Research article

Robustness analysis of stability of Takagi-Sugeno type fuzzy neural network

  • In this paper, inequality techniques, stochastic analysis and algebraic methods are used to analyze the robustness of the stability of recurrent neural networks containing Takagi-Sugeno fuzzy rules. By solving the transcendental equations, the upper bounds of time delay and noise intensity are given, and the dynamic relationship between the two disturbance factors is derived. Finally, numerical examples are given to verify the results of this paper.

    Citation: Wenxiang Fang, Tao Xie. Robustness analysis of stability of Takagi-Sugeno type fuzzy neural network[J]. AIMS Mathematics, 2023, 8(12): 31118-31140. doi: 10.3934/math.20231593

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  • In this paper, inequality techniques, stochastic analysis and algebraic methods are used to analyze the robustness of the stability of recurrent neural networks containing Takagi-Sugeno fuzzy rules. By solving the transcendental equations, the upper bounds of time delay and noise intensity are given, and the dynamic relationship between the two disturbance factors is derived. Finally, numerical examples are given to verify the results of this paper.



    Let Ω={x¯RN+:|x|1}, RN+={x=(x1,x2,,xN)RN:xN>0} and N2. The boundary of Ω is denoted by

    Ω=1i=0Γi,

    where Γ0={xΩ:xN=0} and Γ1={xΩ:xN>0,|x|=1}. We are concerned with the existence an nonexistence of weak solutions to the evolution inequality

    kutkΔ(|u|m1u)+μ|x|2x(|u|m1u)|x|a|u|pin (0,)×Ω, (1.1)

    where u=u(t,x), k1 is an integer, p>m1, μ,aR and is the inner product in RN. Problem (1.1) is considered under the Dirichlet-type boundary conditions

    {u0on (0,)×Γ0,|u|m1ufon (0,)×Γ1, (1.2)

    where f=f(x).

    The issue of existence and nonexistence of solutions to higher order (in time) evolution inequalities has been studied in several papers. For instance, Hamidi and Laptev [1] investigated the nonexistence of weak solutions to higher-order evolution inequalities of the form

    {kutkΔu+λ|x|2u|u|pin (0,)×RN,k1utk1(0,x)0in RN, (1.3)

    where N3, λ(N22)2 and p>1. Namely, it was shown that, if one of the following assumptions is satisfied:

    λ0,1<p1+22k+s;

    or

    (N22)2λ<0,1<p1+22ks,

    where

    s=N22+λ+(N22)2,s=s+2N,

    then (1.3) admits no nontrivial weak solution. In [2], Caristi considered evolution inequalities of the form

    kutk|x|σΔmu|u|qin (0,)×RN, (1.4)

    where m is a positive integer, q>1 and σ2m. In the case σ=2m (critical degeneracy case) it was proven that, if k2, k1utk1(0,)|x|NL1(RN) with a positive average, and one of the following conditions holds:

    (ⅰ) N2(j+1) for j=0,m1 and 1<qk+1;

    (ⅱ) N=2(j+1) with j=0,,m1 and q>1,

    then (1.4) has no weak solution. In the case σ<2m (the subcritical degeneracy case), it was shown that, if k2, jutj(0,)|x|σL1loc(RN) for j=0,k2, k1utk1(0,)|x|σL1(RN) with a positive average, and

    q(k(N2m)+2mσ)Nk+2mσ(k+1),

    then (1.4) has no weak solution. Very recently, Filippucci and Ghergu [3] investigated evolution inequalities of the form

    kutk+(Δ)mu(K|u|p)|u|q,in (0,)×RN, (1.5)

    where N,k,m1 are integers, p,q>0 and KC(R+,R+) satisfies: K(|x|)L1loc(RN) and inf0<r<RK(r)=K(R) for sufficiently large R. Namely, the authors proved the following results:

    (ⅰ) If k is an even integer and q1, then (1.5) admits some positive solutions uC((0,)×RN) which verify k1utk1(0,)<0 in RN;

    (ⅱ) If p+q>2 and

    lim supRK(R)R2N+2mkp+qN+2m(11k)>0,

    then (1.5) has no nontrivial solutions such that

    k1utk10;ork1utk1(0,)L1(RN),RNk1utk1(0,x)dx>0.

    Other contributions related to higher order (in time) evolution equations and inequalities can be found in [4,5,6,7].

    In [8], Zheng and Wang studied the large time behavior of nonnegative solutions to parabolic equations of the form

    utΔumλx|x|2um=|x|σup(u0)in (0,)×RN¯ω, (1.6)

    where kR, σ>2, p>m1 and ω is a bounded domain in RN containing the origin with a smooth boundary ω. Problem (1.6) was investigated under the homogeneous Neumann boundary condition

    umν(t,x)=0on (0,)×ω (1.7)

    and the homogeneous Dirichlet boundary condition

    u(t,x)=0on (0,)×ω. (1.8)

    For problem (1.6) under the boundary condition (1.7), it was shown that (under a certain regularity on the geometry of ω)

    p={m+σ+2N+λ,ifλ>N,,ifλN (1.9)

    is critical in the sense of Fujita. When λ>2N, it was proven that problem (1.6) under the boundary condition (1.8), admits the same Fujita critical exponent p. Other contributions related to parabolic equations involving terms of the form b(x)um can be found in [9,10,11,12,13] (see also the references therein). Notice that in all the above mentioned references, only the parabolic case has been treated. Moreover, the considered solutions have been assumed to be positive. Very recently, in [14], the authors considered evolution inequalities of the form

    kutkΔ(|u|m1u)λx|x|2(|u|m1u)|x|σ|u|pin (0,)×Bc1, (1.10)

    under different types of boundary conditions, where p>m1, B1 denotes the open ball of radius 1 centered at the origin point in RN with N2 and Bc1 denotes the complement of B1. For instance, under the Dirichlet-type boundary condition

    |u(t,x)|m1u(t,x)f(x)on (0,)×B1,

    where fL1(B1) has a positive average, the authors proved that when σ>2, (1.10) admits as Fujita critical exponent

    pcr={m+m(σ+2)λ+N2ifλ>2N,ifλ2N.

    More precisely, the authors proved the following results:

    (ⅰ) If λ2N and fL1(B1) has a positive average, then for all p>m, (1.10) admits no weak solution;

    (ⅱ) If λ>2N and fL1(B1) has a positive average, then for all m<ppcr, (1.10) admits no weak solution;

    (ⅲ) If λ>2N and p>pcr, then (1.10) admits (stationary) solutions for some f>0.

    For more contributions related to the issue of existence and nonexistence of solutions to evolution equations and inequalities in exterior domains, see e.g., [15,16,17,18,19].

    Our aim in this paper is to study the influence of the obstacle domain on the critical behavior of (1.10) by considering the half-unit ball instead of the unit ball. Before presenting our main results, we need to define weak solutions to (1.1) and (1.2).

    Let

    D=(0,)×ΩandDi=(0,)×Γi,i=0,1.

    Notice that DiD for all i=0,1. We introduce the functional space V defined as follows.

    Definition 1.1. A function φ=φ(t,x) belongs to V, if the following conditions are satisfied:

    (ⅰ) φCk,2t,x(D), φ0;

    (ⅱ) supp(φ)⊂⊂D;

    (ⅲ) φ=0 on Di, i=0,1;

    (ⅳ) φνi0 on Di, i=0,1, where νi denotes the outward unit normal vector on Γi, relative to D.

    Using standard integrations by parts, we define weak solutions to (1.1) and (1.2) as follows.

    Definition 1.2. We say that uLploc(D) is a weak solutions to (1.1) and (1.2), if

    D|x|a|u|pφdxdtD1φν1f(x)dσdt(1)kDukφtkdxdtD|u|m1u(Δφ+μdiv(φx|x|2))dxdt (1.11)

    for every φV.

    For μR, let us introduce the parameter

    αμ=N+μ2+μ+(Nμ2)2. (1.12)

    Our main results are stated in the following theorem.

    Theorem 1.3. Let N2, k1 (an integer) and μ,aR.

    (I) Let fL1(Γ1) be such that

    Γ1f(x)xNdσ>0. (1.13)

    Assume that

    p>m,(αμ+N1)pm(αμ+1+a+N)0. (1.14)

    Then (1.1) and (1.2) admits no weak solution.

    (II) If

    p>m,(αμ+N1)pm(αμ+1+a+N)>0, (1.15)

    then (1.1) and (1.2) admits (stationary) solutions in the sense of Definition 1.2, for some f>0.

    The proof of part (Ⅰ) of Theorem 1.3 is based on nonlinear capacity estimates specifically adapted to the domain, the operator Δ+μ|x|2x and the boundary conditions (1.2). Part (Ⅱ) is established by the construction of expilicit solutions.

    Remark 1.4. (ⅰ) Let us point out that the used method in [8] for proving the blow-up of solutions to (1.6) requires the positivity of u. Namely, the authors used that functions of the form

    w(t):=RNωu(t,x)ψl(|x|)dx

    are nondecreasing for sufficiently large , where ψ0 is a certain cut-off function. In this paper, no restriction on the sign of solutions is imposed. Moreover, even in the case of positive solutions, it is difficult to use the method in [8] for proving the blow-up of solutions in the hyperbolic case. Namely, in order to show the blow-up of solutions to (1.6), the authors proved that the function w defined above, satisfies the differential inequality

    dwdtγwp,

    for a certain constant γ>0. A such inequality is related essentially to the parabolic nature of the problem.

    (ⅱ) The emphasis of this paper is on blow up results. The existence result provided by part (Ⅱ) of Theorem 1.3 is a consequence of elliptic results. We refer to [20,21], where some regularization methods to deal with the degeneracy were used to obtain the strong solution with latent singularity. We refer also to [22,23], where global solutions have been obtained following the standard gradient flow method. It will be interested to see if such methods can be adapted to the case of problem (1.1).

    (ⅲ) It is not difficult to show that for all μR, one has

    αμ+N1>0.

    Hence, (1.14) is equivalent to

    m<pm+m(a+2)αμ+N1,a>2.

    (ⅳ) From the above remark, we observe that (1.15) is equivalent to

    a2;orp>m+m(a+2)αμ+N1,a>2.

    Remark 1.5. (ⅰ) From Remark 1.4, we deduce that, if a2, then (1.1) and (1.2) admits no critical behavior. However, if a>2, then (1.1) and (1.2) admits as Fujita critical exponent the real number

    p=p(m,a,μ,N)=m+m(a+2)αμ+N1.

    (ⅱ) It is interesting to observe that p is independent on k. This implies that Theorem 1.3 holds true in the parabolic (k=1) as well as hyperbolic (k=2) case.

    Clearly, Theorem 1.3 yields existence and nonexistence results for the corresponding stationary problem

    Δ(|u|m1u)+μ|x|2x(|u|m1u)|x|a|u|pin Ω (1.16)

    under the Dirichlet-type boundary conditions

    {u0on Γ0,|u|m1ufon Γ1. (1.17)

    Corollary 1.6. Let N2 and μ,aR.

    (I) Let fL1(Γ1) be such that (1.13) holds. If (1.14) is satisfied, then (1.16) and (1.17) admits no weak solution.

    (II) If (1.15) holds, then (1.16) and (1.17) admits solutions for some f>0.

    The rest of the paper is organized as follows. In Section 2, we establish some preliminary lemmas that will be useful in the proof of our main results. Namely, we first prove an a priori estimate for problems (1.1) and (1.2). Next, we construct two families of functions belonging to V. The first family will be used in the proof of part (Ⅰ) of Theorem 1.3 in the case (αμ+N1)pm(αμ+1+a+N)<0, and the second family will be used in the proof of the critical case (αμ+N1)pm(αμ+1+a+N)=0. Finally, Section 3 is devoted to the proof of Theorem 1.3.

    Throughout this paper, the letters C,Ci denote always generic positive constants whose values are unimportant and may vary at different occurrences.

    Let N2, k1 (an integer), p>m1, μ,aR and fL1(Γ1). We denote by Lμ the differential operator given by

    Lμϕ=Δϕ+μdiv(ϕx|x|2).

    For φV, we introduce the integral terms

    ω1(φ)=supp(φ)|x|ap1φ1p1|kφtk|pp1dxdt (2.1)

    and

    ω2(φ)=supp(φ)|x|ampmφmpm|Lμφ|ppmdxdt. (2.2)

    We have the following a priori estimate.

    Lemma 2.1. Let uLploc(D) be a weak solution to (1.1) and (1.2). Then

    D1φν1f(x)dσdtC2i=1ωi(φ), (2.3)

    for every φV, provided that ωi(φ)<, i=1,2.

    Proof. Let uLploc(D) be a weak solution to (1.1) and (1.2) and φV be such that ωi(φ)<, i=1,2. Then, by (1.11), there holds

    D|x|a|u|pφdxdtD1φν1f(x)dσdtD|u||kφtk|dxdt+D|u|m|Lμφ|dxdt. (2.4)

    Making use of Young's inequality, we obtain

    D|u||kφtk|dxdt=D(|x|ap|u|φ1p)(|x|apφ1p|kφtk|)dxdt12D|x|a|u|pφdxdt+Cω1(φ). (2.5)

    Similarly, we obtain

    D|u|m|Lμφ|dxdt12D|x|a|u|pφdxdt+Cω2(φ). (2.6)

    Therefore, combining (2.4)–(2.6), we obtain (2.3).

    Let us introduce the function

    F(x)=xN|x|αμ(1|x|(N+μ)2αμ),xΩ, (2.7)

    where the parameter αμ is given by (1.12). Elementary calculations show that

    F0,LμF=0 in Ω,F|Γ0Γ1=0 (2.8)

    and

    Fν1|Γ1=(N+μ+2αμ)xN<0,Fν0|Γ0=|x|αμ(1|x|(N+μ)2αμ)<0. (2.9)

    Let ξ,ϑ,ιC(R) be three cut-off functions satisfying respectively

    0ξ1,ξ(s)=1 if |s|1,ξ(s)=0 if |s|2, (2.10)
    0ϑ1,ϑ(s)=1 if s0,ϑ(s)=0 if s1 (2.11)

    and

    ι0,supp(ι)⊂⊂(0,1). (2.12)

    For sufficiently large T,R and , let

    ιT(t)=ι(tT),t>0, (2.13)
    ξR(x)=F(x)ξ(|x|2R2),xΩ, (2.14)
    ϑR(x)=F(x)ϑ(ln(|x|R)ln(R)),xΩ. (2.15)

    Next, we consider functions of the form

    φ(t,x)=ιT(t)ξR(x),(t,x)D (2.16)

    and

    ψ(t,x)=ιT(t)ϑR(x),(t,x)D. (2.17)

    Lemma 2.2. For sufficiently large T,R and , the function φ defined by (2.16) belongs to V.

    Proof. By (2.8), (2.10), (2.12)–(2.14) and (2.16), it can be easily seen that properties (i)–(iii) of Definition 1.1 are satisfied. Moreover, for (t,x)Di, i=0,1, one has

    φνi(t,x)=ιT(t)ξRνi(x)=ιT(t)Fνi(x), (2.18)

    which implies by (2.9) that

    φνi(t,x)0,(t,x)Di.

    This shows that property (iv) of Definition 1.1 is also satisfied. Therefore, φV.

    Similarly, using (2.8), (2.9), (2.11), (2.12), (2.15) and (2.17), we obtain the following result.

    Lemma 2.3. For sufficiently large T,R and , the function ψ defined by (2.17) belongs to V.

    For sufficiently large T,R and , let φ be the function defined by (2.16).

    Lemma 2.4. The following estimate holds:

    ω1(φ)CT1kpp1(lnR+Rαμap1+N+1). (2.19)

    Proof. By (2.1) and (2.16), we obtain

    ω1(φ)=(T0ι1p1T|dkιTdtk|pp1dt)(1<|x|<2R,xN>0|x|ap1ξR(x)dx). (2.20)

    On the other hand, by (2.12) and (2.13), we obtain

    |dkιTdtk|CTkιk(tT),0<t<T,

    which yields

    T0ι1p1T|dkιTdtk|pp1dtCTkpp1T0ιkpp1(tT)dt=CT1kpp110ιkpp1(s)ds,

    that is,

    T0ι1p1T|dkιTdtk|pp1dtCT1kpp1. (2.21)

    Moreover, by (2.14), we have

    1<|x|<2R,xN>0|x|ap1ξR(x)dx=1<|x|<2R,xN>0|x|ap1F(x)ξ(|x|2R2)dx. (2.22)

    Using (2.7) and (2.10), we obtain

    1<|x|<2R,xN>0|x|ap1F(x)ξ(|x|2R2)dx1<|x|<2R,xN>0|x|ap1F(x)dx1<|x|<2R|x|αμ+1ap1dx=C2Rr=1rαμap1+NdrC(lnR+Rαμap1+N+1). (2.23)

    Hence, in view of (2.20)–(2.23), we obtain (2.19).

    Lemma 2.5. The following estimate holds:

    ω2(φ)CTR(αμ+N1)pm(αμ+1+a+N)pm. (2.24)

    Proof. By (2.2) and (2.16), we have

    ω2(φ)=(T0ιTdt)(1<|x|<2R,xN>0|x|ampmξmpmR|LμξR|ppmdx). (2.25)

    By (2.13), we obtain

    T0ιTdt=T0ι(tT)dt=T10ι(s)ds,

    that is,

    T0ιTdt=CT. (2.26)

    Moreover, by (2.14), for <|x|<2R,xN>0, we have

    LμξR(x)=Lμ(F(x)ξ(|x|2R2))=Δ(F(x)ξ(|x|2R2))+μdiv((F(x)ξ(|x|2R2)x|x|2)=ξ(|x|2R2)ΔF(x)+F(x)Δ(ξ(|x|2R2))+2F(x)(ξ(|x|2R2))+μξ(|x|2R2)div(F(x)x|x|2)+F(x)|x|2x(ξ(|x|2R2))=ξ(|x|2R2)LμF(x)+F(x)Δ(ξ(|x|2R2))+(2F(x)+F(x)|x|2x)(ξ(|x|2R2))=ξ(|x|2R2)LμF(x)+F(x)Δ(ξ(|x|2R2))+2R2|x|ξ1(|x|2R2)ξ(|x|2R2)(2F(x)x|x|+|x|1F(x)).

    In view of (2.8) (LμF=0), we obtain

    LμξR(x)=F(x)Δ(ξ(|x|2R2))+2R2|x|ξ1(|x|2R2)ξ(|x|2R2)(2F(x)x|x|+|x|1F(x)), (2.27)

    which implies by (2.10) that

    1<|x|<2R,xN>0|x|ampmξmpmR|LμξR|ppmdx=R<|x|<2R,xN>0|x|ampmξmpmR|LμξR|ppmdx. (2.28)

    On the other hand, by (2.7) and (2.10), for R<|x|<2R,xN>0, we obtain

    C1xNRαμF(x)C2xNRαμ,|2F(x)x|x|+|x|1F(x)|CxNRαμ1 (2.29)

    and

    |Δ(ξ(|x|2R2))|CR2ξ2(|x|2R2). (2.30)

    Hence, in view of (2.27), (2.29), (2.30) and using that 0ξ1, there holds

    |LμξR(x)|CxNRαμ2ξ2(|x|2R2),R<|x|<2R,xN>0. (2.31)

    Thus, using (2.14), (2.28), (2.29) and (2.31), we get

    1<|x|<2R,xN>0|x|ampmξmpmR|LμξR|ppmdxCR(αμ2)ppmR<|x|<2R,xN>0|x|ampmFmpm(x)xppmNξ2ppm(|x|2R2)dxCR(αμ2)pαμmpmR<|x|<2R,xN>0xN|x|ampmdxCR(αμ2)pαμmpmR<|x|<2R|x|1ampmdxCR(αμ2)pαμmpmR1ampmRN,

    that is,

    1<|x|<2R,xN>0|x|ampmξmpmR|LμξR|ppmdxCR(αμ+N1)pm(αμ+1+a+N)pm. (2.32)

    Finally, (2.24) follows from (2.25), (2.26) and (2.32).

    For sufficiently large T,R and , let ψ be the function defined by (2.17).

    Lemma 2.6. The following estimate holds:

    ω1(ψ)CT1kpp1(lnR+Rαμap1+N+1). (2.33)

    Proof. By (2.1) and (2.17), we obtain

    ω1(ψ)=(T0ι1p1T|dkιTdtk|pp1dt)(1<|x|<R,xN>0|x|ap1ϑR(x)dx). (2.34)

    Moreover, by (2.15), we have

    1<|x|<R,xN>0|x|ap1ϑR(x)dx=1<|x|<R,xN>0|x|ap1F(x)ϑ(ln(|x|R)ln(R))dx. (2.35)

    Using (2.7) and (2.11), we obtain

    1<|x|<R,xN>0|x|ap1F(x)ϑ(ln(|x|R)ln(R))dx1<|x|<R,xN>0|x|ap1F(x)dxC(lnR+Rαμap1+N+1). (2.36)

    Hence, in view of (2.21), (2.34)–(2.36), we obtain (2.33).

    Lemma 2.7. Let (αμ+N1)p=m(αμ+1+a+N). Then, the following estimate holds:

    ω2(ψ)CT(lnR)mpm. (2.37)

    Proof. By (2.2) and (2.17), we have

    ω2(ψ)=(T0ιTdt)(1<|x|<R,xN>0|x|ampmϑmpmR|LμϑR|ppmdx). (2.38)

    Similar calculations to those done in the proof of Lemma 2.5 give us

    LμϑR(x)=F(x)Δ(ϑ(ln(|x|R)ln(R)))+ln(R)|x|ϑ1(ln(|x|R)ln(R))ϑ(ln(|x|R)ln(R))(2F(x)x|x|+|x|1F(x)), (2.39)

    which implies by (2.11) that

    1<|x|<R,xN>0|x|ampmϑmpmR|LμϑR|ppmdx=R<|x|<R,xN>0|x|ampmϑmpmR|LμϑR|ppmdx. (2.40)

    Moreover, by (2.7) and (2.11), we obtain, as |x|,

    C1xN|x|αμF(x)C2xN|x|αμ,|2F(x)x|x|+|x|1F(x)|CxN|x|αμ1 (2.41)

    and

    |Δ(ϑ(ln(|x|R)ln(R)))|C(lnR)1|x|2ϑ2(ln(|x|R)ln(R)),R<|x|<R,xN>0. (2.42)

    In view of (2.39), (2.41), (2.42) and using that 0ϑ1, we get

    |LμϑR(x)|CxN|x|αμ2(lnR)1ϑ2(ln(|x|R)ln(R)),R<|x|<R,xN>0. (2.43)

    Next, it follows from (2.40), (2.41) and (2.43) that

    1<|x|<R,xN>0|x|ampmϑmpmR|LμϑR|ppmdxC(lnR)ppmR<|x|<R,xN>0|x|(αμ2)pm(a+αμ)pmxNϑ2ppm(ln(|x|R)ln(R))dxC(lnR)ppmR<|x|<R|x|(αμ1)pm(a+αμ+1)pmdx.

    Using that (αμ+N1)p=m(α+1+a+N), we get

    1<|x|<R,xN>0|x|ampmϑmpmR|LμϑR|ppmdxC(lnR)ppmR<|x|<R|x|Ndx=C(lnR)ppmRr=Rr1drC(lnR)mpm. (2.44)

    Finally, (2.37) follows from (2.26), (2.38) and (2.44).

    We use the contradiction argument. Namely, we suppose that uLploc(D) is a weak solutions to (1.1) and (1.2). We first consider the case

    p>m,(αμ+N1)pm(αμ+1+a+N)<0. (3.1)

    By Lemmas 2.1 and 2.2, for sufficiently large T,R and , there holds

    D1φν1f(x)dσdtC2i=1ωi(φ), (3.2)

    where φ is the function defined by (2.16). On the other hand, by (2.9), (2.18) and (2.26), we have

    D1φν1f(x)dσdt=D1ιT(t)f(x)Fν1(x)dσdt=(N+μ+2αμ)(T0ιT(t)dt)Γ1f(x)xNdσ=CTΓ1f(x)xNdσ. (3.3)

    Then, using Lemmas 2.4 and 2.5, (3.2) and (3.3), we obtain

    TΓ1f(x)xNdσC(T1kpp1(lnR+Rαμap1+N+1)+TR(αμ+N1)pm(αμ+1+a+N)pm),

    that is,

    Γ1f(x)xNdσC(Tkpp1(lnR+Rαμap1+N+1)+R(αμ+N1)pm(αμ+1+a+N)pm).

    Next, taking T=Rθ, where

    θ>max{0,p1kp(αμap1+N+1)}, (3.4)

    the above estimate reduces to

    Γ1f(x)xNdσC(Rθkpp1lnR+Rζ1+Rζ2), (3.5)

    where

    ζ1=αμap1+N+1θkpp1,ζ2=(αμ+N1)pm(αμ+1+a+N)pm.

    Notice that due to (3.4), one has ζ1<0. Moreover, by (3.1), we get ζ2<0. Therefore, passing to the limit as R in (3.5), we obtain Γ1f(x)xNdσ0, which contradicts (1.13).

    Next, we consider the case

    p>m,(αμ+N1)pm(αμ+1+a+N)=0. (3.6)

    By Lemmas 2.1 and 2.3, for sufficiently large T,R and , there holds

    D1ψν1f(x)dσdtC2i=1ωi(ψ), (3.7)

    where ψ is the function defined by (2.17). As in the previous case, using Lemmas 2.6 and 2.7, (2.9), (2.17) and (3.7), we obtain

    TΓ1f(x)xNdσC(T1kpp1(lnR+Rαμap1+N+1)+T(lnR)mpm),

    that is,

    Γ1f(x)xNdσC(Tkpp1(lnR+Rαμap1+N+1)+(lnR)mpm). (3.8)

    Hence, taking T=Rθ, where the parameter θ satisfies (3.4), and passing to the limit as R in (3.8), we reach a contradiction with (1.13). This completes the proof of part (Ⅰ) of Theorem 1.3.

    Assume that (1.15) holds. Let us consider a parameter δ satisfying

    max{μαμ,1+m(a+2)pm,1}<δ<N+αμ. (3.9)

    Notice that μαμ<N+αμ and 1<N+αμ. Moreover, due to (1.15), one has 1+m(a+2)pm<N+αμ. Hence, the set of δ satisfying (3.9) is nonempty. Let

    0<ε<[(N+αμδ)(δ+μ+αμ)]1pm. (3.10)

    We consider functions of the form

    uδ,ε(x)=εx1mN|x|δm,xΩ. (3.11)

    Elementary calculations show that

    Δumδ,ε+μ|x|2xumδ,ε=εm(N+αμδ)(δ+μ+αμ)xN|x|δ2,xΩ.

    Hence, using (3.9)–(3.11), for all xΩ, we obtain

    Δumδ,ε+μ|x|2xumδ,ε=(|x|aεpxpmN|x|δpm)εmp(N+αμδ)(δ+μ+αμ)x1pmN|x|δ2a+δpm=|x|aupδ,ε(x)εmp(N+αμδ)(δ+μ+αμ)x1pmN|x|δ2a+δpm|x|aupδ,ε(x)|x|(δ1)(pm1)(a+2)|x|aupδ,ε(x).

    Therefore, uδ,ε is a stationary solution to (1.1) and (1.2) with f(x)=εmxN, xΓ1. This completes the proof of part (Ⅱ) of Theorem 1.3.

    We investigated The existence and nonexistence of weak solutions to the evolution inequality (1.1) under the Dirichlet-type boundary conditions (1.2). When a2, we proved that (1.1) and (1.2) admit no critical behavior, namely, for all p>m1, (1.1) and (1.2) admit stationary solutions for some f>0. When a>2, we proved that (1.1) and (1.2) admit a critical exponent

    p=p(m,a,μ,N)=m+m(a+2)αμ+N1,

    in the following sense:

    (ⅰ) If

    m<pp,

    then (1.1) and (1.2) admit no weak solution, provided that fL1(Γ1) and

    Γ1f(x)xNdσ>0.

    (ⅱ) If

    p>p,

    then (1.1) and (1.2) admit (stationary) solutions, for some f>0.

    It is interesting to observe that in the case a>2, the critical exponent p depends only on m,a,μ and N, but it is independent of k, the order of the time-derivative. Therefore, our obtained results hold in both parabolic and hyperbolic cases. Finally, let us mention that comparing with previous existing results in the literature, in this study no restriction on the sign of solutions is imposed.

    The authors extend their appreciation to the Deanship of Scientific Research at the Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no. RP-21-09-02.

    The authors declare no conflict of interest.



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