Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js
Research article

Existence and blow up for viscoelastic hyperbolic equations with variable exponents

  • Received: 22 November 2023 Revised: 27 February 2024 Accepted: 07 August 2024 Published: 14 October 2024
  • 35L35, 35B40, 35B44

  • In this article, we consider a nonlinear viscoelastic hyperbolic problem with variable exponents. By using the FaedoGalerkin method and the contraction mapping principle, we obtain the existence of weak solutions under suitable assumptions on the variable exponents m(x) and p(x). Then we prove that a solution blows up in finite time with positive initial energy as well as nonpositive initial energy.

    Citation: Ying Chu, Bo Wen, Libo Cheng. Existence and blow up for viscoelastic hyperbolic equations with variable exponents[J]. Communications in Analysis and Mechanics, 2024, 16(4): 717-737. doi: 10.3934/cam.2024032

    Related Papers:

    [1] Reinhard Racke . Blow-up for hyperbolized compressible Navier-Stokes equations. Communications in Analysis and Mechanics, 2025, 17(2): 550-581. doi: 10.3934/cam.2025022
    [2] Huiyang Xu . Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials. Communications in Analysis and Mechanics, 2023, 15(2): 132-161. doi: 10.3934/cam.2023008
    [3] Yue Pang, Xiaotong Qiu, Runzhang Xu, Yanbing Yang . The Cauchy problem for general nonlinear wave equations with doubly dispersive. Communications in Analysis and Mechanics, 2024, 16(2): 416-430. doi: 10.3934/cam.2024019
    [4] Xiulan Wu, Yaxin Zhao, Xiaoxin Yang . On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity. Communications in Analysis and Mechanics, 2024, 16(3): 528-553. doi: 10.3934/cam.2024025
    [5] Yuxuan Chen . Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity. Communications in Analysis and Mechanics, 2023, 15(4): 658-694. doi: 10.3934/cam.2023033
    [6] Tingfu Feng, Yan Dong, Kelei Zhang, Yan Zhu . Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth. Communications in Analysis and Mechanics, 2025, 17(1): 263-289. doi: 10.3934/cam.2025011
    [7] Isaac Neal, Steve Shkoller, Vlad Vicol . A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy. Communications in Analysis and Mechanics, 2025, 17(1): 188-236. doi: 10.3934/cam.2025009
    [8] Mahmoud El Ahmadi, Mohammed Barghouthe, Anass Lamaizi, Mohammed Berrajaa . Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions. Communications in Analysis and Mechanics, 2024, 16(3): 509-527. doi: 10.3934/cam.2024024
    [9] Ming Liu, Binhua Feng . Grand weighted variable Herz-Morrey spaces estimate for some operators. Communications in Analysis and Mechanics, 2025, 17(1): 290-316. doi: 10.3934/cam.2025012
    [10] Yang Liu, Xiao Long, Li Zhang . Long-time dynamics for a coupled system modeling the oscillations of suspension bridges. Communications in Analysis and Mechanics, 2025, 17(1): 15-40. doi: 10.3934/cam.2025002
  • In this article, we consider a nonlinear viscoelastic hyperbolic problem with variable exponents. By using the FaedoGalerkin method and the contraction mapping principle, we obtain the existence of weak solutions under suitable assumptions on the variable exponents m(x) and p(x). Then we prove that a solution blows up in finite time with positive initial energy as well as nonpositive initial energy.



    In this paper, we study the initial boundary value problem of the nonlinear viscoelastic hyperbolic problem with variable exponents:

    {utt+2u+2uttt0g(tτ)2u(τ)dτ+|ut|m(x)2ut=|u|p(x)2u,(x,t)Ω×(0,T),u(x,t)=uν(x,t)=0,(x,t)Ω×(0,T),u(x,0)=u0(x), ut(x,0)=u1(x),xΩ, (1.1)

    where ΩRn(n1) is a bounded domain in Rn with a smooth boundary Ω, ν is the unit outer normal to Ω, the exponents m(x) and p(x) are continuous functions on ¯Ω with the logarithmic module of continuity:

    x,yΩ,|xy|<1,|m(x)m(y)|+|p(x)p(y)|ω(|xy|), (1.2)

    where

    limτ0+supω(τ)ln1τ=C<. (1.3)

    In addition to this condition, the exponents satisfy the following:

    2m:=essinfxΩm(x)m(x)m+:=esssupxΩm(x)<2(n2)n4, (1.4)
    2p:=essinfxΩp(x)p(x)p+:=esssupxΩp(x)<2(n2)n4, (1.5)

    g:R+R+ is a C1 function satisfying

    g(0)>0, g(τ)0,10g(τ)dτ=l>0. (1.6)

    The equation of Problem (1.1) arises from the modeling of various physical phenomena such as the viscoelasticity and the system governing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Boltzmann's model, or electro-rheological fluids, viscoelastic fluids, processes of filtration through a porous medium, and fluids with temperature-dependent viscosity and image processing which give rise to equations with nonstandard growth conditions, that is, equations with variable exponents of nonlinearities. More details on these problems can be found in previous studies [1,2,3,4,5,6].

    When m(x) and p(x) are constants, Messaoudi [7] discussed the nonlinear viscoelastic wave equation

    uttu+t0g(tτ)u(τ)dτ+|ut|m2ut=|u|p2u,

    he proved that any weak solution with negative initial energy blows up in finite time if p>m, and a global existence result for pm. The results were improved later by Messaoudi [8], where the blow-up result in finite time with positive initial energy was obtained. Moreover, Song [9] showed the finite-time blow-up of some solutions whose initial data had arbitrarily high initial energy. In the same year, Song [10] studied the initial-boundary value problem

    |ut|ρuttu+t0g(tτ)u(τ)dτ+|ut|m2ut=|u|p2u,

    and proved the nonexistence of global solutions with positive initial energy. Cavalcanti, Domingos, and Ferreira [11] were concerned with the non-linear viscoelastic equation

    |ut|ρuttuutt+t0g(tτ)u(τ)dτγut=0,

    and proved the global existence of weak solutions. Moreover, they obtained the uniform decay rates of the energy by assuming a strong damping ut acting in the domain and providing the relaxation function which decays exponentially.

    In 2017, Messaoudi [12] considered the following nonlinear wave equation with variable exponents:

    uttu+a|ut|m(x)2ut=b|u|p(x)2u,

    where a,b are positive constants. By using the FaedoGalerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents m(x) and p(x). Then this paper also proved the finite-time blow-up of solutions and gave a two-dimensional numerical example to illustrate the blow up result. Park [13] showed the blow up of solutions for a viscoelastic wave equation with variable exponents

    uttu+t0g(ts)u(s)ds+a|ut|m(x)2ut=b|u|p(x)2u,

    where the exponents of nonlinearity p(x) and m(x) are given functions and a,b>0 are constants. For nonincreasing positive function g, they prove the blow-up result for the solutions with positive initial energy as well as nonpositive initial energy. Alahyane [14] discussed the nonlinear viscoelastic wave equation with variable exponents

    uttu+t0g(tτ)u(τ)dτ+μut=|u|p(x)2u,

    where μ is a nonnegative constant and the exponent of nonlinearity p(x) and g are given functions. Under arbitrary positive initial energy and specific conditions on the relaxation function g, they prove a finite-time blow-up result and give some numerical applications to illustrate their theoretical results. Ouaoua and Boughamsa [15] considered the following boundary value problem:

    utt+2uu+|ut|m(x)2ut=|u|p(x)2u,

    the authors established the local existence by using the FaedoGalerkin method with positive initial energy and suitable conditions on the variable exponents m(x) and r(x). In addition, they also proved that the local solution is global and obtained the stability estimate of the solution. Ding and Zhou [16] considered a Timoshenko-type equation

    utt+2uM(||u||22)u+|ut|p(x)2ut=|u|q(x)2u,

    they prove that the solutions blow up in finite time with positive initial energy. Therefore, the existence of finite-time blow-up solutions with arbitrarily high initial energy is established, and the upper and lower bounds of the blow-up time are derived. More related references can be found in [17,18,19,20,21,22].

    Motivated by [7,13,14], we considered the existence of the solutions and their blow-up for the nonlinear damping and viscoelastic hyperbolic problem with variable exponents. Our aim in this work is to prove the existence of the weak solutions and to find sufficient conditions on m(x) and p(x) for which the blow-up takes place.

    This article consists of three sections in addition to the introduction. In Section 2, we recall the definitions and properties of Lp(x)(Ω) and the Sobolev spaces W1,p(x)(Ω). In Section 3, we prove the existence of weak solutions for Problem (1.1). In Section 4, we state and prove the blow-up result for solutions with positive initial energy as well as nonpositive initial energy.

    In this section, we review some results regarding Lebesgue and Sobolev spaces with variable exponents first. All of these results and a comprehensive study of these spaces can be found in [23]. Here (,) and , denote the inner product in space L2(Ω) and the duality pairing between H2(Ω) and H20(Ω).

    The variable exponent Lebesgue space Lp(x)(Ω) is defined by

    Lp(x)(Ω)={u(x):uismeasurableinΩ, ρp(x)(u)=Ω|u|p(x)dx<},

    this space is endowed with the norm

    up(x)=inf {λ>0:Ω|u(x)λ|p(x)dx1}.

    The variable exponent Sobolev space W1,p(x)(Ω) is defined by

    W1,p(x)(Ω)={uLp(x)(Ω) suchthatuexistsand|u|Lp(x)(Ω)},

    the corresponding norm for this space is

    u1,p(x)=up(x)+up(x),

    define W1,p(x)0(Ω) as the closure of  C0(Ω) with respect to the W1,p(x)(Ω) norm. The spaces  Lp(x)(Ω),W1,p(x)(Ω) and W1,p(x)0(Ω) are separable and reflexive Banach spaces when 1<pp+<, where p:=essinfΩp(x) and p+:=esssupΩp(x). As usual, we denote the conjugate exponent of p(x) by p(x)=p(x)/(p(x)1) and the Sobolev exponent by

    p(x)={np(x)nkp(x),if p(x)<n,,if p(x)n.

    Lemma 2.1. If p1(x), p2(x)C+(¯Ω)={hC(¯Ω):minx¯Ωh(x)>1}, p1(x)p2(x) for any xΩ, then there exists the continuous embedding Lp2(x)(Ω)Lp1(x)(Ω), whose norm does not exceed |Ω|+1.

    Lemma 2.2. Let p(x), q(x)C+(¯Ω). Assuming that q(x)<p(x), there is a compact and continuous embedding Wk,p(x)(Ω)Lq(x)(Ω).

    Lemma 2.3. (Hölder's inequality) [24] For any uLp(x)(Ω) and vLq(x)(Ω), then the following inequality holds:

    |Ωuvdx|(1p+1q)||u||p(x)||v||q(x)2||u||p(x)||v||q(x).

    Lemma 2.4. For uLp(x)(Ω), the following relations hold:

    u0(up(x)=λρp(x)(uλ)=1),
    up(x)<1(=1;>1)ρp(x)(u)<1(=1;>1),
    up(x)>1upp(x)ρp(x)(u)up+p(x),
    up(x)<1up+p(x)ρp(x)(u)upp(x).

    Next, we give the definition of the weak solution to Problem (1.1).

    Definition 2.1. A function u(x, t) is called a weak solution for Problem (1.1), if uC(0,T;H20(Ω)) C1(0,T;H20(Ω))C2(0,T;H2(Ω)) with uttL2(0,T;H20(Ω)) and u satisfies the following conditions:

    (1) For every ωH20(Ω) and for a.e.t(0,T)

    utt,ω+(u,ω)+(utt,ω)t0g(tτ)(u(τ),ω)dτ+(|ut|m(x)2ut,ω)=(|u|p(x)2u,ω),

    (2) u(x,0)=u0(x)H20(Ω),ut(x,0)=u1(x)H20(Ω).

    In this section, we prove the existence of a weak solution for Problem (1.1) by making use of the Faedo–Galerkin method and the contraction mapping principle. For a fixed T>0, we consider the space H=C(0,T;H20(Ω))C1(0,T;H20(Ω)) with the norm ||v||2H=max0tT(||vt||22+l||v||22).

    Lemma 3.1. Assume that (1.4), (1.5), and (1.6) hold, let (u0,u1)H20(Ω)×H20(Ω), for any T>0, vH, then there exists uC(0,T;H20(Ω))C1(0,T;H20(Ω))C2(0,T;H2(Ω)) with  uttL2(0,T;H20(Ω)) satisfying

    {utt+2u+2uttt0g(tτ)2u(τ)dτ+|ut|m(x)2ut=|v|p(x)2v,(x,t)Ω×(0,T),u(x,t)=uν(x,t)=0,(x,t)Ω×(0,T),u(x,0)=u0(x), ut(x,0)=u1(x),xΩ. (3.1)

    Proof. Let {ωj}j=1 be the orthogonal basis of H20(Ω), which is the standard orthogonal basis in L2(Ω) such that

    ωj=λjωj  in  Ω,ωj=0  on  Ω,

    we denote by Vk=span{ω1,ω2,,ωk} the subspace generated by the first k vectors of the basis {ωj}j=1. By normalization, we have ||ωj||2=1. For all k1, we seek k functions ck1(t),ck2(t),,ckk(t)C2[0,T] such that

    uk(x,t)=kj=1ckj(t)ωj(x),

    satisfying the following approximate problem

    {(uktt,ωi)+(uk,ωi)+(uktt,ωi)t0g(tτ)(uk,ωi)dτ+(|ukt|m(x)2ukt,ωi)=Ω|v|p(x)2vωidx,uk(0)=uk0,   ukt(0)=uk1,     i=1,2,k, (3.2)

    where

    uk0=ki=1(u0,ωi)ωiu0   in  H20(Ω),
    uk1=ki=1(u1,ωi)ωiu1   in  H20(Ω),

    thus, (3.2) generates the initial value problem for the system of second-order differential equations with respect to cki(t):

    {(1+λ2i)ckitt(t)+λ2icki(t)=Gi(ck1t(t),,ckkt(t))+gi(cki(t)),   i=1,2,,k,cki(0)=Ωu0ωidx,      ckit(0)=Ωu1ωidx,              i=1,2,,k. (3.3)

    where

    Gi(ck1t(t),,ckkt(t))=Ω|kj=1ckjt(t)ωj(x)|m(x)2kj=1ckjt(t)ωj(x)ωi(x)dx,

    and

    gi(cki(t))=λ2it0g(tτ)cki(τ)dτ+Ω|v|p(x)2vωidx,

    by Peano's Theorem, we infer that the Problem (3.3) admits a local solution cki(t)C2[0,T].

    The first estimate. Multiplying (3.2) by ckit(t) and summing with respect to i, we arrive at the relation

    ddt(12||ukt||22+12||uk||22+12||ukt||22)+Ω|ukt|m(x)dxt0g(tτ)Ωuk(τ)uktdxdτ=Ω|v|p(x)2vuktdx. (3.4)

    By simple calculation, we have

    t0g(tτ)ΩΔuk(τ)Δuktdxdτ=12ddt(guk)12(guk)12ddtt0g(τ)dτ||Δuk||22+12g(t)||uk||22, (3.5)

    where

    (φψ)=t0φ(tτ)||ψ(t)ψ(τ)||22dτ,

    inserting (3.5) into (3.4), using Hölder's inequality and Young's inequality, we obtain

    ddt[12||ukt||22+12||ukt||22+12(guk)+12(1t0g(τ)dτ)||Δuk||22]=12(guk)12g(t)||uk||22+Ω|v|p(x)2vuktdxΩ|ukt|m(x)dxΩ|v|p(x)2vuktdx|v|p(x)2v2||ukt||2η2Ω|v|2(p(x)1)dx+12η||ukt||22, (3.6)

    using the embedding H20(Ω)L2(p(x)1)(Ω) and Lemma 2.4, we easily obtain

    Ω|v|2(p(x)1)dxmax{||v||2(p1)2(p(x)1),||v||2(p+1)2(p(x)1)}Cmax{||v||2(p1)2,||v||2(p+1)2}C, (3.7)

    where C is a positive constant. We denote by C various positive constants that may be different at different occurrences.

    Combining (3.6) and (3.7), we obtain

    ddt[12||ukt||22+12||ukt||22+12(guk)+12(1t0g(τ)dτ)||Δuk||22]η2C+12η||ukt||22,

    by Gronwall's inequality, there exists a positive constant CT such that

    ||ukt||22+||ukt||22+(guk)+l||Δuk||22CT, (3.8)

    therefore, there exists a subsequence of {uk}k=1, which we still denote by {uk}k=1, such that

    uku weakly star in L(0,T;H20(Ω)),uktut weakly star in L(0,T;H20(Ω)),uku weakly in L2(0,T;H20(Ω)),uktut weakly in L2(0,T;H20(Ω)). (3.9)

    The second estimate. Multiplying (3.2) by ckitt(t) and summing with respect to i, we obtain

    ||uktt||22+||Δuktt||22+ddt(Ω1m(x)|ukt|m(x)dx)=Ωukukttdx+t0g(tτ)ΩΔuk(τ)Δukttdxdτ+Ω|v|p(x)2vukttdx. (3.10)

    Note that we have the estimates for ε>0

    |Ωukukttdx|ε||uktt||22+14ε||uk||22, (3.11)
    Ω|v|p(x)2vukttdx|v|p(x)2v2uktt2ε||uktt||22+14εΩ|v|2(p(x)1)dx, (3.12)

    and

    |t0g(tτ)Ωuk(τ)ukttdxdτ|14εΩ(t0g(tτ)uk(τ)dτ)2dx+ε||uktt||22ε||uktt||22+14εt0g(s)dst0g(tτ)Ω|uk(τ)|2dxdτε||uktt||22+(1l)g(0)4εt0||uk(τ)||22dτ, (3.13)

    similar to (3.6) and (3.7), from H20(Ω)L2(Ω), we have

    Ω|v|p(x)2vukttdxεC||uktt||22+C4ε. (3.14)

    Taking into account (3.10)(3.14), we obtain

    ||uktt||22+(12εCε)||uktt||22+ddt(Ω1m(x)|ukt|m(x)dx)14ε||uk||22+(1l)g(0)4εt0||uk(τ)||22dτ+C4ε, (3.15)

    integrating (3.15) over (0,t), we obtain

    t0||uktt||22dτ+(12εCε)t0||uktt||22dτ+Ω1m(x)|ukt|m(x)dxC4εt0(||uk||22+τ0||uk(s)||22ds)dτ+CT, (3.16)

    taking ε small enough in (3.16), for some positive constant CT, we obtain

    t0||uktt||22dτ+t0||uktt||22dτCT, (3.17)

    we observe that estimate (3.17) implies that there exists a subsequence of {uk}k=1, which we still denote by {uk}k=1, such that

    ukttutt weakly in L2(0,T;H20(Ω)). (3.18)

    In addition, from (3.9), we have

    (uktt,ωi)=ddt(ukt,ωi)ddt(ut,ωi)=(utt,ωi)  weakly  star  in  L(0,T;H2(Ω)). (3.19)

    Next, we will deal with the nonlinear term. Combining (3.9), (3.18), and Aubin–Lions theorem [25], we deduce that there exists a subsequence of {uk}k=1 such that

    uktut strongly in C(0,T;L2(Ω)), (3.20)

    then

    |ukt|m(x)2ukt|ut|m(x)2ut  a.e. (x,t)Ω×(0,T), (3.21)

    using the embedding H20(Ω)L2(m(x)1)(Ω) and Lemma 2.4, we have

    |ukt|m(x)2ukt22=Ω|ukt|2(m(x)1)dxmax{||ukt||2(m1)2,||ukt||2(m+1)2}C, (3.22)

    hence, using (3.21) and (3.22), we obtain

    |ukt|m(x)2ukt|ut|m(x)2ut  weakly  star in L(0,T;L2(Ω)). (3.23)

    Setting up k in (3.2), combining with (3.9), (3.18), (3.19), and (3.23), we obtain

    utt,ω+(u,ω)+(utt,ω)t0g(tτ)(u(τ),ω)dτ+(|ut|m(x)2ut,ω)=(|v|p(x)2v,ω).

    To handle the initial conditions. From (3.9) and Aubin–Lions theorem, we can easily get uku in C(0,T;L2(Ω)), thus uk(0)u(0) in L2(Ω), and we also have that uk(0)=uk0u0 in H20(Ω), hence u(0)=u0 in H20(Ω). Similarly, we get that ut(0)=u1.

    Uniqueness. Suppose that (3.1) has solutions u and z, then ω=uz satisfies

    {ωtt+2ω+2ωttt0g(tτ)2ω(τ)dτ+|ut|m(x)2ut|zt|m(x)2zt=0,(x,t)Ω×(0,T),ω(x,t)=ων(x,t)=0,(x,t)Ω×(0,T),ω(x,0)=0, ωt(x,0)=0,xΩ.

    Multiplying the first equation of Problem (3.1) by ωt and integrating over Ω, we have

    12ddt[||ωt||22+(1t0g(τ)dτ)||ω||22+||ωt||22+(gω)]+12g(t)||ω||22=Ω(|ut|m(x)2ut|zt|m(x)2zt)(utzt)dx+12(gω),

    from the inequality

    (|a|m(x)2a|b|m(x)2b)(ab)0, (3.24)

    for all a,bRn and a.e. xΩ, we obtain

    ||ωt||22+l||ω||22+||ωt||22=0,

    which implies that ω=0. This completes the proof.

    Theorem 3.1. Assume that (1.4) and (1.6) hold, let the initial date (u0,u1)H20(Ω)×H20(Ω), and

    2pp(x)p+2(n3)n4,

    then there exists a unique local solution of Problem (1.1).

    Proof. For any T>0, consider MT={uH:u(0)=u0,ut(0)=u1,||u||HM}. Lemma 3.1 implies that for vMT, there exists u=S(v) such that u is the unique solution to Problem 3.1. Next, we prove that for a suitable T>0, S is a contractive map satisfying S(MT)MT.

    Multiplying the first equation of the Problem (3.1) by ut and integrating it over (0,t), we obtain

    ||ut||22+||ut||22+(gu)+l||Δu||22||u1||22+||u1||22+||Δu0||22+2t0Ω|v|p(x)2vutdxdτ, (3.25)

    using Hölder's inequality and Young's inequality, we have

    |Ω|v|p(x)2vutdx|γ||ut||22+14γΩ|v|2p(x)2dxγ||ut||22+14γ[Ω|v|2p2dx+Ω|v|2p+2dx]γ||ut||22+C4γ[||v||2p22+||v||2p+22],

    thus, (3.25) becomes

    ||ut||22+||ut||22+l||Δu||22λ0+2t0Ω|v|p(x)2vutdxdτλ0+2γTsup(0,T)||ut||22+TC2γsup(0,T)[||v||2p22+||v||2p+22],

    hence, we have

    sup(0,T)||ut||22+sup(0,T)||ut||22+lsup(0,T)||Δu||22λ0+2γTsup(0,T)||ut||22+TC2γsup(0,T)[||v||2p2H+||v||2p+2H],

    where λ0=||u1||22+||u1||22+||Δu0||22, choosing γ=12T such that

    ||u||2Hλ0+T2Csup(0,T)[||v||2p2H+||v||2p+2H].

    For any vMT, by choosing M large enough so that

    ||u||2Hλ0+2T2CM2(p+1)M2,

    and T>0, sufficiently small so that

    TM2λ02CM2(p+1),

    we obtain ||u||HM, which shows that S(MT)MT.

    Let v1,v2MT,u1=S(v1),u2=S(v2),u=u1u2, then u satisfies

    {utt+2u+2uttt0g(tτ)2u(τ)dτ+|u1t|m(x)2u1t|u2t|m(x)2u2t=|v1|p(x)2v1|v2|p(x)2v2,(x,t)Ω×(0,T),u(x,t)=uν(x,t)=0,(x,t)Ω×(0,T),u(x,0)=0, ut(x,0)=0,xΩ.

    Multiplying by ut and integrating over Ω×(0,t), we obtain

    12||ut||22+12(1t0g(τ)dτ)||u||22+12||ut||22+12(gu)+t0Ω[|u1t|m(x)2u1t|u2t|m(x)2u2t](u1tu2t)dxdτt0Ω(f(v1)f(v2))utdxdτ, (3.26)

    where f(v)=|v|p(x)2v. From (1.6) and (3.24), we obtain

    12||ut||22+l2||u||22+12||ut||22+12(gu)t0Ω(f(v1)f(v2))utdxdτ. (3.27)

    Now, we evaluate

    I=Ω|(f(v1)f(v2))||ut|dx=Ω|f(ξ)||v||ut|dx,

    where and , . Thanks to Young's inequality and Hölder's inequality, we have

    (3.28)

    Inserting into , choosing small enough, we obtain

    taking small enough so that , we conclude

    thus, the contraction mapping principle ensures the existence of a weak solution to Problem . This completes the proof.

    In this section, we show that the solution to Problem blows up in finite time when the initial energy lies in positive as well as nonpositive. For this task, we define

    (4.1)

    by the definition of , we also have

    (4.2)

    Now, we set

    and

    (4.3)

    where the constant will be discussed later, and is the best constant of the Sobolev embedding . It follows from that

    (4.4)

    and is a nondecreasing function.

    To prove Theorem , we need the following two lemmas:

    Lemma 4.1. Suppose that holds and the exponents and satisfy condition and . Assume further that

    then there exists a constant such that

    (4.5)

    Proof. Using , , Lemma , and the embedding , we find that

    (4.6)

    where Analyzing directly the properties of , we deduce that satisfies the following properties:

    It is easily verified that is strictly increasing for , strictly decreasing for , as , and . Since , there exists a such that . By , we see that , which implies since the condition To prove , we suppose by contradiction that for some , . The continuity of illustrates that we could choose such that , then we have . This is a contradiction. The proof is completed.

    Lemma 4.2. Let the assumption in Lemma be satisfied. For , we have

    Proof. indicates that is nondecreasing with respect to , thus

    It follows from , , and Lemma that

    The proof is completed.

    Our blow-up result reads as follows:

    Theorem 4.3. Suppose that

    and

    (4.7)

    hold, if the following conditions

    are satisfied, then there exists such that

    (4.8)

    Proof. Assume by contradiction that does not hold true, then for and all , we get

    (4.9)

    where is a positive constant.

    Now, we define as follows:

    (4.10)

    where , small enough to be chosen later, and

    The remaining proof will be divided into two steps.

    By taking the derivative of and using , we obtain

    applying Hölder's inequality and Young's inequality, we have

    where , then

    rewriting to , using and to substitute for , choosing sufficiently small, we obtain

    (4.11)

    It follows from the condition in Theorem that

    here, we can take sufficiently small and choose sufficiently close to such that

    (4.12)

    Therefore, we obtain by combining and ,

    (4.13)

    Applying Young's inequality with , the embedding , Lemma and Lemma , we easily have

    (4.14)

    where , . Next, we have

    and

    which illustrate

    where . Recalling and Lemma , apparently,

    (4.15)

    it follows from , , and that

    let us fix the constant so that

    and then choose so small that . Therefore, we obtain

    (4.16)

    where

    Inequalities and Lemma imply Therefore, for a sufficiently small , we have

    Applying Hölder's inequality, Young's inequality and the embedding , we easily obtain

    (4.17)

    where Choosing , then further, can be rewritten as

    (4.18)

    recalling , we obtain

    (4.19)

    with . Inserting into , we obtain

    (4.20)

    We now estimate

    (4.21)

    therefore, combining and , we obtain

    (4.22)

    where

    Combining and , we arrive at

    (4.23)

    A simple integration of over yields

    this shows that blows up in finite time

    furthermore, one gets from that

    it easily follows that

    and using Lemma , we obtain

    this leads to a contradiction with . Thus, the solution to Problem blows up in finite time.

    Ying Chu: Methodology, Wring-original draft, Writing-review editing; Bo Wen and Libo Cheng: Methodology, Writing-original draft.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work was supported by Science and Technology Development Plan Project of Jilin Province, China (20240101307JC).

    The authors declare there is no conflict of interest.



    [1] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, Siam. J. Appl. Math., 4 (2006), 1383–1406. https://doi.org/10.1137/050624522 doi: 10.1137/050624522
    [2] R. Aboulaich, D. Meskine, A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874–882. https://doi.org/10.1016/j.camwa.2008.01.017 doi: 10.1016/j.camwa.2008.01.017
    [3] S. Lian, W. Gao, C. Cao, H. Yuan, Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, J. Math. Anal. Appl., 342 (2008), 27–38. https://doi.org/10.1016/j.jmaa.2007.11.046 doi: 10.1016/j.jmaa.2007.11.046
    [4] S. Antontsev, S. Shmarev, Blow up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234 (2010), 2633–2645. https://doi.org/10.1016/j.cam.2010.01.026 doi: 10.1016/j.cam.2010.01.026
    [5] S. Antontsev, S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up, Atlantis Press Paris, 2015.
    [6] B. Tahir, K. Mohamed, B. Masoud, Global existence, blow-up and asymptotic behavior of solutions for a class of -choquard diffusion equations in , J. Math. Anal. Appl., 506 (2021), 125720. https://doi.org/10.1016/j.jmaa.2021.125720 doi: 10.1016/j.jmaa.2021.125720
    [7] S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr., 231 (2001), 105–111.
    [8] S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902–915. https://doi.org/10.1016/j.jmaa.2005.07.022 doi: 10.1016/j.jmaa.2005.07.022
    [9] H. T. Song, Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation, Nonlinear Anal-Real, 26 (2015), 306–314. https://doi.org/10.1016/j.nonrwa.2015.05.015 doi: 10.1016/j.nonrwa.2015.05.015
    [10] H. T. Song, Global nonexistence of positive initial energy solutions for a viscoelastic wave equation, Nonlinear Anal, 125 (2015), 260–269. https://doi.org/10.1016/j.na.2015.05.015 doi: 10.1016/j.na.2015.05.015
    [11] M. M. Cavalcanti, C. Domingos, J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24 (2001), 1043–1053. https://doi.org/10.1002/mma.250 doi: 10.1002/mma.250
    [12] S. A. Messaoudi, A. A.Talahmeh, J. H. Al-Smail, Nonlinear damped wave equation: existence and blow-up, Comput. Math. Appl., 74 (2017), 3024–3041. https://doi.org/10.1016/j.camwa.2017.07.048 doi: 10.1016/j.camwa.2017.07.048
    [13] S. H. Park, J. R. Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Meth. Appl. Sci., 42 (2019), 2083–2097. https://doi.org/10.1002/mma.5501 doi: 10.1002/mma.5501
    [14] M. Alahyane, A. A. Talahmeh, S. A. Messaoudi, Theoretical and numerical study of the blow up in a nonlinear viscoelastic problem with variable exponents and arbitrary positive energy, Acta. Math. Sci., 42 (2022), 141–154. https://doi.org/10.1007/s10473-022-0107-y doi: 10.1007/s10473-022-0107-y
    [15] A. Ouaoua, W. Boughamsa, Well-posedness and stability results for a class of nonlinear fourth-order wave equation with variable-exponents, J. Nonlinear. Anal. Appl., 14 (2023), 1769–1785. https://doi.org/10.22075/ijnaa.2022.27129.3507 doi: 10.22075/ijnaa.2022.27129.3507
    [16] H. Ding, J. Zhou, Blow-up for the Timoshenko-type equation with variable exponentss, Nonlinear Anal-Real, 71 (2023), 103801. https://doi.org/10.1016/j.nonrwa.2022.103801 doi: 10.1016/j.nonrwa.2022.103801
    [17] M. Liao, B. Guo, X Zhu, Bounds for blow-up time to a viscoelastic hyperbolic equation of Kirchhoff type with variable sources, Acta. Appl. Math., 170 (2020), 755–772. https://doi.org/10.1007/s10440-020-00357-3 doi: 10.1007/s10440-020-00357-3
    [18] M. Liao, Study of a viscoelastic wave equation with a strong damping and variable exponents, Mediterr. J. Math., 18 (2021), 186. https://doi.org/10.1007/s00009-021-01826-1 doi: 10.1007/s00009-021-01826-1
    [19] W. Lian, V. Radulescu, R. Xu, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589–611. https://doi.org/10.1515/acv-2019-0039 doi: 10.1515/acv-2019-0039
    [20] Y. Luo, R. Xu, C. Yang, Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities, Cal. Var. Partial Dif., 61 (2022), 210. https://doi.org/10.1007/s00526-022-02316-2 doi: 10.1007/s00526-022-02316-2
    [21] M. Liao, Z. Tan, 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
    [22] Y. Pang, V. Radulescu, R. Xu, Global Existence and Finite Time Blow-up for the m-Laplacian Parabolic Problem, Acta. Math. Sin., 39 (2023), 1497–1524. https://doi.org/10.1007/s10114-023-1619-7 doi: 10.1007/s10114-023-1619-7
    [23] L. Diening, P. Harjulehto, P. Hästö, M. Ržička, Lebesgue and Sobolev spaces with variable exponents, Springer Berlin, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
    [24] X. L. Fan, D. Zhao, On the spaces and , J. Math. Anal. Appl., 263 (2001), 424–446. https://doi.org/10.1006/jmaa.2000.7617 doi: 10.1006/jmaa.2000.7617
    [25] J. L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, 1969.
  • This article has been cited by:

    1. Tahir Boudjeriou, Ngo Tran Vu, Nguyen Van Thin, High Energy Blowup for a Class of Wave Equations With Critical Exponential Nonlinearity, 2025, 0170-4214, 10.1002/mma.10873
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(955) PDF downloads(82) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog