Newtonian fluids. These fluids include electro-rheological fluids, which have the ability to drastically change when applying some external electromagnetic field. The variable exponents of non-linearity is a given function of density, temperature, saturation, electric field, etc.
As a consequence, the topic of long-time behavior of solutions for non-linear equations with source terms has attracted many researchers. For example, there is an extensive literature on the existence, asymptotic behavior and nonexistence of solutions for the following wave equation:
{ψtt−Δψ+g(ψt)=f(ψ), x∈Ω, t>0,ψ(x,t)=0, x∈∂Ω, t≥0,ψ(x,0)=ψ0(x), ψt(x,0)=ψ1(x), x∈Ω. | (1.2) |
For when the source function f(ψ) is a polynomial type function, we refer the reader to see [12,13,14,15,16,17,18,19,20,21]. Antontsev et al. [22] studied the following Petrovsky equation:
ψtt+Δ2ψ−Δψt+|ψt|m(x)−2ψt=|ψ|p(x)−2ψ. | (1.3) |
They proved the existence of local weak solutions by using the Banach fixed-point theorem, and gave a blow-up result for negative-initial-energy solutions, under suitable assumptions. In [23], Liao and Tan treated the following similar problem:
ψtt+Δ2ψ−M(‖∇ψ‖22)−Δψt+|ψt|m(x)−2ψt=|ψ|p(x)−2ψ, |
where M(s)=a+bsγ is a positive C1-function, a>0,b>0,γ≥1, and m,p are given measurable functions. They established some uniform decay estimates and the upper and lower bounds of the blow-up time.
Concerning the existence, asymptotic behavior and nonexistence of solutions of coupled systems, Andrade and Mognon [24] treated the following problem:
{utt−Δu+∫t0g1(t−s)Δu(s)ds+f1(u,v)=0,in[0,T]×Ω,vtt−Δv+∫t0g2(t−s)Δv(s)ds+f2(u,v)=0,in[0,T]×Ω, | (1.4) |
with
f1(u,v)=|u|p−2u|v|pandf2(u,v)=|v|p−2v(t)|u|p, |
where p>1 if N=1,2 and 1<p≤N−1N−2 if N=3. They proved the well posedness under some assumptions on the relaxation functions. Also, we point out the work of Agre and Rammaha [25], where they considered a system of wave equations of the form
{utt−Δu+|ut|m−1ut=f1(u,v),vtt−Δv+|vt|r−1vt=f2(u,v), | (1.5) |
in Ω×(0,T), with initial and boundary conditions of Dirichlet type, and the nonlinear functions f1 and f2 are given by
{f1(u,v)=a|u+v|2(ρ+1)(u+v)+b|u|ρu|v|ρ+2,f2(u,v)=a|u+v|2(ρ+1)(u+v)+b|v|ρv|u|ρ+2. | (1.6) |
They proved, under some appropriate conditions on f1, f2 and the initial data, several results on local and global existence. They also showed that any weak solution with negative initial energy blows up in finite time. Wang et al. [26] considered the following system:
{utt−M(‖∇u‖2+‖∇v‖2)Δu+ut=|u|k−2ulnuvtt−M(‖∇u‖2+‖∇v‖2)Δv+vt=|v|k−2vlnv, | (1.7) |
where k≥2, M(s)=α+βsγ for any α≥1, β≥0 and γ>0. By employing the potential well method, the concavity method and the unstable invariant set, they proved the global existence and a finite time blow up. In [27], Bouhoufani and Hamchi discussed the following coupled system of two nonlinear hyperbolic equations with variable-exponents:
{ψtt−div(A∇ψ)+|ψt|m(x)−2ψt=h1(ψ,φ)in Ω×(0,T),φtt−div(B∇φ)+|φt|r(x)−2φt=h2(ψ,φ)in Ω×(0,T), | (1.8) |
with initial and Dirichlet-boundary conditions, where h1 and h2 are the coupling terms introduced in (1.1). Under suitable assumptions on the variable exponents m,r and p, the authors proved the global existence of a weak solution and established decay rates of the solution in a bounded domain. In [28], Messaoudi et al. considered the following system:
{ψtt−Δψ+|ψt|m(x)−2ψt+h1(ψ,φ)=0in Ω×(0,T),φtt−Δφ+|φt|r(x)−2φt+h2(ψ,φ)=0in Ω×(0,T), | (1.9) |
with initial and Dirichlet-boundary conditions (here, h1 and h2 are the coupling terms introduced in (1.1)). The authors proved the existence of global solutions, obtained explicit decay rate estimates, under suitable assumptions on the variable exponents m,r and p, and presented some numerical tests. For more studies on the existence and asymptotic behavior of solutions for other nonlinear coupling systems, we refer to [29,30,31,32,33,34].
However, to the best of our knowledge, there are no investigations on the existence and asymptotic behavior of solutions for a nonlinear coupling system of two plate equations of type (P). Therefore, our aim in the present work is to prove the local and global existence of the solutions of this problem and study the long-time behavior of the energy associated with this problem. So, the originating motivation in the study of problem (P) is twofold:
● On the one hand, we consider the non-linear system of plate equations with a nonstandard internal forcing terms caused by the smart nature of the medium.
● On the other hand, we investigate the effect of replacing the classical power-type nonlinearity with the logarithmic nonlinearity, which is a natural extension done for many problems.
The following remark states the main difference of our result with the present ones in the literature. This will clarify our main contributions in the present work.
Remark 1.1. Notice that our work is an extension of all the above works. For example, the works of [24,25] only treated nonlinear systems of two wave equations where the coupling functions are polynomials of constant exponents. The work of [26] only treated nonlinear systems where the coupling functions are only a polynomial of constant type. The works of [27,28,33] only treated nonlinear systems of two wave equations without logarithmic source terms. So, in the present work, we treated the nonlinear system of two plate equations with logarithmic source terms, and the coupling functions are nonlinear polynomials of variable exponents type, which are more general than the ones in the literature. We note here that though the logarithmic nonlinearity is somehow weaker than the polynomial nonlinearity, both the existence and stability result are not obtained by straightforward application of the method used for polynomial nonlinearity.
The rest of this paper is organized as follows: In Section 2, we present some definitions and basic properties of the logarithmic nonlinearity and the variable-exponent Lebesgue and Sobolev spaces. The local and global existence results are given in Section 3. In Section 4, we prove the stability result. A conclusion is given in Section 5.
In this section, we present some notations and material needed in the proof of our results. We use the standard Lebesgue space L2(Ω) and Sobolev space H20(Ω) with their usual scalar products and norms. Throughout this paper, c is used to denote a generic positive constant, and we assume the following hypotheses:
(H1) p(⋅) is a given continuous function on ¯Ω satisfying the log-Hölder continuity condition:
|p(x)−p(y)|≤−Mlog|x−y|,for allx,y∈Ω,with|x−y|<δ, | (2.1) |
where M>0, 0<δ<1, and
0<p1=ess infx∈Ωp(x)≤p(x)≤p2=ess supx∈Ωp(x). | (2.2) |
(H2) The constant α in (P) satisfies 0<α<α0, where α0 is the positive real number satisfying
√2πcpα0=e−32−1α0, | (2.3) |
and cp is the smallest positive number satisfying
‖∇u‖22≤cp‖Δu‖22,∀u∈H20(Ω), |
where ‖.‖2=‖.‖L2(Ω).
Lemma 2.1. [35,36] (Logarithmic Sobolev inequality) Let v be any function in H10(Ω) and a>0 be any number. Then,
∫Ωv2ln|v|dx≤12‖v‖22ln‖v‖22+a22π‖∇v‖22−(1+lna)‖v‖22. | (2.4) |
Corollary 2.2. Let u be any function in H20(Ω) and a be any positive real number. Then,
∫Ωu2ln|u|dx≤12‖u‖22ln‖u‖22+cpa22π‖Δu‖22−(1+lna)‖u‖22. | (2.5) |
Remark 2.1. The function f(s)=√2πcps−e−32−1s is continuous and decreasing on (0,∞), with
lims→0+f(s)=∞andlims→∞f(s)=−e−32. |
Therefore, there exists a unique α0>0 such that f(α0)=0; that is,
√2πα0cp=e−32−1α0. | (2.6) |
Moreover,
e−32−1s<√2πcps,∀s∈(0,α0). | (2.7) |
Because f(2πe3cp)>0, f(α)>0, and so (2.7) holds for s=α.
Lemma 2.3. [37] (Young's inequality) Let p,s:Ω⟶[1,∞) be a measurable functions, such that
1s(y)=1p(y)+1q(y), for a.e. y∈Ω. |
Then, for all a,b≥0, we have
(ab)s(.)s(.)≤ap(.)p(.)+bq(.)q(.). |
By taking s=1 and 1<p,q<+∞, it follows that for any ε>0,
ab≤εap+Cεbq, where Cε=1/q(εp)qp. | (2.8) |
Lemma 2.4. [37] If 1<q−≤q(x)≤q+<∞ holds, then
∫Ω|v|q(⋅)dx≤‖v‖q−q−+‖v‖q+q+, | (2.9) |
for any v∈Lq(⋅)(Ω).
Lemma 2.5. [37] (Embedding property) Let Ω⊂Rn be a bounded domain with a smooth boundary ∂Ω. If q∈C(¯Ω), and p:Ω⟶(1,∞) is a continuous function such that
essinfx∈Ω(q∗(x)−p(x))>0 with q∗(x)={nq(x)esssupx∈Ω(n−q(x)),if q+<n,∞,if q+≥n, |
then the embedding W1,q(.)(Ω)↪Lp(.)(Ω) is continuous and compact.
Lemma 2.6. [5] (Logarithmic Gronwall inequality) Let c>0, u∈L1(0,T;R+), and assume that the function v:[0,T]→[1,∞) satisfies
v(t)≤c(1+∫t0u(s)v(s)lnv(s)ds),0≤t≤T. | (2.10) |
Then,
v(t)≤cexp(c∫t0u(s)ds),0≤t≤T. | (2.11) |
Remark 2.2. By recalling the definitions of h1(ψ,φ) and h2(ψ,φ) in (1.1), it is easily seen that
ψh1(ψ,φ)+φh2(ψ,φ)=2(p(x)+2)H(ψ,φ),∀(ψ,φ)∈R2, | (2.12) |
where
H(ψ,φ)=12(p(x)+2)[c1|ψ+φ|2(p(x)+2)+2c2|ψφ|p(x)+2]. |
We define the energy functional E(t) associated to System (P) as follows:
E(t):=12[‖ψt‖22+‖φt‖22]+12[‖Δψ‖22+‖Δφ‖22]+α+24[‖ψ‖22+‖φ‖22]+∫ΩH(ψ,φ)dx−12∫Ωψ2ln|ψ|dx−12∫Ωφ2ln|φ|dx. | (2.13) |
By multiplying the two equations in (P) by ψt and φt, respectively, integrating over Ω, using integration by parts and adding results together, we get
ddtE(t)=−∫Ω|ψt(t)|2dx−∫Ω|φt(t)|2dx≤0. | (2.14) |
In this section, we state and prove the local and global existence results of system (P).
In this subsection, we state and prove the local existence of the solutions of system (P) using the Faedo-Galerkin method and Banach fixed point theorem.
Definition 3.1. Let X be a Banach space and Y be its dual space. Then,
Cw([0,T],X)={z(t):[0,T]→X:t→⟨y,z(t)⟩ is continuous on [0,T],∀y∈Y}. |
Definition 3.2. Let (ψ0,ψ1),(φ0,φ1)∈H20(Ω)×L2(Ω). Any pair of functions
(ψ,φ)∈Cw([0,T],H20(Ω)), (ψt,φt)∈Cw([0,T],L2(Ω)) |
is called a weak solution of (P) if
{ddt∫Ωψt(x,t)¯ψ(x)dx+∫ΩΔψ(x,t)Δ¯ψ(x)dx+∫Ωψt(x,t)¯ψ(x)dx+∫Ωψ(x,t)¯ψ(x)dx+∫Ω¯ψ(x)h1dx=∫Ωα¯ψ(x)ψln|ψ|dxddt∫Ωφt(x,t)¯φ(x)dx+∫ΩΔφ(x,t)Δ¯φ(x)dx+∫Ωφt(x,t)¯φ(x)dx+∫Ωφ(x,t)¯φ(x)dx+∫Ω¯φ(x)h2dx=∫Ωα¯φ(x)φln|φ|dxψ(0)=ψ0, ψt(0)=ψ1, φ(0)=φ0, φt(0)=φ1, | (3.1) |
for a.e. t∈[0,T] and all test functions ¯ψ,¯φ∈H20(Ω).
In order to establish an existence result of a local weak solution for system (P), we, first, consider the following initial-boundary-value problem:
{ψtt+Δ2ψ+ψ+ψt+˜h(x,t)=ψln|ψ|αin Ω×(0,T),φtt+Δ2φ+φ+φt+˜k(x,t)=φln|φ|αin Ω×(0,T),ψ=φ=∂ψ∂ν=∂φ∂ν=0on ∂Ω×(0,T),ψ(0)=ψ0,ψt(0)=ψ1,φ(0)=φ0,φt(0)=φ1in Ω, | (S) |
for given ˜h,˜k∈L2(Ω×(0,T)) and T>0.
We have the following theorem of existence for problem (S).
Theorem 3.1. Let (ψ0,ψ1),(φ0,φ1)∈H20(Ω)×L2(Ω). Assume that assumptions (H1)–(H2) hold. Then, problem (S) admits a weak solution on [0,T).
The proof of Theorem 3.1 will be carried out through several steps and lemmas. We use the standard Faedo-Galerkin method to prove this theorem.
Step 1: (Approximate solution) Consider {wj}∞j=1 an orthonormal basis of H20(Ω). Let Vk=span{w1,w2,...,wk}, and the projections of initial data on the finite-dimensional subspace Vk are given by
ψk0=k∑j=1ajwj,φk0=k∑j=1bjwj,ψk1=k∑j=1cjwj,φk1=k∑j=1djwj, |
where,
{ψk0→ψ0 and φk0→φ0inH20(Ω)andψk1→ψ1 and φk1→φ1inL2(Ω). | (3.2) |
We search for solutions of the form
ψk(x)=k∑j=1rj(t)wj(x) and φk(x)=k∑j=1gj(t)wj(x) |
for the following approximate system in Vk:
{⟨ψktt,wj⟩L2(Ω)+⟨Δψk,Δwj⟩L2(Ω)+⟨ψk,wj⟩L2(Ω)+⟨ψkt,wj⟩L2(Ω)+⟨˜h(x,t),wj⟩L2(Ω)=⟨αψkln|ψk|,wj⟩L2(Ω),j=1,2,...,k,⟨φktt,wj⟩L2(Ω)+⟨Δφk,Δwj⟩L2(Ω)+⟨φk,wj⟩L2(Ω)+⟨φkt,wj⟩L2(Ω)+⟨˜k(x,t),wj⟩L2(Ω)=⟨αφkln|φk|,wj⟩L2(Ω),j=1,2,...,k,ψk(0)=ψk0,ψkt(0)=ψk1,φk(0)=φk0,φkt(0)=φk1. | (3.3) |
This leads to a system of ODE's for unknown functions rj and gj. Based on standard existence theory for ODEs, system (3.3) admits a solution (ψk,φk) on a maximal time interval [0,tk),0<tk<T, for each k∈N.
Lemma 3.2. There exists a constant T>0 such that the approximate solutions (ψk,φk) satisfy, for all k≥1,
{(ψk)and(φk)are bounded sequences inL∞(0,T;H20(Ω)),(ψkt)and(φkt)are bounded sequences inL∞(0,T;L2(Ω))∩L2(Ω×(0,T)). | (3.4) |
Proof. We multiply the first equation by r′j and the second equation by g′j in (3.3), sum over j=1,2,...k and add the two equations to obtain
12ddt(‖ψkt‖22+‖φkt‖22+‖Δψk‖22+‖Δφk‖22+α+22(‖ψk‖22+‖φk‖22)−∫Ω(ψk)2ln|ψk|dx−∫Ω(φk)2ln|φk|dx)=−∫Ω|ψkt(s)|2dx−∫Ω|φkt(s)|2dx−∫Ω(ψkt˜h(x,t)+φkt˜k(x,y))dx. | (3.5) |
The integration of (3.5), over (0,t), leads to
12(||ψkt||22+||Δψk||22+||φkt||22+||Δφk||22+α+22(‖ψk‖22+‖φk‖22)−∫Ω(ψk)2lnψkdx)−12∫Ω(φk)2lnφkdx+∫t0∫Ω|ψkt(s)|2dxds+∫t0∫Ω|φkt(s)|2dxds=12(||Δψk0||22+||ψk1||22+||Δφk0||22+||φk1||22)+α+24(||ψk0||22+||φk0||22)−∫t0∫Ω(ψkt˜h(x,s)+φkt˜k(x,s))dxds+12∫Ω(ψk0)2ln|ψk0|dx+12∫Ω(φk0)2ln|φk0|dx. | (3.6) |
Combining (3.6) and convergence (3.2) implies
12(||ψkt||22+||Δψk||22+||φkt||22+||Δφk||22+α+22(‖ψk‖22+‖φk‖22)−∫Ω(ψk)2ln|ψk|dx)−12∫Ω(φk)2ln|φk|dx+∫t0∫Ω|ψkt(s)|2dxds+∫t0∫Ω|φkt(s)|2dxds≤C0−∫t0∫Ω(ψkt˜h(x,s)+φkt˜k(x,s))dxds. | (3.7) |
Applying the logarithmic Sobolev inequality to (3.7), we obtain
‖ψkt‖22+‖φkt‖22+(1−αa22π)(‖Δψk‖22+‖Δφk‖22)+[α+22+α(1+lna)](‖ψk‖22+‖ψk‖22)+∫t0∫Ω|ψkt(s)|2dxds+∫t0∫Ω|φkt(s)|2dxds≤C0−∫t0∫Ω(ψkt˜h(x,s)+φkt˜k(x,s))dxds+α2(‖ψk‖22ln‖ψk‖22+‖φk‖22ln‖φk‖22). | (3.8) |
Now, we select
e−32−1α<a<√2παcp, | (3.9) |
to make
1−cpαa22π>0,andα+22+α(1+lna)>0. | (3.10) |
Exploiting Young's inequality and using (3.10), (3.8) becomes, for some C>0,
‖ψkt‖22+‖φkt‖22+‖Δψk‖22+‖Δφk‖22+‖ψk‖22+‖ψk‖22≤C+(1+ε)∫Tk0(‖ukt(s)‖22+‖vkt(s)‖22)ds+α2(‖ψk‖22ln‖ψk‖22+‖φk‖22ln‖φk‖22)+Cε∫T0∫Ω(|˜h(x,s)|2+|˜k(x,s)|2)dxds. |
Using the fact that ˜h,˜k∈L2(Ω×(0,T)), we infer that
‖ψkt‖22+‖φkt‖22+‖Δψk‖22+‖Δφk‖22+‖ψk‖22+‖ψk‖22≤C(1+∫Tk0(‖ukt(s)‖22+‖vkt(s)‖22)ds+‖ψk‖22ln‖ψk‖22+‖φk‖22ln‖φk‖22). | (3.11) |
Let us note that
ψk(.,t)=ψk(.,0)+∫t0∂ψk∂s(.,s)ds, and φk(.,t)=φk(.,0)+∫t0∂φk∂s(.,s)ds. |
Then, applying the Cauchy-Schwarz' inequality, we get
‖ψk(t)‖22≤2‖ψk(0)‖22+2||∫t0∂ψk∂s(s)ds||22≤2‖ψk(0)‖22+2T∫t0‖ψkt(s)‖22ds,‖φk(t)‖22≤2‖φk(0)‖22+2||∫t0∂φk∂s(s)ds||22≤2‖φk(0)‖22+2T∫t0‖φkt(s)‖22ds. | (3.12) |
Hence, inequality (3.11) leads to
‖ψk‖22+‖φk‖22≤2‖ψk(0)‖22+2‖φk(0)‖22+2cT(1+∫t0‖ψk‖22ln‖ψk‖22ds+∫t0‖φk‖22ln‖φk‖22ds)≤2C(1+∫t0‖ψk‖22ln‖ψk‖22ds+∫t0‖φk‖22ln‖φk‖22ds)≤2C1(1+∫t0(C1+‖ψk‖22)ln(C1+‖ψk‖22)ds+∫t0(C1+‖φk‖22)ln(C1+‖φk‖22)ds), | (3.13) |
where, without loss of generality, C1≥1. The logarithmic Gronwall inequality implies that
‖ψk‖22+‖φk‖22≤2C1e2C1T:=C2. | (3.14) |
‖ψk‖22ln‖ψk‖22+‖φk‖22ln‖φk‖22≤C. |
After combining (3.13) and (3.14), we obtain
sup(0,Tk)[‖ψkt‖22+‖φkt‖22+‖Δψk‖22+‖Δφk‖22+‖ψk‖22+‖ψk‖22]≤Cε+Tεsup(0,Tk)(‖ukt‖22+‖vkt‖22). | (3.15) |
Choosing ε<12T, estimate (3.15) yields, for all Tk≤T,
sup(0,Tk)[‖ψkt‖22+‖φkt‖22+‖Δψk‖22+‖Δφk‖22+‖ψk‖22+‖ψk‖22]≤CT, |
which completes the proof of (3.4).
Lemma 3.3. The approximate solutions (ψk,φk) satisfy, for all k≥1,
ψkln|ψk|α→ψln|ψ|α strongly in L2(0,T;L2(Ω)),φkln|φk|α→φln|φ|α strongly in L2(0,T;L2(Ω)). | (3.16) |
Proof. The arguments in (3.4) imply that there exist subsequences of (ψk) and (φk), still denoted by (ψk) and (φk), such that
{ψk⇀∗ψ and φk⇀∗φinL∞(0,T;H20(Ω)),ψkt⇀∗ψt and φkt⇀∗φtinL∞(0,T;L2(Ω)). | (3.17) |
Making use of the Aubin-Lions theorem, we find, up to subsequences, that
ψk→ψ and φk→φ strongly in L2(0,T;L2(Ω)), |
and
ψk→ψ and φk→φ a.e. in Ω×(0,T). | (3.18) |
We use (3.18) and the fact that the map s→sln|s|α is continuous on R, then, we have the convergence
ψkln|ψk|α→ψln|ψ|α a.e. in Ω×(0,T). |
Using the embedding of H20(Ω) in L∞(Ω) (since Ω⊂R2), it is clear that ψkln|ψk|α is bounded in L∞(Ω×(0,T)). Next, taking into account the Lebesgue bounded convergence theorem (Ω is bounded), we get
ψkln|ψk|α→ψln|ψ|α strongly in L2(0,T;L2(Ω)). | (3.19) |
Similarly, we can establish the second argument of (3.17).
Step 2: (Limiting process) Integrate (3.3) over (0,t) to obtain
∫Ωψkt(t)wjdx−∫Ωψk1wjdx+∫Ωψk(t)wjdx+∫t0∫ΩΔψk(t)Δwjdxds+∫t0∫Ωψkt(t)wjdxds+∫t0∫Ωwj˜h(x,s)dxds=∫t0∫Ωαwjψkln|ψk|ds∫Ωφkt(t)wjdx−∫Ωφk1wjdx+∫Ωφk(t)wjdx+∫t0∫ΩΔφk(t)Δwjdxds+∫t0∫Ωφkt(t)wjdxds+∫t0∫Ωwj˜k(x,s)dxds=∫t0∫Ωαwjφkln|φk|ds,∀ j=1,2,...,k. | (3.20) |
Convergence (3.2) and (3.17) allow us to pass to the limit in both equations in (3.20), as k→+∞, and get
∫Ωψt(t)wjdx−∫Ωψ1wjdx+∫Ωψ(t)wjdx+∫t0∫ΩΔψ(t)Δwjdxds+∫t0∫Ωψt(t)wjdxds+∫t0∫Ωwj˜h(x,s)dxds=∫t0∫Ωαwjψln|ψ|ds∫Ωφt(t)wjdx−∫Ωφ1wjdx+∫Ωφ(t)wjdx+∫t0∫ΩΔφ(t)Δwjdxds∫t0∫Ωφt(t)wjdxds+∫t0∫Ωwj˜k(x,s)dxds=∫t0∫Ωαwjφln|φ|ds,∀ j=1,2,...,k, | (3.21) |
which implies that (3.21) is valid for any w∈H20(Ω). Using the fact that the left hand sides of both equations in (3.21) are absolutely continuous functions, they are differentiable for a.e. t∈(0,∞). Therefore, for a.e. t∈[0,T], the equations in (3.21) become
ddt∫Ωψt(x,t)w(x)dx+∫ΩΔψ(x,t)Δw(x)dx+∫Ωψw(x)dx+∫Ωψtw(x)dx+∫Ωw˜h(x,s)dx=∫Ωαw(x)ψ(x,t)ln|ψ(x,t)|dx,∀w∈H20(Ω)ddt∫Ωφt(x,t)w(x)dx+∫ΩΔφ(x,t)Δw(x)dx+∫Ωφw(x)dx+∫Ωφtw(x)dx+∫Ωw˜k(x,s)dx=∫Ωαw(x)φ(x,t)ln|φ(x,t)|dx,∀w∈H20(Ω). |
Step 3: (Initial conditions) To handle the initial conditions, we note that
ψk⇀ψ and φk⇀φweakly inL2(0,T;H20(Ω)),ψkt⇀ψt and φkt⇀φtweakly inL2(0,T;L2(Ω)). | (3.22) |
Thus, using the Lions lemma and (3.2), we easily obtain
ψ(x,0)=ψ0(x) and φ(x,0)=φ0(x). |
As in [38], multiply (3.3) by ˜ψ∈C∞0(0,T) and integrate over (0,T), and we obtain for any w∈Vk
−∫T0∫Ωψktw˜ψ′(t)dxdt=−∫T0∫ΩΔψkΔw˜ψdxdt−∫T0∫Ωψkw˜ψdxdt+∫T0∫Ωαwψk˜ψln|ψk|dxdt−∫T0∫Ωψktw˜ψdxdt−∫T0∫Ωw˜ψ˜h(x,s)dxdt−∫T0∫Ωφktw˜ψ′(t)dxdt=−∫T0∫ΩΔφkΔw˜ψdxdt−∫T0∫Ωφkw˜ψdxdt+∫T0∫Ωαwφk˜ψln|φk|dxdt−∫T0∫Ωφktw˜ψdxdt−∫T0∫Ωw˜ψ˜k(x,s)dxdt. | (3.23) |
As k→+∞, we have, for any w∈H20(Ω) and any ˜ψ∈C∞0((0,T)),
−∫T0∫Ωψtw˜ψ′(t)dxdt=−∫T0∫ΩΔψΔw˜ψdxdt−∫T0∫Ωψw˜ψdxdt+∫T0∫Ωαwψ˜ψln|ψ|dxdt−∫T0∫Ωw˜ψ˜h(x,s)dxdt−∫T0∫Ωψtw˜ψdxdt−∫T0∫Ωψtw˜ψ′(t)dxdt=−∫T0∫ΩΔψΔw˜ψdxdt−∫T0∫Ωφw˜ψdxdt+∫T0∫Ωαwφ˜ψln|φ|dxdt−∫T0∫Ωw˜ψ˜k(x,s)dxdt−∫T0∫Ωφtw˜ψdxdt. | (3.24) |
This means (see [38])
ψtt,φtt∈L2([0,T),H−2(Ω)). |
Recalling that ψt,φt∈L2((0,T),L2(Ω)), we obtain
ψt,φt∈C([0,T),H−2(Ω)). |
So, ψkt(x,0) makes sense, and
ψkt(x,0)→ψt(x,0) and φkt(x,0)→φt(x,0) in H−2(Ω). |
However,
ψkt(x,0)=ψk1(x)→ψ1(x) and φkt(x,0)=φk1(x)→φ1(x) in L2(Ω). |
Hence,
ψt(x,0)=ψ1(x) and φt(x,0)=φ1(x). |
This completes the proof. Therefore, (ψ,φ) is a local solution of (S).
Now, we state and the existence result related to system (P).
Theorem 3.4. System (P) admits a weak solution (ψ,φ), in the sense of Definition (3.2), for T small enough.
Proof. By recalling the definition of h1 and h2 in (1.1) and using Lemma 2.4 and Young's inequality, we have
∫Ω|h1(y,z)|2dx≤2[c21∫Ω|y+z|2(2p(x)+3)dx+c22∫Ω|y|2p(x)+2|z|2p(x)+4dx]≤C0[∫Ω|y+z|2(2p2+3)dx+∫Ω|y+z|2(2p1+3)dx+∫Ω|y|3(2p(x)+2)dx+∫Ω|z|3(p(x)+2)dx], | (3.25) |
where C0=2max{c21,3c22}>0. By the embedding, we have, for n=2,
1<3(p1+2)≤3(p2+2)≤2(2p2+3)≤3(2p2+2)<∞, |
since 3≤2p1+3≤2p(x)+3≤2p2+3<∞. Therefore, estimate (3.25) and Lemma 2.4 lead to
∫Ω|h1(y,z)|2dx≤C1[‖Δ(y+z)‖2(2p2+3)2+‖Δ(y+z)‖2(2p1+3)2+‖Δy‖3(2p2+2)2+‖Δy‖3(2p1+2)2]+C1[‖Δz‖3(p2+2)2+‖Δz‖3(p1+2)2]<+∞, | (3.26) |
where C1=C0Ce. Consequently, under the assumption (2.2), we have,
∫Ω|h1(y,z)|2dx<∞ |
and, similarly,
∫Ω|h2(y,z)|2dx<∞. |
Thus,
h1(y,z),h2(y,z)∈L2(Ω×(0,T)), ∀ y,z∈H20(Ω). |
Now, let
WT={w∈L∞((0,T),H20(Ω))/wt∈L∞((0,T),L2(Ω))}, |
and define the map K:WT×WT:⟶WT×WT by K(y,z)=(ψ,φ), where (ψ,φ) is the solution of
{ψtt+Δ2ψ+ψ+ψt+h1(y,z)=αψln|ψ|,in Ω,t>0,φtt+Δ2φ+φ+φt+h2(y,z)=αφln|φ|,in Ω,t>0,ψ(⋅,t)=φ(⋅,t)=∂ψ∂ν=∂φ∂ν=0,on∂Ω,t≥0,(ψ(0),φ(0))=(ψ0,φ0),(ψt(0),φt(0))=(ψ1,φ1),in Ω. | (3.27) |
We note that WT is a Banach space with respect to the norm
||w||2WT=sup(0,T)∫Ω|Δw|2dx+sup(0,T)∫Ω|wt|2dx, |
and K is well defined by virtue of Theorem (3.1). As in [39] and [33], it is a routine work to prove that K is a contraction mapping from a bounded ball B(0,d) into itself, where
B(0,d)={(y,z)∈WT×WT/‖(y,z)‖WT0×WT0≤d}, |
for d>1 and some T0>0. Then, the Banach-fixed-point theorem guarantees the existence of a solution (ψ,φ)∈B(0,d), such that K(ψ,φ)=(ψ,φ), which is a local weak solution of (P).
In this subsection, we state and prove a global existence result using the potential wells corresponding to the logarithmic nonlinearity. For this purpose, we define the following functionals:
J(ψ,φ)=12(‖Δψ‖22+‖Δφ‖22−∫Ωψ2ln|ψ|αdx−∫Ωφ2ln|φ|αdx)+α+24(‖ψ‖22+‖φ‖22). | (3.28) |
I(ψ,φ)=‖Δψ‖22+‖Δϕ‖22+‖ψ‖22+‖ϕ‖22−∫Ωψ2ln|ψ|αdx−∫Ωφ2ln|ϕ|αdx. | (3.29) |
Remark 3.1. 1) From the above definitions, it is clear that
J(ψ,φ)=12I(ψ,φ)+α4(‖ψ‖22+‖φ‖22), | (3.30) |
E(t)=12(‖ψt‖22+‖φt‖22)+∫ΩH(ψ,ϕ)dx+J(ψ,φ). | (3.31) |
2) According to the logarithmic Sobolev inequality, J(ψ,φ) and I(ψ,φ) are well defined.
We define the potential well (stable set)
W={(ψ,ϕ)∈H20(Ω)×H20(Ω),I(ψ,ϕ)>0}∪{(0,0)}. |
The potential well depth is defined by
0<d=inf(ψ,φ){supλ≥0J(λψ,λφ):(ψ,φ)∈H20(Ω)×H20(Ω),‖Δψ‖2≠0 and ‖Δφ‖2≠0}, | (3.32) |
and the well-known Nehari manifold
N={(ψ,φ):(ψ,φ)∈H20(Ω)×H20(Ω)/I(ψ,φ)=0,‖Δψ‖2≠0 and ‖Δφ‖2≠0}. | (3.33) |
Proceeding as in [21,40], one has
0<d=inf(ψ,φ)∈NJ(ψ,φ). | (3.34) |
Lemma 3.5. For any (ψ,φ)∈H20(Ω)×H20(Ω), ‖ψ‖2≠0 and ‖φ‖2≠0, let g(λ)=J(λψ,λφ). Then, we have
I(λψ,λφ)=λg′(λ){>0, 0≤λ<λ∗,=0, λ=λ∗,<0, λ∗<λ<+∞, |
where
λ∗=exp(‖Δψ‖22+‖Δφ‖22−∫Ωψ2ln|ψ|αdx−∫Ωφ2ln|φ|αdxα(‖ψ‖22+‖φ‖22)) |
Proof.
g(λ)=J(λψ,λφ)=12λ2(‖Δψ‖22+‖Δφ‖22)−12λ2(∫Ωψ2ln|ψ|αdx+∫Ωφ2ln|φ|αdx)+λ2(α+24−α2ln|λ|)(‖ψ‖22+‖φ‖22). |
Since ‖ψ‖2≠0 and ‖φ‖2≠0, g(0)=0, g(+∞)=−∞, and
I(λψ,λφ)=λdJ(λψ,λφ)dλ=λg′(λ)=λ2(‖Δψ‖22+‖Δφ‖22)−λ2(∫Ωψ2ln|ψ|αdx+∫Ωφ2ln|ϕ|αdx)+λ2(1−αln|λ|)(‖ψ‖22+‖φ‖22). |
This implies that \frac{d}{d\lambda}J(\lambda u)_{\lambda = \lambda^*} = 0, and J(\lambda u) is increasing on 0 < \lambda\le \lambda^* , decreasing on \lambda^*\le \lambda < \infty and takes the maximum at \lambda = \lambda^* . In other words, there exists a unique \lambda^* \in(0, \infty) such that I(\lambda^* u) = 0 , and so, we have the desired result.
Lemma 3.6. Let (\psi, \varphi)\in H^2_0(\Omega)\times H^2_0(\Omega) and \beta_0 = \sqrt{\frac{2\pi}{\alpha}}e^{\frac{1}{\alpha}+1} . If 0 < \|\psi\|_2\le \beta_0 and 0 < \|\varphi\|_2\le \beta_0 , then I(\psi, \varphi)\ge 0 .
Proof. Using the logarithmic Sobolev inequality (2.5), for any a > 0 , we have
\begin{equation} \begin{aligned} &I(\psi, \phi) = \vert \vert\Delta \psi\vert\vert_2^2+\vert \vert\Delta \varphi\vert\vert_2^2-\int_{\Omega}\psi^2\ln{\vert \psi\vert^\alpha}dx-\int_{\Omega}\varphi^2\ln{\vert \varphi\vert^\alpha}dx\\ & \quad \ge \left(1-\frac{c_{p}\alpha a^2 }{2\pi}\right)\left( \vert \vert\Delta \psi\vert\vert_2^2+\vert \vert\Delta \varphi\vert\vert_2^2\right)+\alpha(1+\ln{a})\| \psi\|_2^2-\frac{\alpha}{2}\| \psi\|_{2}^2\ln{\| \psi\|_2^2}\\ & \quad +\alpha(1+\ln{a})\| \phi\|_2^2-\frac{\alpha}{2}\| \phi\|_{2}^2\ln{\| \phi\|_2^2}. \end{aligned} \end{equation} | (3.35) |
Taking a = \sqrt{\frac{2\pi}{c_{p}\alpha}} in (3.35), we obtain
\begin{equation} \begin{aligned} I(\psi, \phi)\ge & \left(1+\alpha\left(1+\ln{\sqrt{\frac{2 \pi}{\alpha}}}\right)-\frac{\alpha}{2}\ln{\| \psi\|_2^2}\right)\| \psi\|_2^2\\ &+\left(1+\alpha\left(1+\ln{\sqrt{\frac{2 \pi}{\alpha}}}\right)-\frac{\alpha}{2}\ln{\| \varphi\|_2^2}\right)\| \varphi\|_2^2. \end{aligned} \end{equation} | (3.36) |
If 0 < \|\psi\|_2\le \beta_0 and 0 < \|\varphi\|_2\le \beta_0 , then
1+\alpha\left(1+\ln{\sqrt{\frac{2 \pi}{\alpha}}}\right)-\frac{\alpha}{2}\ln{\| \psi\|_2^2}\ge 0 \text{ and }1+\alpha\left(1+\ln{\sqrt{\frac{2 \pi}{\alpha}}}\right)-\frac{\alpha}{2}\ln{\| \varphi\|_2^2}\ge 0, |
which gives I(\psi, \varphi)\ge 0 .
Lemma 3.7. The potential well depth d satisfies
\begin{equation} d\ge \frac{\pi}{\alpha} e^{2+\frac{2}{\alpha}}. \end{equation} | (3.37) |
Proof. The proof of this lemma is similar to the proof of Lemma 4.3. in [10].
Lemma 3.8. Let (\psi_{0}, \psi_{1}), (\varphi_{0}, \varphi_{1})\in H_{0}^{1}(\Omega)\times L^2(\Omega) such that 0 < E(0) < d and I(\psi_0, \varphi_0) > 0 . Then, any solution of (P), (\psi, \varphi)\in \mathcal{W} .
Proof. Let T be the maximal existence time of a weak solution of (\psi, \varphi) . From (2.14) and (3.31), we have
\begin{equation} \frac{1}{2}\left(\| \psi_t\|^2+\| \varphi_t\|^2\right)+ J(\psi, \varphi) \le \frac{1}{2}\left(\| \psi_1\|^2 +\| \varphi_1\|^2\right)+ J(\psi_0, \varphi_0) < d, \text{ for any }t\in [0, T). \end{equation} | (3.38) |
Then, we claim that (\psi(t), \varphi(t))\in \mathcal{W} for all t\in [0, T) . If not, then there is a t_0 \in (0, T) such that I(\psi(t_0), \varphi(t_0)) < 0 . Using the continuity of I(\psi(t), \varphi(t)) in t , we deduce that there exists a t_*\in (0, T) such that I(\psi(t_*), \varphi(t_*)) = 0 . Then, using the definition of d in (3.32) gives
d\le J(\psi(t_*), \varphi(t_*))\le E(\psi(t_*), \varphi(t_*))\le E(0) < d, |
which is a contradiction.
In this section, we discuss the decay of the solutions of system (P).
Lemma 4.1. The functional
L(t) = E(t)+\varepsilon \int_{\Omega}\psi \psi_{t}dx +\varepsilon \int_{\Omega} \varphi \varphi_{t}dx+\frac{\varepsilon}{2}\int_{\Omega}\psi^2 dx + \frac{\varepsilon}{2}\int_{\Omega}\varphi^2 dx |
satisfies, along the solutions of (P),
\begin{equation} L \sim E, \end{equation} | (4.1) |
and
\begin{equation} \begin{aligned} L^{\prime}(t)&\le -(1-\varepsilon)\int_{\Omega} \left(\vert \psi_{t} \vert^{2}+\vert \varphi_{t} \vert^{2}\right)dx-\varepsilon \int_{\Omega} \left( \vert \Delta \psi \vert^2 + \vert \Delta\varphi \vert^2 \right)dx- \varepsilon \int_{\Omega}2(p(x)+2) H(\psi, \varphi)dx \\&+ \varepsilon \alpha \int_{\Omega} \psi^2 \ln \vert \psi \vert dx + \varepsilon \alpha \int_{\Omega} \varphi^2 \ln \vert \varphi \vert dx-\varepsilon \int_{\Omega}\psi^2dx-\varepsilon \int_{\Omega}\varphi^2dx.\end{aligned} \end{equation} | (4.2) |
Proof. We differentiate L(t) and use (P) to get
\begin{equation*} \begin{aligned} L^{\prime}(t)& = -\int_{\Omega} \psi_{t}^{2}dx-\int_{\Omega} \varphi_{t}^{2}dx+\varepsilon \int_{\Omega} \psi \psi_t dx+\varepsilon \int_{\Omega} \varphi \varphi_t dx\\&+\varepsilon \int_{\Omega} \psi_{t}^2 dx + \varepsilon \int_{\Omega} \psi \Big[-\Delta^2 \psi -\psi_t-\psi-h_1(\psi, \varphi)+\alpha \psi \ln \vert \psi \vert \Big] dx \\&+\varepsilon \int_{\Omega} \varphi_{t}^2 dx+ \varepsilon \int_{\Omega} \varphi \Big[-\Delta^2 \varphi-\varphi-\varphi_t-h_2(\psi, \varphi)+\alpha \varphi \ln \vert \varphi \vert \Big]dx\\ & = -(1-\varepsilon) \int_{\Omega}\left(\vert \psi_{t} \vert^{2}+\vert \varphi_{t} \vert^{2}\right)dx-\varepsilon \int_{\Omega}\left( \vert \Delta \psi \vert^2 + \vert \Delta \varphi \vert^2 \right)dx\\ &- \varepsilon \int_{\Omega} \psi h_1(\psi, \varphi)dx- \varepsilon \int_{\Omega} \varphi h_2(\psi, \varphi)dx - \varepsilon \int_{\Omega} \psi^2 dx-\varepsilon \int_{\Omega} \varphi^2 dx \\ &- \varepsilon \int_{\Omega} \psi \psi_t dx - \varepsilon \int_{\Omega} \varphi \varphi_t dx +\varepsilon \int_{\Omega} \psi \psi_t dx + \varepsilon \int_{\Omega} \varphi \varphi_t dx\\ &+ \varepsilon \alpha \int_{\Omega} \psi^2 \ln \vert \psi \vert dx + \varepsilon \alpha \int_{\Omega} \varphi^2 \ln \vert \varphi \vert dx. \end{aligned} \end{equation*} |
Recalling the definition of H , (4.2) is established.
Theorem 4.2. Assume that (H_1) and (H_2) hold and let (\psi_0, \psi_1), (\varphi_0, \varphi_1) \in H^{2}_{0}(\Omega) \times L^{2}(\Omega) . Assume further that 0 < E(0) < \ell \tau < d , where
\begin{equation} \tau = \frac{\pi}{\alpha} e^{2+\frac{2}{\alpha}}, \; \; 0 < \sqrt{\frac{2 \ell }{\alpha}}e^{\frac{1}{\alpha}} < 1. \end{equation} | (4.3) |
Then there exist two positive constants \kappa_1 and \kappa_2 such that the energy defined in (2.13) satisfies
\begin{equation} 0 < E(t) \leq \kappa_1 e^{- \kappa_2 t}, \; t \geq 0. \end{equation} | (4.4) |
Proof. By adding (\pm \varepsilon \sigma E) to the right hand side of (4.2), we have, for a positive constant \sigma ,
\begin{equation*} \begin{aligned} L^{\prime}(t)& = -\varepsilon \sigma E(t) + \left(\frac{\sigma \varepsilon}{2}+\varepsilon-1\right)\int_{\Omega} \left(\vert \psi_{t} \vert^{2}+\vert \varphi_{t} \vert^{2}\right)dx+\varepsilon \left(\frac{\sigma}{2}-1\right)\int_{\Omega}\left(\vert \Delta \psi \vert^{2}+\vert \Delta \varphi \vert^{2}\right)dx\\ &+\varepsilon \sigma \int_{\Omega} H(\psi, \varphi)dxdx- \varepsilon \int_{\Omega}2(p(x)+2) H(\psi, \varphi)dx\\ &+\frac{ \varepsilon \sigma (\alpha+2)}{4}\Big[\| \psi\|_{2}^{2}+\| \varphi\|_{2}^{2}\Big]- \varepsilon \int_{\Omega} \psi^2 dx - \varepsilon \int_{\Omega} \varphi^2 dx\\ &+ \varepsilon (1-\frac{\sigma}{2}) \int_{\Omega} \psi^2 \ln \vert \psi \vert^\alpha dx + \varepsilon (1-\frac{\sigma}{2}) \int_{\Omega} \varphi^2 \ln \vert \varphi \vert^\alpha dx. \end{aligned} \end{equation*} |
Using the logarithmic Sobolev inequality, we get
\begin{equation} \begin{aligned} &L^{\prime}(t)\le -\sigma \varepsilon E(t)+ \left(\frac{\sigma\varepsilon}{2}+\varepsilon-1\right) \int_{\Omega}\left(\vert \psi_{t} \vert^{2}+\vert \varphi_{t} \vert^{2}\right)dx+\varepsilon \left(\frac{\sigma}{2}-1\right)\int_{\Omega}\left(\vert \Delta \psi \vert^{2}+\vert \Delta \varphi \vert^{2}\right)dx\\ & \quad +\varepsilon \left[\sigma -2(p(x)+2)\right]\int_{\Omega} H(\psi, \varphi)dx\\ & \quad +\frac{ \varepsilon \sigma (\alpha+2)}{4}\Big[\| \psi\|_{2}^{2}+\| \varphi\|_{2}^{2}\Big]- \varepsilon \int_{\Omega} \psi^2 dx - \varepsilon \int_{\Omega} \varphi^2 dx\\ & \quad + \varepsilon \alpha (1-\frac{\sigma}{2}) \left[\frac{1}{2}{\| \psi\|}^{2}_{2}\ln{\|\psi\|}^2_{2}+\frac{ a^2}{2\pi}{\| \Delta \psi\|}^{2}_{2} -(1+\ln{a}){\| \psi\|}^{2}_{2} \right]\\ & \quad + \varepsilon \alpha (1-\frac{\sigma}{2}) \left[\frac{1}{2}{\| \varphi\|}^{2}_{2}\ln{\|\varphi\|}^2_{2}+\frac{ a^2}{2\pi}{\| \Delta\varphi\|}^{2}_{2} -(1+\ln{a}){\| \varphi\|}^{2}_{2} \right]\\ &\le -\sigma\varepsilon E(t)+ \left(\frac{\sigma\varepsilon}{2}+\varepsilon-1\right) \int_{\Omega}\left(\vert \psi_{t} \vert^{2}+\vert \varphi_{t} \vert^{2}\right)dx\\ & \quad +\varepsilon \left[\sigma -2(p_1+2)\right] \int_{\Omega} H(\psi, \varphi)dx\\ & \quad -\varepsilon \left(1-\frac{\sigma}{2}\right)\left(1-\frac{\alpha a^2}{2 \pi}\right) \int_{\Omega}\left(\vert\Delta \psi \vert^{2}+\vert \Delta \varphi \vert^{2}\right)dx\\ & \quad +\varepsilon \left[\frac{\sigma(\alpha+2)}{4}-1+\alpha\left(1-\frac{\sigma}{2}\right)\left(\frac{1}{2}\ln{\|\psi\|}^2_{2}-(1+\ln a)\right) \right] \|\psi\|^2_{2}\\ & \quad +\varepsilon \left[\frac{\sigma(\alpha+2)}{4}-1+\alpha\left(1-\frac{\sigma}{2}\right)\left(\frac{1}{2}\ln{\|\varphi\|}^2_{2}-(1+\ln a)\right) \right] \|\varphi\|^2_{2}. \end{aligned} \end{equation} | (4.5) |
Using (2.13), (2.14) and the fact that u\in W , we find that
\begin{equation} \begin{aligned} \ln{\|\psi\|}^2_{2}& < \ln {\left(\frac{4 E(t)}{\alpha}\right)} < \ln \left(\frac{4 E(0)}{\alpha}\right) \\ & < \ln \frac{4 \ell \tau}{\alpha} = \ln \frac{4 \pi \ell e^{2+\frac{2}{\alpha}}}{\alpha^2}. \end{aligned} \end{equation} | (4.6) |
Similarly, we obtain
\begin{equation} \begin{aligned} \ln{\|\varphi\|}^2_{2}& < \ln \frac{4 \pi \ell e^{2+\frac{2}{\alpha}}}{\alpha^2}. \end{aligned} \end{equation} | (4.7) |
By picking 0 < \sigma < \min\{2(p_1 +2), \frac{4}{\alpha+2}\} and taking a such that
\begin{equation} \frac{ 2 \sqrt{\pi \ell}}{\alpha} e^{\frac{1}{\alpha}} < a < \sqrt{\frac{2\pi}{\alpha}}, \end{equation} | (4.8) |
we guarantee the following:
\frac{\sigma(\alpha+2)}{4}-1 < 0, |
\begin{equation*} \left(1-\frac{\sigma}{2}\right) \left[\frac{1}{2}\ln{\|\psi\|}^2_{2}-(1+\ln a)\right] < 0, \end{equation*} |
and
\begin{equation*} \left(1-\frac{\sigma}{2}\right) \left[\frac{1}{2}\ln{\|\varphi\|}^2_{2}-(1+\ln a)\right] < 0. \end{equation*} |
Now, selecting \varepsilon > 0 small enough so that (4.1) remains true, and
\begin{equation*} \left(\frac{\sigma\varepsilon}{2}+\varepsilon-1\right) < 0. \end{equation*} |
Using all the above inequalities, we see that
\begin{equation} L^{\prime}(t)\le -\sigma \varepsilon E(t). \end{equation} | (4.9) |
Now, using (4.1) and integrating (4.9) over (0, t) , the proof of Theorem 4.2 is completed.
This paper has proved the local and global existence and established an exponential decay estimate for a nonlinear system with nonlinear forcing terms of variable exponent type and logarithmic source terms. These results are new and generalize many related problems in the literature. In addition, the results in this paper have shown how to overcome the difficulties coming from the variable exponent and logarithmic nonlinearities.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project (No. INCB2311), is also greatly acknowledged.
The authors declare that there is no conflict of interest regarding the publication of this paper.
[1] |
K. Bartkowski, P. Górka, One-dimensional klein-gordon equation with logarithmic nonlinearities, J. Phys. A Math. Theor., 41 (2008), 355201. https://doi.org/10.1088/1751-8113/41/35/355201 doi: 10.1088/1751-8113/41/35/355201
![]() |
[2] | I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62–93. https://doi.org/10.1016/0003-4916(76)90057-9 |
[3] |
J. D. Barrow, P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576–5587. https://doi.org/10.1103/PhysRevD.52.5576 doi: 10.1103/PhysRevD.52.5576
![]() |
[4] |
K. Enqvist, J. McDonald, Q-balls and baryogenesis in the mssm, Phys. Lett. B, 425 (1998), 309–321. https://doi.org/10.1016/S0370-2693(98)00271-8 doi: 10.1016/S0370-2693(98)00271-8
![]() |
[5] | T. Cazenave, A. Haraux, Équations d'évolution avec non linéarité logarithmique, Annales de la Faculté des sciences de Toulouse, 2 (1980), 21–51. https://doi.org/10.5802/AFST.543 |
[6] |
T. Hiramatsu, M. Kawasaki, F. Takahashi, Numerical study of q-ball formation in gravity mediation, J. Cosmol. Astropart. P., 2010 (2010), 008. https://doi.org/10.1088/1475-7516/2010/06/008 doi: 10.1088/1475-7516/2010/06/008
![]() |
[7] | P. Górka, Logarithmic klein-gordon equation, Acta Phys. Pol. B, 40 (2009), 59–66. |
[8] |
X. Han, Global existence of weak solutions for a logarithmic wave equation arising from q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275–283. https://doi.org/10.4134/BKMS.2013.50.1.275 doi: 10.4134/BKMS.2013.50.1.275
![]() |
[9] |
W. Lian, R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613–632. https://doi.org/10.1515/anona-2020-0016 doi: 10.1515/anona-2020-0016
![]() |
[10] |
M. M. Al-Gharabli, S. A. Messaoudi, The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term, J. Math. Anal. Appl., 454 (2017), 1114–1128. https://doi.org/10.1016/j.jmaa.2017.05.030 doi: 10.1016/j.jmaa.2017.05.030
![]() |
[11] |
A. M. Al-Mahdi, Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity, Bound. Value Probl., 2020 (2020), 84. https://doi.org/10.1186/s13661-020-01382-9 doi: 10.1186/s13661-020-01382-9
![]() |
[12] |
H. A. Levine, J. Serrin, A global nonexistence theorem for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341–361. https://doi.org/10.1007/s002050050032 doi: 10.1007/s002050050032
![]() |
[13] | D. R. Pitts, M. A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations, Indiana U. Math. J., 51 (2002), 1479–1509. |
[14] |
J. Serrin, G. Todorova, E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differ. Integral Equ., 16 (2003), 13–50. https://doi.org/10.57262/die/1356060695 doi: 10.57262/die/1356060695
![]() |
[15] |
G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Anal. Theor., 41 (2000), 891–905. https://doi.org/10.1016/S0362-546X(98)00317-4 doi: 10.1016/S0362-546X(98)00317-4
![]() |
[16] |
X. Han, M. Wang, General decay estimate of energy for the second order evolution equations with memory, Acta Appl. Math., 110 (2010), 195–207. https://doi.org/10.1007/s10440-008-9397-x doi: 10.1007/s10440-008-9397-x
![]() |
[17] |
V. K. Kalantarov, O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, J. Math. Sci., 10 (1978), 53–70. https://doi.org/10.1007/BF01109723 doi: 10.1007/BF01109723
![]() |
[18] | H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form p u_tt = -a u+ f (u), T. Am. Math. Soc., 192 (1974), 1–21. |
[19] |
V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equations, 109 (1994), 295–308. https://doi.org/10.1006/jdeq.1994.1051 doi: 10.1006/jdeq.1994.1051
![]() |
[20] |
Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differ. Equations, 192 (2003), 155–169. https://doi.org/10.1016/S0022-0396(02)00020-7 doi: 10.1016/S0022-0396(02)00020-7
![]() |
[21] |
Y. Liu, J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. Theor., 64 (2006), 2665–2687. https://doi.org/10.1016/j.na.2005.09.011 doi: 10.1016/j.na.2005.09.011
![]() |
[22] | S. Antontsev, J. Ferreira, E. Piskin, Existence and blow up of solutions for a strongly damped petrovsky equation with variable-exponent nonlinearities, Electron. J. Differ. Eq., 2021 (2021), 06. |
[23] |
M. Liao, Z. Tan, On behavior of solutions to a petrovsky equation with damping and variable-exponent sources, Sci. China Math., 66 (2023), 285–302. https://doi.org/10.1007/s11425-021-1926-x doi: 10.1007/s11425-021-1926-x
![]() |
[24] | D. Andrade, A. Mognon, Global solutions for a system of klein-gordon equations with memory, Bol. Soc. Paran. Mat., 21 (2003), 127–138. |
[25] |
K. Agre, M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differ. Integral Equ., 19 (2006), 1235–1270. https://doi.org/10.57262/die/1356050301 doi: 10.57262/die/1356050301
![]() |
[26] |
X. Wang, Y. Chen, Y. Yang, J. Li, R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019), 475–499. https://doi.org/10.1016/j.na.2019.06.019 doi: 10.1016/j.na.2019.06.019
![]() |
[27] |
O. Bouhoufani, I. Hamchi, Coupled system of nonlinear hyperbolic equations with variable-exponents: global existence and stability, Mediterr. J. Math., 12 (2020), 166. https://doi.org/10.1007/s00009-020-01589-1 doi: 10.1007/s00009-020-01589-1
![]() |
[28] |
S. A. Messaoudi, A. A. Talahmeh, M. M. Al-Gharabli, M. Alahyane, On the existence and stability of a nonlinear wave system with variable exponents, Asymptotic Anal., 128 (2021), 1–28. https://doi.org/10.3233/ASY-211704 doi: 10.3233/ASY-211704
![]() |
[29] |
S. A. Messaoudi, N. E. Tatar, Uniform stabilization of solutions of a nonlinear system of viscoelastic equations, Appl. Anal., 87 (2008), 247–263. https://doi.org/10.1080/00036810701668394 doi: 10.1080/00036810701668394
![]() |
[30] |
X. Han, M. Wang, Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal. Theor., 71 (2009), 5427–5450. https://doi.org/10.1016/j.na.2009.04.031 doi: 10.1016/j.na.2009.04.031
![]() |
[31] |
B. Said-Houari, S. Messaoudi, A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonlinear Differ. Equ. Appl., 18 (2011), 659–684. https://doi.org/10.1007/s00030-011-0112-7 doi: 10.1007/s00030-011-0112-7
![]() |
[32] |
M. I. Mustafa, Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations, Nonlinear Anal. Real, 13 (2012), 452–463. https://doi.org/10.1016/j.nonrwa.2011.08.002 doi: 10.1016/j.nonrwa.2011.08.002
![]() |
[33] |
S. Messoaudi, M. Al-Gharabli, A. Al-Mahdi, On the existence and decay of a viscoelastic system with variable-exponent nonlinearity, Discrete Cont. Dyn. S, 16 (2023), 1557–1595. https://doi.org/10.3934/dcdss.2022183 doi: 10.3934/dcdss.2022183
![]() |
[34] |
S. A. Messaoudi, M. M. Al-Gharabli, A. M. Al-Mahdi, M. A. Al-Osta, A coupled system of laplacian and bi-laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities: existence, uniqueness, blow-up and a large-time asymptotic behavior, AIMS Mathematics, 8 (2023), 7933–7966. https://doi.org/10.3934/math.2023400 doi: 10.3934/math.2023400
![]() |
[35] |
L. Gross, Logarithmic sobolev inequalities, Am. J. Math., 97 (1975), 1061–1083. https://doi.org/10.2307/2373688 doi: 10.2307/2373688
![]() |
[36] |
H. Chen, P. Luo, G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 442 (2015), 84–98. https://doi.org/10.1016/j.jmaa.2014.08.030 doi: 10.1016/j.jmaa.2014.08.030
![]() |
[37] | L. Diening, P. Harjulehto, P. Hästö, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-18363-8 |
[38] | M. T. Lacroix-Sonrier, Distributions, espaces de sobolev: applications, Paris: Ellipses, 1998. |
[39] | O. Bouhoufani, S. Messaoudi, M. Alahyane, Exsistence, blow up and numerical approximations of solutions for a biharmonic coupled system with variable exponents, 2022. Available from: https://doi.org/10.22541/au.166010582.26966044/v1 |
[40] |
H. Chen, G. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper., 3 (2012), 329–349. https://doi.org/10.1007/S11868-012-0046-9 doi: 10.1007/S11868-012-0046-9
![]() |
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