Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

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

  • In this paper, we consider a coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities. This system is supplemented with initial and mixed boundary conditions. First, we establish the existence and uniqueness results of a weak solution, under suitable assumptions on the variable exponents. Second, we show that the solutions with positive-initial energy blow-up in a finite time. Finally, we establish the global existence as well as the energy decay results of the solutions, using the stable-set and the multiplier methods, under appropriate conditions on the variable exponents and the initial data.

    Citation: Salim A. Messaoudi, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Mohammed 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[J]. AIMS Mathematics, 2023, 8(4): 7933-7966. doi: 10.3934/math.2023400

    Related Papers:

    [1] Zhiqiang Li . The finite time blow-up for Caputo-Hadamard fractional diffusion equation involving nonlinear memory. AIMS Mathematics, 2022, 7(7): 12913-12934. doi: 10.3934/math.2022715
    [2] Sen Ming, Xiaodong Wang, Xiongmei Fan, Xiao Wu . Blow-up of solutions for coupled wave equations with damping terms and derivative nonlinearities. AIMS Mathematics, 2024, 9(10): 26854-26876. doi: 10.3934/math.20241307
    [3] 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. AIMS Mathematics, 2023, 8(9): 19971-19992. doi: 10.3934/math.20231018
    [4] Adel M. Al-Mahdi . The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type. AIMS Mathematics, 2023, 8(11): 27439-27459. doi: 10.3934/math.20231404
    [5] Sen Ming, Jiayi Du, Yaxian Ma . The Cauchy problem for coupled system of the generalized Camassa-Holm equations. AIMS Mathematics, 2022, 7(8): 14738-14755. doi: 10.3934/math.2022810
    [6] Ahmed Himadan . Well defined extinction time of solutions for a class of weak-viscoelastic parabolic equation with positive initial energy. AIMS Mathematics, 2021, 6(5): 4331-4344. doi: 10.3934/math.2021257
    [7] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Maher Nour, Mostafa Zahri . Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study. AIMS Mathematics, 2022, 7(8): 15370-15401. doi: 10.3934/math.2022842
    [8] Mohammad Kafini, Shadi Al-Omari . Local existence and lower bound of blow-up time to a Cauchy problem of a coupled nonlinear wave equations. AIMS Mathematics, 2021, 6(8): 9059-9074. doi: 10.3934/math.2021526
    [9] Khaled Zennir, Abderrahmane Beniani, Belhadji Bochra, Loay Alkhalifa . Destruction of solutions for class of wave p(x)bi-Laplace equation with nonlinear dissipation. AIMS Mathematics, 2023, 8(1): 285-294. doi: 10.3934/math.2023013
    [10] Salah Boulaaras, Abdelbaki Choucha, Bahri Cherif, Asma Alharbi, Mohamed Abdalla . Blow up of solutions for a system of two singular nonlocal viscoelastic equations with damping, general source terms and a wide class of relaxation functions. AIMS Mathematics, 2021, 6(5): 4664-4676. doi: 10.3934/math.2021274
  • In this paper, we consider a coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities. This system is supplemented with initial and mixed boundary conditions. First, we establish the existence and uniqueness results of a weak solution, under suitable assumptions on the variable exponents. Second, we show that the solutions with positive-initial energy blow-up in a finite time. Finally, we establish the global existence as well as the energy decay results of the solutions, using the stable-set and the multiplier methods, under appropriate conditions on the variable exponents and the initial data.



    The biharmonic equation, besides providing a benchmark problem for various analytical and numerical methods, arises in many practical applications. For example, the bending behavior of a thin elastic rectangular plate, as might be encountered in ship design and manufacture, or the equilibrium of an elastic rectangle, can be formulated in terms of the two-dimensional biharmonic equation, e.g., Timoshenko & Woinowsky-Krieger [1]. Also, Stokes flow of a viscous fluid in a rectangular cavity under the influence of the motion of the walls, can be described in terms of the solution of this equation, e.g., Pan and Acrivos (1967), Shankar [2], Srinivasan [3], Meleshko [4] or Shankar and Deshpande [5]. A more recent application of the biharmonic equation has been in the area of geometric and functional design, where it has been used as a mapping to produce efficient mathematical descriptions of surfaces in physical space, e.g., Sevant et al. [6] and Bloor and Wilson [7]. Interest in solutions of the biharmonic equation and their mathematical properties go back over 130 years, and comprehensive reviews of this work have been given by Meleshko [8,9]. In his review article, he concentrates upon the method of superposition in which the solution is described in terms of a sum of separable solutions of the biharmonic equation. In another work, Meleshko [4] obtained some results for Stokes flow in a rectangular cavity in which the solution is based upon the sum of terms consisting of the product of exponential and sinusoidal functions, where the coefficients in the series are determined from the requirement that the prescribed boundary conditions are satisfied, and Meleshko [10] described the work which has been done in trying to solve this problem, e.g., Meleshko and Gomilko [11]. Other physical phenomena like flows of electro-rheological fluids, fluids with temperature dependent viscocity, filtration processes through a porous media, image processing and thermorheological fluids give rise to mathematical models of hyperbolic, parabolic and biharmonic equations with variable exponents of nonlinearity. More details can also be found in references [12,13]. Recently, the hyperbolic equations with nonlinearities of variable exponents type had received a considerable amount of attention. We refer the reader to [14,15,16,17] and the references therein. Only few works concerning coupled systems of wave equations in the variable-exponents case have been found in the literature. For examples, Bouhoufani and Hamchi [18] obtained the global existence of a weak solution and established decay rates of the solutions, in a bounded domain, of a coupled system of nonlinear hyperbolic equations with variable-exponents. Messaoudi et al. [15] studied a system of wave equations with nonstandard nonlinearities and proved a theorem of existence and uniqueness of a weak solution, established a blow-up result for certain solutions with positive-initial energy and gave some numerical applications for their theoretical results. In [16], Messaoudi et al. considered the following system

    uttΔu+|ut|m(x)2ut+f1(u,v)=0in Ω×(0,T),vttΔv+|vt|r(x)2vt+f2(u,v)=0in Ω×(0,T), (1.1)

    with initial and Dirichlet-boundary conditions (here, f1 and f2 are the coupling terms introduced in (1.3). 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. In this work, we consider the following initial-boundary-value problem

    {utt+Δ2u+|ut|m(x)2ut=f1(u,v)in Ω×(0,T),vttΔv+|vt|r(x)2vt=f2(u,v)in Ω×(0,T),u=v=uη=0on Ω×(0,T),u(0)=u0 and ut(0)=u1in Ω,v(0)=v0 and vt(0)=v1in Ω, (1.2)

    where Ω is a smooth and bounded domain of Rn,(n=1,2,3), the exponents m and r are continuous functions on ¯Ω satisfying some conditions to be specified later, uη denotes the external normal derivatives of u on the boundary Ω and the coupling terms f1 and f2 are given as follows: for all x¯Ω and (u,v)R2,

    f1(x,u,v)=uF(x,u,v) and  f2(x,u,v)=vF(x,u,v), (1.3)

    with

    F(x,u,v)=a|u+v|p(x)+1+2b|uv|p(x)+12, (1.4)

    where a,b>0 are two positive constants and p is a given continuous function on ¯Ω satisfying the condition (H.2) (below).

    This section presents some material needed to prove the main result. Let q:Ω[1,) be a continuous function. We define the Lebesgue space with a variable exponent by

    Lq(.)(Ω)={f:ΩR measurable in Ω: ϱq(.)(λf)<+, for some λ>0},

    where

    ϱq(.)(f)=Ω|f(x)|q(x)dx.

    Lemma 2.1. [13,19] If 1<qq(x)q+<+ holds then, for any fLq(.)(Ω),

    min{fqq(.),fq+q(.)}ϱq(.)(f)max{fqq(.),fq+q(.)},

    where

    q=essinfxΩ q(x) and q+=esssupxΩ q(x).

    Lemma 2.2. (Embedding property [20]) Let q:¯Ω[1,) be a measurable function and k1 be an integer. Suppose that r is a log-Hölder continuous function on Ω, such that, for all xΩ, we have

    {kqq(x)q+<nr(x)nkr(x),if r+<nk,kqq+<,if r+nk.

    Then, the embedding Wk,r(.)0(Ω)Lq(.)(Ω) is continuous and compact.

    Throughout this paper, we denote by V the following space

    V={uH2(Ω): u=uη=0 on Ω}=H20(Ω).

    So, V is a separable Hilbert space endowed with the inner product and norm, respectively,

    (w,z)V=ΩΔwΔzdx  and  wV=Δw2,

    where Δwk=ΔwLk(Ω).

    We assume the following hypotheses:

    (H.1) The exponents m and r are continuous on ¯Ω such that

    2m(x), if  n=1,2,2m1m(x)m26, if  n=3 (2.1)

    and

    2r(x), if  n=1,2,2r1r(x)r26, if  n=3, (2.2)

    for all x¯Ω, where

    m1= infx¯Ω m(x), m2= supx¯Ω m(x), r1= infx¯Ω r(x) and r2= supx¯Ω r(x).

    (H.2) The variable exponent p is a given continuous function on ¯Ω such that

    3pp(x)p+<+, if  n=1,2,p(x)=3, if  n=3, (2.3)

    for all x¯Ω.

    In this section, we prove the local existence of the solutions of (1.2). For this purpose, we introduce the definition of a weak solution for system (1.2). We multiply the first equation in (1.2) by ΦC0(Ω) and the second equation by ΨC0(Ω), integrate each result over Ω, use Green's formula and the boundary conditions to obtain the following definition:

    Definition 3.1. Let (u0,v0)V×H10(Ω),(u1,v1)L2(Ω)×L2(Ω). Any pair of functions (u,v), such that

    {uL([0,T);V),vL([0,T);H10(Ω)),utL([0,T);L2(Ω))Lm(.)(Ω×(0,T)),vtL([0,T);L2(Ω))Lr(.)(Ω×(0,T)), (3.1)

    is called a weak solution of (1.2) on [0,T), if

    {ddtΩutΦdx+ΩΔuΔΦdx+Ω|ut|m(x)2utΦdx=Ωf1Φdx,ddtΩvtΨdx+ΩvΨdx+Ω|vt|r(x)2vtΨdx=Ωf2Ψdx,u(0)=u0,ut(0)=u1,v(0)=v0,vt(0)=v1,

    for a.e. t(0,T) and all test functions ΦV and ΨH10(Ω). Note that C0(Ω) is dense in V and in H10(Ω) as well. In addition, the spaces V, H10(Ω)Lm(.)(Ω)Lr(.)(Ω), under the conditions (H.1) and (H.2).

    In order to establish an existence result of a local weak solution for the system (1.2); we, first, consider the following auxiliary problem:

    {utt+Δ2u+ut|ut|m(x)2=f(x,t)in Ω×(0,T),vttΔv+vt|vt|r(x)2=g(x,t)in Ω×(0,T),u=v=uη=0on Ω×(0,T),u(0)=u0,ut(0)=u1,v(0)=v0,vt(0)=v1in Ω, (S)

    for given f,gL2(Ω×(0,T)) and T>0.

    We have the following theorem of existence and uniqueness for Problem (S).

    Theorem 3.1. Let n=1,2,3 and (u0,v0)V×H10(Ω),(u1,v1)H10(Ω)×L2(Ω). Assume that assumptions (H.1) and (H.2) hold. Then, the problem (S) admits a unique weak solution on [0,T).

    Proof. Let {ωj}j=1 be an orthogonal basis of V and define, for all k1, (uk,vk) a sequence in Vk=span{ω1,ω2,...,ωk}V, given by

    uk(x,t)=Σkj=1aj(t)ωj(x) and vk(t)=Σkj=1bj(t)ωj(x)

    for all xΩ and  t(0,T) and solves the following approximate problem:

    {Ωuktt(x,t)ωjdx+ΩΔuk(x,t)Δωjdx+Ω|ukt(x,t)|m(x)2ukt(x,t)ωjdx=Ωf(x,t)ωj,Ωvktt(x,t)ωjdx+Ωvk(x,t)ωjdx+Ω|vkt(x,t)|r(x)2vkt(x,t)ωjdx=Ωg(x,t)ωj, (Sk)

    for all j=1,2,...,k, with

    uk(0)=uk0=Σki=1u0,ωiωi, ukt(0)=uk1=Σki=1u1,ωiωivk(0)=vk0=Σki=1v0,ωiωi, vkt(0)=vk1=Σki=1v1,ωiωi, (3.2)

    such that

    uk0u0 and  vk0v0  in H10(Ω),uk1u1 and  vk1v1  in L2(Ω). (3.3)

    For any k1, problem (Sk) generates a system of k nonlinear ordinary differential equations. The ODE standard existence theory assures the existence of a unique local solution (uk,vk) for (Sk) on [0,Tk), with 0<TkT. Next, we have to show that Tk=T,k1. Multiplying (Sk)1 and (Sk)2 by aj(t) and bj(t), respectively, and then summing each result over j=1,...,k, we obtain, for all 0<tTk,

    12ddt[Ω(|ukt(x,t)|2+(Δuk)2(x,t))dx]+Ω|ukt(x,t)|m(x)dx=Ωf(x,t)ukt(x,t)dx (3.4)

    and

    12ddt[Ω(|vkt(x,t)|2+|vk|2(x,t))dx]+Ω|vkt(x,t)|r(x)dx=Ωg(x,t)vkt(x,t)dx. (3.5)

    The addition of (3.4) and (3.5), and then the integration of the result, over (0,t), lead to

    12[ukt(t)22+uk(t)2V+vkt(t)22+vk(t)22]+t0Ω(|ukt(x,s)|m(x)+|vkt(x,s)|r(x))dxds=12[uk122+uk02V+vk122+vk022]+t0Ω[f(x,s)ukt(x,s)+g(x,s)vkt(x,s)]dxds. (3.6)

    Using Young's inequality and the convergence (3.3), then Eq (3.6) becomes, for some C>0,

    12[ukt(t)22+vkt(t)22+uk(t)2V+vk(t)22]+Tk0Ω(|ukt(x,s)|m(x)+|vkt(x,s)|r(x))dxdsC+εTk0(ukt(s)22+vkt(s)22)ds+CεT0Ω(|f(x,s)|2+|g(x,s)|2)dxds.

    Using the fact that f,gL2(Ω×(0,T)) and choosing ε=14T, we infer

    12sup(0,Tk)[ukt22+vkt22+uk2V+vk22]+Tk0Ω(|ukt(x,s)|m(x)+|vkt(x,s)|r(x))dxdsCε+Tεsup(0,Tk)(ukt22+vkt22)CT, (3.7)

    where CT>0 is a constant depending on T only. Consequently, the solution (uk,vk) can be extended to (0,T), for any k1. In addition, we have

    {(uk) is bounded in L((0,T),V),(vk) is bounded in L((0,T),H10(Ω)),(ukt) is bounded in L((0,T),L2(Ω))Lm(.)(Ω×(0,T)),(vkt) is bounded in L((0,T),L2(Ω))Lr(.)(Ω×(0,T)).

    Therefore, we can extract two subsequences, denoted by (ul) and (vl), respectively, such that, when l, we have

    {ulu  weakly * in L((0,T),V),vlv  weakly * in L((0,T),H10(Ω)),ultut weakly * in L((0,T),L2(Ω)) and weakly in Lm(.)(Ω×(0,T)),vltvt weakly * in L((0,T),L2(Ω)) and weakly in Lr(.)(Ω×(0,T)).

    Under the assumptions (H.1) and (H.2) and using similar ideas and arguments as in [[15], Theorem 3.2, p.6], one can see that

    ultm(.)2ult utm(.)2ut weakly in  Lm(.)m(.)1(Ω×(0,T)),
    vltr(.)2vlt vtr(.)2vt weakly in Lr(.)r(.)1(Ω×(0,T))

    and establish that (u,v) satisfies the two differential equations in (S), on Ω×(0,T).

    To handle the initial conditions, we follow the same procedures as in [15], and we easily conclude that (u,v) satisfies the initial conditions. For the uniqueness, Assume that (S) has two weak solutions (u1,v1) and (u2,v2), in the sense of Definition 3.1. Let (Φ,Ψ)=(u1tu2t,v1tv2t), then (u,v)=(u1u2,v1v2) satisfies the following identities, for all t(0,T),

    ddt[Ω(|ut|2+(Δu)2)dx]+2Ω(|u1t|m(x)2u1t|u2t|m(x)2u2t)(u1tu2t)dx=0 (3.8)

    and

    ddt[Ω(|vt|2+|v|2)dx]+2Ω(|v1t|r(x)2v1t|v2t|r(x)2v2t)(v1tv2t)dx=0. (3.9)

    Integrating (3.8) and (3.9) over (0,t), with tT, we obtain

    ut22+u2V+2t0Ω(|u1t|m(x)2u1t|u2t|m(x)2u2t)(u1tu2t)dxdτ=0 (3.10)

    and

    vt22+v22+2t0Ω(|v1t|r(x)2v1t|v2t|r(x)2v2t)(v1tv2t)dxdτ=0. (3.11)

    But we have, for all xΩ,Y,ZR and q(x)2,

    (|Y|q(x)2Y|Z|Zq(x)2)(YZ)0, (3.12)

    then, estimates (3.10) and (3.11) yield

    ut2+u2V=vt2+v22=0.

    Thus, ut(.,t)=vt(.,t)=0 and u(.,t)=v(.,t)=0, for all t(0,T). Thanks to the boundary conditions, we conclude u=v=0 on Ω×(0,T), which proves the uniqueness of the solution. Therefore, (u,v) is the unique local solution of (S), in the sense of Definition 3.1, having the regularity (3.1).

    Lemma 3.1. Let yL((0,T),V) and zL((0,T),H10(Ω)). Then

    f1(y,z),f2(y,z)L2(Ω×(0,T)). (3.13)

    Proof. From (1.3) and (1.4), we have, for all (u,v)R2,

    f1(u,v)=(p(x)+1)[a|u+v|p(x)1(u+v)+bu|u|p(x)32|v|p(x)+12] (3.14)

    and

    f2(u,v)=(p(x)+1)[a|u+v|p(x)1(u+v)+bv|v|p(x)32|u|p(x)+12]. (3.15)

    Let yL((0,T),V) and zL((0,T),H10(Ω)). Applying Young's inequality and the Sobolev embedding, we obtain, for all t(0,T) and some C1,C2>0, the following results:

    Ω|f1(y,z)|2dx2[a2Ω|y+z|2p(x)dx+b2Ω|y|p(x)1|z|p(x)+1dx]C0[Ω|y+z|2p+dx+Ω|y+z|2pdx+Ω|y|3(p(x)1)dx+Ω|z|32(p(x)+1)dx], (3.16)

    where C0=2max{a2,3b2}>0. By the embeddings, we have for n=1,2,

    1<32(p+1)32(p++1)2p+3(p+1)<,

    since 3pp(x)p+<. Therefore, estimate (3.16) leads to

    Ω|f1(y,z)|2dxC1[(y+z)2p+2+(y+z)2p2+Δy3(p+1)2+Δy3(p1)2]+C1[z32(p++1)2+z32(p+1)2]<+, (3.17)

    where C1=C0Ce.

    ● For n=3, we use the embedding H10(Ω) in L6(Ω) to obtain (3.17), since p3 on ¯Ω.

    So, under the assumption (H.2), we have

    Ω|f1(y,z)|2dx<,

    and similarly

    Ω|f2(y,z)|2dx<,

    for all t(0,T). Which completes the proof.

    Corollary 3.1. There exists a unique (u,v) solution of the problem:

    {utt+Δ2u+|ut|m(x)2ut=f1(y,z),in Ω×(0,T),vttΔv+|vt|r(x)2vt=f2(y,z),in Ω×(0,T),u=v=uη=0on Ω×(0,T),u(0)=u0 and ut(0)=u1in Ω,v(0)=v0 and vt(0)=v1,in Ω, (R)

    in the sense of Definition 3.1 and having the regularity 3.1.

    Proof. A combination of Theorem 3.1 and Lemma 3.1 implies this corollary.

    Now, consider the following Banach spaces

    AT={wL((0,T),V)/wtL((0,T),L2(Ω))},

    equipped with the norm:

    w2AT=sup(0,T)w2V+sup(0,T)wt22

    and

    BT={wL((0,T),H10(Ω))/wtL((0,T),L2(Ω))},

    equipped with the norm:

    w2BT=sup(0,T)w22+sup(0,T)wt22

    and define a map F:AT×BT:⟶AT×BT by F(y,z)=(u,v).

    Lemma 3.2. F maps D(0,d) into itself where

    D(0,d)={(w,w)AT×BTsuch that||(w,w)||AT×BTd}.

    Proof. Let (y,z) be in D(0,d) and (u,v) be the corresponding solution of problem (R) (i.e., F(y,z)=(u,v)). Taking (Φ,Ψ)=(ut,vt) in Definition 3.1 and integrating each identity over (0,t), we obtain, for all tT,

    12[ut22u122+Δu22Δu022]+t0Ω|ut(x,s)|m(x)dxds=t0Ωutf1(y,z)dxds (3.18)

    and

    12[vt22v122+v22v022]+t0Ω|vt(x,s)|r(x)dxds=t0Ωvtf2(y,z)dxds. (3.19)

    The addition of (3.18) and (3.19) lead to

    12[ut22+vt22+Δu22+v22]12[u122+v122+Δu022+v022]+t0(|Ωutf1(y,z)dx|+|Ωvtf2(y,z)dx|)ds.

    for all t(0,T). Therefore,

    sup0tT(ut22+vt22+u2V+v22)γ+2sup0tTt0(|Ωutf1(y,z)dx|+|Ωvtf2(y,z)dx|)dτ, (3.20)

    where γ=u122+v122+u02V+v022. Under the assumption (2.3) and applying Young's inequality and the Sobolev embedding (Lemma 2.2), we obtain for all t(0,T),

    |Ωutf1(y,z)dx|(p++1)[aΩ|ut||y+z|p(x)dx+bΩ|ut|.|y|p(x)12|z|p(x)+12dx](p++1)[ε(a+b)2Ω|ut|2dx+2aεΩ|y+z|2p(x)dx+2bεΩ|y|p(x)1|z|p(x)+1dx]c1[ε2ut22+Cε(Ω|y+z|2p++Ω|y+z|2p+Ω|y|3(p(x)1)+Ω|z|32(p(x)+1))]c2[εut22+Δy2p2+z2p2+Δy2p+2+z2p+2]+c2[Δy3(p1)2+Δy3(p+1)2+z32(p+1)2+z32(p++1)2], (3.21)

    where ε,c1,c2 are positive constants. Likewise, we get

    |Ωvtf2(y,z)dx|(p++1)[aΩ|vt||y+z|p(x)dx+bΩ|vt|.|z|p(x)12|y|p(x)+12dx]c2[εvt22+Δy2p2+z2p2+Δy2p+2+z2p+2]+c2[z3(p1)2+z3(p+1)2+Δy32(p+1)2+Δy32(p++1)2]. (3.22)

    Combining (3.21) and (3.22), yields

    sup(0,T)t0(|Ωutf1(y,z)dx|+|Ωvtf2(y,z)dx|)dsεTc2(u,v)2AT×BT+2Tc2((y,z)2pAT×BT+(y,z)2p+AT×BT)+Tc2((y,z)3(p1)AT×BT+(y,z)3(p+1)AT×BT+(y,z)32(p+1)AT×BT+(y,z)32(p++1)AT×BT). (3.23)

    By substituting (3.23) into (3.20), we obtain, for some c3>0,

    12(u,v)2AT×BTγ0+εTc3(u,v)2AT×BT+2Tc3((y,z)2pAT×BT+(y,z)2p+AT×BT)+Tc3((y,z)3(p1)AT×BT+(y,z)3(p+1)AT×BT+(y,z)32(p+1)AT×BT+(y,z)32(p++1)AT×BT). (3.24)

    Choosing ε such that εTc3=14 and recalling that (y,z)AT×BTd, for some d>1 (large enough), inequality (3.24) implies

    (u,v)2AT×BT4γ0+8Tc3((y,z)2pAT×BT+(y,z)2p+AT×BT)+4Tc3((y,z)3(p1)AT×BT+(y,z)3(p+1)AT×BT+(y,z)32(p+1)AT×BT+(y,z)32(p++1)AT×BT)4γ0+Tc4d3(p+1), c4>0,

    So, if we take d such that d2>>4γ0 and TT0=d24γ0c4d3(p+1), we find

    4γ0+Tc4d3(p+1)d2.

    Therefore,

    (u,v)2AT×BTd2.

    Thus, F maps D(0,d) to D(0,d).

    Lemma 3.3. F:D(0,d)D(0,d) is a contraction.

    Proof. Let (y1,z1) and (y2,z2) be in D(0,d) and set (u1,v1)=F(y1,z1) and (u2,v2)=F(y2,z2). Clearly, (U,V)=(u1u2,v1v2) is a weak solution of the following system

    {Utt+Δ2U+|u1t|m(x)2u1t|u2t|m(x)2u2t=f1(y1,z1)f1(y2,z2)in Ω×(0,T),VttΔV+|v1t|r(x)2v1t|v2t|r(x)2v2t=f2(y1,z1)f2(y2,z2)in Ω×(0,T),U=V=0on Ω×(0,T),(U(0),V(0))=(Ut(0),Vt(0))=(0,0)in Ω,

    in the sense of Definition 3.1. So, taking (Φ,Ψ)=(Ut,Vt), in this definition, using Green's formula together with the boundary conditions and then, integrating each result over (0,t), we obtain, for a.e. tT,

    12(Ut22+ΔU22)+t0Ω(u1t|u1t|m(x)2u2t|u2t|m(x)2)Utdxdst0Ω|f1(y1,z1)f1(y2,z2)||Ut|dxds

    and

    12(Vt22+V22)+t0Ω(v1t|v1t|r(x)2v2t|v2t|r(x)2)Vtdxdst0Ω|f2(y1,z1)f2(y2,z2)||Vt|dxds.

    Under the condition (H.2), using Hölder's inequality and inequality (3.12), these two estimates give, for n=1,2,3,

    Ut22+U2V4t0Ut2f1(y1,z1)f1(y2,z2)2ds (3.25)

    and

    Vt22+V224t0Vt2f2(y1,z1)f2(y2,z2)2ds. (3.26)

    The addition of (3.25) and (3.26) imply

    Ut22+Vt22+U2V+V224t0Ut2f1(y1,z1)f1(y2,z2)2ds+4t0Vt2f2(y1,z1)f2(y2,z2)2ds, (3.27)

    for all t(0,T). Now, we estimate the terms:

    f1(y1,z1)f1(y2,z2)2 and f2(y1,z1)f2(y2,z2)2.

    Using appropriate algebraic inequalities (see [21]), we obtain for two constants C1,C2>0 and for all xΩ and t(0,T),

    Ω|f1(y1,z1)f1(y2,z2)|2dxI1+I2+I3+I4, (3.28)

    where

    I1=C1Ω|y1y2|2(|y1|2(p(x)1)+|z1|2(p(x)1))dx+C1Ω|y1y2|2(|y2|2(p(x)1)+|z2|2(p(x)1))dx,I2=C1Ω|z1z2|2(|y1|2(p(x)1)+|z1|2(p(x)1))dx+C1Ω|z1z2|2(|y1|2(p(x)1)+|z2|2(p(x)1))dx,I3=C2Ω|z1z2|2|y1|p(x)1(|z1|p(x)1+|z2|p(x)1)dx,I4=C2Ω|y1y2|2|z2|p(x)+1(|y1|p(x)3+|y2|p(x)3)dx.

    By using Hölder's and Young's inequalities and the Sobolev embedding (Lemma 2.2), we get the following estimate for a typical term in I1 and I2,

    Ω|y1y2|2|y1|2(p(x)1)dx2(Ω|y1y2|6dx)13(Ω|y1|3(p(x)1))23C||y1y2||26[(Ω|y1|3(p+1)dx)23+(Ω|y1|3(p1)dx)23]C||Δ(y1y2)||22(||y1||2(p+1)3(p+1)+||y1||2(p1)3(p1))C||ΔY||22(||Δy1||2(p+1)2+||Δy1||2(p1)2)C||ΔY||22(||(y1,z1)||2(p+1)AT×BT+||(y1,z1)||2(p1)AT×BT), (3.29)

    since

    13(p1)3(p+1)<, when n=1,2.

    13(p1)=3(p+1)=6=2nn2, when n=3.

    Likewise, we obtain

    Ω|z1z2|2|y2|2(p(x)1)dxC||Z||22(||(y2,z2)||2(p+1)AT×BT+||(y2,z2)||2(p1)AT×BT). (3.30)

    Since (y1,z1),(y2,z2)D(0,d) and d>1, estimates (3.29) and (3.30) lead to

    I1C||ΔY||22d2(p+1) and  I2C||Z||22d2(p+1).

    Hence,

    I1+I2Cd2(p+1)(||ΔY||22+||Z||22). (3.31)

    Similarly, a typical term in I3 can be handled as follows

    Ω|z1z2|2|y1|p(x)1|z1|p(x)1dx2(Ω|z1z2|6dx)13(Ω|y1|32(p(x)1)|z1|32(p(x)1))23C||z1z2||26[(Ω|y1|32(p(x)1)dx)23+(Ω|z1|32(p(x)1)dx)23]C||(z1z2)||22(||y1||(p+1)32(p+1)+||y1||(p1)32(p1)+||z1||(p+1)32(p+1)+||z1||(p1)32(p1))C||(z1z2)||22(||Δy1||(p+1)2+||Δy1||(p1)2+||z1||(p+1)2+||z1||(p1)2)2C||Z||22(||(y1,z1)||(p+1)AT×BT+||(y1,z1)||(p1)AT×BT),

    since

    132(p1)32(p+1)<, when n=1,2.

    132(p1)=32(p+1)=6=2nn2, when n=3.

    Therefore,

    I3Cdp+1||Z||22, (3.32)

    since (y1,z1),(y2,z2)D(0,d). Using the same arguments, a typical term in I4, can be estimated as follows:

    Case 1: If n=1,2, we have 3pp+<. So,

    Ω|y1y2|2|z2|p(x)+1|y1|p(x)3dx2(Ω|y1y2|3dx)23(Ω|z2|3(p(x)+1)|y1|3(p(x)3))13C||y1y2||23[(Ω|z2|6(p(x)+1)dx)13+(Ω|y1|6(p(x)3)dx)13]C||ΔY||22(||z2||2(p++1)2+||z2||2(p+1)2+||Δy1||2(p+3)2+||Δy1||2(p3)2)4C||ΔY||22d2(p++1),

    since (y1,z1),(y2,z2)D(0,d) and d>1.

    Case 2: If n=3, then p3 on ¯Ω. Hence,

    Ω|y1y2|2|z2|p(x)+1|y1|p(x)3dx=Ω|y1y2|2|z2|4dxC(Ω|y1y2|6dx)13(Ω|z2|6dx)23C||y1y2||26.||z2||46C||ΔY||22.||(y2,z2)||4AT×BT.

    So, for all t(0,T), we deduce that

    I4C||ΔY||22d2(p++1). (3.33)

    Finally, by substituting (3.31)–(3.33) in (3.28), the following can be obtained

    Ω|f1(y1,z1)f1(y2,z2)|2dxCd2(p++1)(||ΔY||22+||Z||22), (3.34)

    for all t(0,T). Similarly, we get

    Ω|f2(y1,z1)f2(y2,z2)|2dxCd2(p++1)(||ΔY||22+||Z||22). (3.35)

    Now, we use (3.34) and (3.35) in (3.27) to obtain

    (u,v)2AT×BTCd2(p++1)sup(0,T)t0(ΔY(s)22+Z(s)22)dsCd2(p++1)T(Y,Z)2AT×BT.

    Hence, if we take T small enough, we get for, 0<γ<1,

    (u,v)2AT×BTγ(Y,Z)2AT×BT.

    Thus,

    K(y1,z1)K(y2,z2)2AT×BTγ(y1,z1)(y2,z2)2AT×BT.

    This proves that F:D(0,d)D(0,d) is a contraction.

    Theorem 3.2. Let n=1,2,3. Under the assumptions (H.1) and (H.2) and for any (u0,v0)V×H10(Ω),(u1,v1)H10(Ω)×L2(Ω) the problem (1.2) admits a unique weak solution (u,v), in the sense of Definition 3.1, having the regularity (3.1), for T small enough.

    Proof. The above Lemmas and the Banach-fixed-point theorem guarantee the existence of a unique (u,v)D(0,d), such that F(u,v)=(u,v), which is a local weak solution of (1.2).

    Remark 3.1. From the definitions (1.3) and (1.4), one can easily see that, for all (u,v)R2,

    u f1(x,u,v)+vf2(x,u,v)=(p(x)+1)F(x,u,v). (3.36)

    We, also, have the following results.

    Lemma 3.1. [22] There exist C1,C2>0 such that, for all  x¯Ω and (u,v)R2, we have

    C1(|u|p(x)+1+|v|p(x)+1)F(x,u,v)C2(|u|p(x)+1+|v|p(x)+1). (3.37)

    Corollary 3.2. For all  x¯Ω and (u,v)R2, we have

    C1(ζ(u)+ζ(v))ΩF(x,u,v)dxC2(ζ(u)+ζ(v)), (3.38)

    where

    ζ(u)=Ω|u |p(x)+1dx and ζ(v)=Ω|v |p(x)+1dx.

    Now, we introduce the energy functional associated with our problem

    E(t)=12(ut22+vt22+Δu22+v22)ΩF(x,u,v)dx, (3.39)

    for all t[0,T). A direct computation implies, for a.e. t(0,T),

    E(t)=Ω|ut|m(x)dxΩ|vt|r(x)dx0. (3.40)

    In this section, our goal is to prove that any solution of Problem (1.2) blows-up in some finite time T, if

    max{m+,r+}<p and 0<E(0)<E1, (4.1)

    where

    E1=(121p+1)γ21,    γ1=(d(p+1))11p,  (4.2)
    d=(2(p+1)a+2b)cp+1

    and c is a positive constant, which comes from the Sobolev embedding.

    Remark 4.1. The following well-known inequalities are needed in the proof of the lemmas.

    (1) For  A,B0 and d1, we have

    (A+B)d2d1(Ad+Bd). (4.3)

    (2) For z0, 0<δ1 and a>0, we have

    zδz+1(1+1a)(z+a). (4.4)

    (3) For X, Y0,  δ>0 and 1λ+1β=1, Young's inequality gives

    XYδλλXλ+δββYβ. (4.5)

    (4) The embedding Lemma 2.2, Hölder's and Young's inequalities and (4.3) imply that

    u+vp(.)+12c[(Δu22+v22)]1/2 (4.6)

    and

    uvp(.)+12c2(Δu22+v22). (4.7)

    Lemma 4.1. For any solution (u,v) of the system (1.2), with initial energy

    E(0)<E1 (4.8)

    and

    γ1<(Δu022+v022)1/212c (4.9)

    there exists γ2>γ1 such that

    γ2(Δu22+v22)1/2, t[0,T). (4.10)

    Proof. Let γ=(Δu22+v22)1/2, then using (3.39), we have

    E(t)12γ2ΩF(x,u,v)dx. (4.11)

    The use of Lemma 2.1, (4.6) and (4.7) leads to

    ΩF(x,u,v)dx=aΩ|u+v|p(x)+1dx+2bΩ|uv|p(x)+12dxamax{u+vp+1p(.)+1,u+vp++1p(.)+1}+2bmax{uvp+12p(.)+12,uvp++12p(.)+12}amax{(2cγ)p+1,(2cγ)p++1 }+2bmax{(cγ)p+1,(cγ)p++1}. (4.12)

    Combining (4.11) and (4.12), we obtain

    E(t)12γ2amax{(2cγ)p+1,(2cγ)p++1}2bmax{(cγ)p+1,(cγ)p++1}. (4.13)

    For γ in [0,12c], one can easily check that

    c2γ22c2γ21.

    Consequently, we have

    (2cγ)p+1(2cγ)p++1 and  (cγ)p+1(2cγ)p++1.

    Thus, (4.13) reduces to

    E(t)12γ2(2(p+1)a+2b)cp+1γp+1.

    If we set

    h(γ)=12γ2kγp+1,wherek=(2(p+1)a+2b)cp+1,

    then

    E(t)h(γ), for all γ[0,12c]. (4.14)

    It is clear that h is strictly increasing on [0,γ1) and strictly decreasing on [γ1,+). Since E(0)<E1 and E1=h(γ1), then, we can find γ2>γ1 such that h(γ2)=E(0). But,

    α0=(Δu022+v022)1/2,

    therefore, by (4.14), we get

    h(γ2)=E(0)h(γ0).

    This implies that γ0γ2. Hence, γ2(γ1,12c]. To prove (4.10), we assume that there is a t0[0,T) such that

    (Δu(.,t0)22+v(.,t0)22)1/2<γ2.

    Since  (Δu22+v22)1/2 is continuous and γ2>γ1, t0 can be selected so that

    [Δu(.,t0)22+v(.,t0)22]1/2>γ1.

    Using (4.14) and the fact that h  is decreasing on [γ1,12c], we obtain

    E(t0)h((Δu(.,t0)22+v(.,t0)22)1/2)>h(γ2)=E(0),

    which contradicts the fact that E(t)E(0), for all t[0.T). Thus, (4.10) is established.

    Lemma 4.2. Let H(t)=E1E(t), for all t[0, T). Then, we have

    0<H(0)H(t)ΩF(x,u,v)dx, for all t[0, T)   (4.15)

    and

    ΩF(x,u,v)dxdγp+12. (4.16)

    Proof. Using (3.40), (4.8) and (4.11), we have

    0<E1E(0) =H(0)H(t)E112γ2+ΩF(x,u,v)dx. (4.17)

    From the fact that h(γ1)=12γ21dγp+11=E1, we have

    E112γ21=dγp+11,

    then since  γγ2>γ1, we obtain

    H(t)dγp+11+ΩF(x,u,v)dxΩF(x,u,v)dx.

    Thus, (4.15) is established. To establish (4.16), we use (4.15) to obtain

    E(0)12γ2ΩF(x,u,v)dx,

    which implies,

    ΩF(x,u,v)dx12γ2E(0).

    But E(0)=h(γ2) and γγ2, so

    ΩF(x,u,v)dx12γ22h(γ2)=dγp+12.

    Lemma 4.3. There exist C3,C4,C5>0 such that any solution of (1.2) satisfies

    up+1p+1+vp+1p+1C3(ζ(u)+ζ(v)), (4.18)
    Ω|u|m(x)dxC4[(ζ(u)+ζ(v))m+p+1+(ζ(u)+ζ(v))mp+1] (4.19)

    and

    Ω|v|r(x)dxC5[(ζ(u)+ζ(v))r+p+1+(ζ(u)+ζ(v))rp+1], (4.20)

    where ζ(u) and ζ(v) are defined in Corollary 3.2.

    Proof. We define the following partition of Ω

     Ω+={xΩ / |u(x,t)|1} and Ω={xΩ / |u(x,t)|<1}.

    The properties of p(.) and Hölder's inequality imply that, for some c1>0,

    ζ(u)=Ω+|u|p(x)+1dx +Ω|u|p(x)+1dxΩ+|u|p+1dx +Ω|u|p++1dxΩ+|u|p+1dx +c1(Ω|u|p+1dx)p++1p+1.

    Hence,

    ζ(u)Ω+|u|p+1dx and (ζ(u)c1)p+1p++1Ω|u|p+1dx. (4.21)

    Use (4.21) to obtain, for some c2>0.

    up+1p+1ζ(u)+c2(ζ(u))p+1p++1ζ(u)+ζ(v)+c2(ζ(u)+ζ(v))p+1p++1=(ζ(u)+ζ(v))[1+c2(ζ(u)+ζ(v))p p+p++1].

    Recalling (3.38) and (4.15), we deduce that

    0<H(0)H(t)C2(ζ(u)+ζ(v)). (4.22)

    Therefore,

    up+1p+1(ζ(u)+ζ(v))[1+c2(H(0)/C2)p p+p++1]c(ζ(u)+ζ(v)).

    Similarly, we arrive at

    vp+1p+1c(ζ(u)+ζ(v)).

    Therefore, (4.18) is established. To establish (4.19), we recall that p max {m+,r+}, to conclude that

    Ω|u|m(x)dxΩ+|u|m+dx +Ω|u|mdxc(Ω+|u|p+1dx)m+p+1 +c(Ω|u|p+1dx)mp+1c(um+p+1+ump+1), c>0.

    Using similar calculations as above, we obtain (4.19) and (4.20).

    Lemma 4.4. Let G(t)=H1σ(t)+εΩ(uut+vvt)dx,t>0, where ε>0 to be fixed later. Then, there exists ρ>0, such that

    G(t)ερ(H(t)+ut22+vt22+ζ(u)+ζ(v)) (4.23)

    and hence,

    G(t)G(0)>0,  for t>0,

    where

    0<σmin{pm++1(p+1)(m+1) pr++1(p+1)(r+1) p12(p+1)}. (4.24)

    Proof. Differentiate G and use (1.2) to have

    G(t)=(1σ)Hσ(t)H(t)+ε(ut22+vt22)+εΩ(uf1(x,u,v)+vf2(x,u,v))dxε(Δu22+v22)εΩ(|ut|m(x)2utu+|vt|r(x)2vtv)dx. (4.25)

    By the definition of H and E, we get

    Δu22+v22=2ΩF(x,u,v)dxut22vt22+2E1 2H(t). (4.26)

    Combining (3.36), (4.25) and (4.26), we obtain

    G(t)(1σ)Hσ(t)H(t)+2ε(ut22+vt22)+2εH(t)2εE1+ε(p1)ΩF(x,u,v)dxεΩ(|u||ut|m(x)1+|v||vt|r(x)1)dx. (4.27)

    A combination of (4.16) and (4.27) leads to

    G(t)(1σ)Hσ(t)H(t)+2ε(ut22+vt22) (4.28)
    +ε(p12(dγp+12)1E1)ΩF(x,u,v)dx+2εH(t)εΩ(|u||ut|m(x)1+|v||vt|r(x)1)dx, (4.29)

    where p12(dαp+12)1E1>0, since γ2>γ1.

    Now, the last two terms of (4.29) can be estimated by applying (4.5) with X=|u|, Y=|ut|m(x)1, λ=m(x), β=m(x)m(x)1, as follows:

    Ω|u||ut|m(x)1dxΩδm(x)m(x)|u|m(x)dx +Ωm(x)1m(x)δm(x)/(m(x)1)|ut|m(x)dx. (4.30)

    Let  ˜k  be a positive constant to be selected later and take δ=[˜kHσ(t)]1m(x)m(x) to obtain

    Ω|u||ut|m(x)1dx˜k1mmΩ[H(t)]σ(m(x)1)|u|m(x)dx +m+1m˜kHσ(t)Ω|ut|m(x)dx. (4.31)

    The properties of m(x) and H(t) give

    Ω[H(t)]σ(m(x)1)|u|m(x)dx=Ω[H(t)H(0)]σ(m(x)1)[H(0)]σ(m(x)1)|u|m(x)dx~c2[H(t)]σ(m+1)Ω[H(0)]σ(m(x)1)|u|m(x)dx,

    where ~c2=1/[H(0)]σ(m+1). But [H(0)]σ(m(x)1)c3, for all xΩ, where c3>0. So, for some c4>0, we get

    Ω[H(t)]σ(m(x)1)|u|m(x)dxc4[H(t)]σ(m+1)Ω|u|m(x)dx. (4.32)

    Combining (4.31) and (4.32) to obtain

    Ω|u||ut|m(x)1dxc4˜k1mm[H(t)]σ(m+1)Ω|u|m(x)dx +m+1m˜kHσ(t)Ω|ut|m(x)dx. (4.33)

    Applying Similar calculations, we arrive at

    Ω|vt|r(x)1vdxc5˜k1rr[H(t)]σ(r+1)Ω|v|r(x)dx +r+1r˜kHσ(t)Ω|vt|r(x)dx. (4.34)

    Adding (4.33) and (4.34), we have

    Ω(|u||ut|m(x)1+|v||vt|r(x)1)dxc4˜k1mm[H(t)]σ(m+1)Ω|u|m(x)dx+c5˜k1rr[H(t)]σ(r+1)Ω|v|r(x)dx+˜αHσ(t)(Ω|ut|m(x)dx+Ω|vt|r(x)dx), (4.35)

    where ˜α=max{m+1m˜k,r+1r˜k}. Using (3.43), we have

    H(t)=Ω|ut|m(x)dx+Ω|vt|r(x)dx.

    Hence, (4.35) becomes

    Ω(|u||ut|m(x)1+|v||vt|r(x)1)dxc4˜k1mm[H(t)]σ(m+1)Ω|u|m(x)dx+c5˜k1rr[H(t)]σ(r+1)Ω|v|r(x)dx+˜αHσ(t)H(t). (4.36)

    Using (3.38) and (4.15), we have

    [H(t)]σ(m+1)c(ζ(u)+ζ(v))σ(m+1).

    Using the last inequality and (4.19), it can be concluded that

    [H(t)]σ(m+1)Ω|u|m(x)dxc6(ζ(u)+ζ(v))σ(m+1)+m+p+1+c6(ζ(u)+ζ(v))σ(m+1)+mp+1, (4.37)

    Applying (4.4) with z=ζ(u)+ζ(v), a=H(0), δ=σ(m+1)+m+p+1  and then with δ=σ(m+1)+mp+1, respectively, we get

    (ζ(u)+ζ(v))σ(m+1)+m+p+1[1+1H(0)](ζ(u)+ζ(v)+H(0))α(ζ(u)+ζ(v)+H(t)) (4.38)

    and

     (ζ(u)+ζ(v))σ(m+1)+mp+1α(ζ(u)+ζ(v)+H(t)) (4.39)

    where α=1+1H(0).

    A combination of (4.37)–(4.39) implies that, for some c7>0,

    [H(t)]σ(m+1)Ω|u|m(x)dxc7(ζ(u)+ζ(v)+H(t)). (4.40)

    Similar calculations give, for some c8>0,

    [H(t)]σ(r+1)Ω|v|r(x)dxc8(ζ(u)+ζ(v)+H(t)). (4.41)

    Using (4.35), (4.40) and (4.41), we obtain, for c9,c10>0,

    Ω(|u||ut|m(x)1+|v||vt|r(x)1)dx˜k1mmc9(ζ(u)+ζ(v)+H(t))+˜k1rrc10(ζ(u)+ζ(v)+H(t))+r+1r˜kHσ(t)H(t). (4.42)

    Inserting (4.42) into (4.29), we have

    G(t)(1σε˜R)Hσ(t)H(t)+2ε(ut22+vt22)+ε(2˜k1mmc9˜k1rrc10)H(t)+ε(c11˜k1mmc9˜k1rrc10)(ζ(u)+ζ(v)).

    where c11>0 and ˜R=˜k(m+1m+r+1r). Now, we select ˜k large enough so that

    G(t)(1σε˜R)Hσ(t)H(t)+εc12(ut22+vt22+H(t)+ζ(u)+ζ(v)),

    where c12>0. Once ˜k is fixed, we select ε small enough so that

    1σε˜R0 and G(0)=H1σ(0)+εΩ(u0u1+v0v1)dx>0.

    Using the fact that H is a non-decreasing function, therefore (4.23) is established.

    Theorem 4.1. Under the assumptions (4.1) and (4.9), any solution of the system (1.2) blows-up in a finite time.

    Proof. Using (4.3) and the definition of G, we have

    G1/(1σ)(t)(H1σ(t)+εΩ|uut+vvt|dx)1/(1σ)2σ/(1σ)(H(t)+(εΩ(|uut|+|vvt|)dx)1/(1σ))c13(H(t)+(Ω(|u||ut|+|v||vt|)dx)1/(1σ)), (4.43)

    where c13=2σ/(1σ)max{1,ε1/(1σ)}.

    The embedding Lemma 2.2, Lemma 4.2, Hölder's and Young's inequalities give

    (Ω(|u||ut|+|v||vt|)dx)1/(1σ)2σ/(1σ)(Ω|u||ut|dx)1/(1σ)+2σ/(1σ)  (Ω|v||vt|dx)1/(1σ)2σ/(1σ)(u1/(1σ)2ut1/(1σ)2+v1/(1σ)2vt1/(1σ)2)c14(u1/(1σ)p+1ut1/(1σ)2+v1/(1σ)p+1vt1/(1σ)2)c15(u2/(12σ)p+1+ut22+v2/(12σ)p+1+vt22)c15((ζ(u)+ζ(v))τ+ut22+vt22), (4.44)

    where τ=2/(p+1)(12σ).

    Using (4.15), (3.38) and since τ1, we get, for some c18>0,

    (Ω(|u||ut|+|v||vt|)dx)1/(1σ)c16(ζ(u)+ζ(v)+ut22+vt22+H(t)).

    Inserting the last estimate in (4.43), we obtain

    G1/(1σ)(t)c17(ζ(u)+ζ(v)+H(t)+ut22+vt22). (4.45)

    Combining (4.23) and (4.45), we deduce that

    G(t)˜cG1/(1σ)(t), for all  t>0.

    where ˜c=ερc16. A simple integration over (0,t) yields

    Gσ/(1σ)(t)1Gσ1σ(0)σ˜ct1σ,

    which implies that G(t)+, as tT, where T1σσ˜c[Gσ(1σ)(0)]. Consequently, the solution of Problem (1.2) blows-up in a finite time.

    In this section, we establish the existence of global solutions for initial data in a certain stable set. Then, we show that the decay estimates of the solution energy are exponential or polynomial, depending on the max{m+,r+}.

    To state and prove our first result, we introduce the two functionals defined for all t(0,T) by

    I(t)=I(u(t))=Δu22+v22(p++1)ΩF(x,u,v)dx, (5.1)
    J(t)=J(u(t))=12(Δu22+v22)ΩF(x,u,v)dx (5.2)

    and give the following Lemma.

    Lemma 5.1. Under the assumptions (H.1) and (H.2), we suppose that

    I(0)>0 and β<1,

    where

    β=C2(p++1)max{cp+1(2(p++1)p+1E(0))p12,cp++1(2(p++1)p+1E(0))p+12}.

    Then,

    I(t)>0, for all t(0,T). (5.3)

    Proof. From the continuity of I and the fact that I(0)>0, there exists tk in ]0,T) such that

    I(t)0, t(0,tk). (5.4)

    We have to show that this inequality is strict.

    Recalling (5.1) and (5.2), we have

    J(t)=p+12(p++1)(Δu22+v22)+1p++1I(t),

    Combining with (5.4), this gives

    J(t)p+12(p++1)(Δu22+v22),t(0,tk). (5.5)

    From the definition of the energy, we have

    E(t)=J(t)+12(ut22+vt22), (5.6)

    for all t(0,T). Consequently,

    Δu22+v222(p++1)(p+1)E(t).

    Thus, the decreasing property of E leads to

    max{Δu22,v22}2(p++1)(p+1)E(0),t(0,tk). (5.7)

    On the other hand, from Lemma 2.1 and the Sobolev embedding H20(Ω)Lp()+1(Ω), we have

    Ω|u|p(x)+1dxmax{cp+1Δup+12,cp++1Δup++12}max {cp+1Δup12,cp++1Δup+12}Δu22.

    Combining with (5.7), this yields, for all t(0,tk),

    Ω|u|p(x)+1dxmax{cp+1(2(p++1)(p+1)E(0))p12,cp++1(2(p++1)(p+1)E(0))p+12}Δu22.

    Therefore,

    Ω|u|p(x)+1dxβC2(p++1)Δu22. (5.8)

    Similarly, we have

    Ω|v|p(x)+1dxβC2(p++1)v22. (5.9)

    The addition of (5.8) and (5.9) gives

    Ω(|u|p(x)+1+|v|p(x)+1)dxβC2(p++1)(Δu22+v22). (5.10)

    Combining (5.10) with (3.41), we infer that

    ΩF(x,u,v)dxβp++1(Δu22+v22)<1p++1(Δu22+v22), (5.11)

    for all t(0,tk). From the definition of I, this leads to

    I(t)>0. t(0,tk).

    By repeating the above procedure and using the decreasing property of E, we can extend tk to T and obtain (5.3).

    Theorem 5.1. Suppose that all assumptions of Lemma 5.1 are fulfilling. Then, the local solution (u,v) of the system (1.2) exists globally.

    Proof. Substituting (5.5) into (5.6) and thanks to (5.3), it yields

    E(t)p+12(p++1)(Δu22+v22)+12(ut22+vt22),

    for all t(0,T). Then, we have

    Δu22+v22+ut22+vt22C3E(t)C3E(0), (5.12)

    for C3=max{2,2(p++1)p+1}. This means that the norm in (5.12) is bounded independently of t. Therefore, the solution (u,v) exists globally.

    To prove the decay result, we need the following Lemma.

    Lemma 5.2. Suppose that the assumptions of Lemma 5.1 hold. Then, there exists a positive constant C4, such that the global solution (u,v) satisfies

    Ω(|u(t)|m(x)+|v(t)|r(x))dxC4E(t) for all  t0. (5.13)

    Proof. The result is immediate by replacing p with m and r in (5.8) and (5.9), respectively, and by recalling (5.12).

    Theorem 5.2. Under the assumptions of Lemma 5.1, the solution of the system (1.2) satisfies the following decay estimates, for all t0,

    E(t){k(1+t)2/(λ+2), if α>2,keωt, if α=2, (5.14)

    where α=max {m+,r+} and k,w>0 are two positive constants.

    Proof. Multiplying (1.2)1 by u(t)Eη(t) and (1.2)2 by v(t)Eη(t) and then, integrating each result over Ω×(s,T), for s(0,T) and η0 to be specified later, we arrive at

    TsΩEη(t)[u(t)utt(t)+u(t)Δ2u(t)+u(t)|ut|m(x)2ut(t)]dxdt=TsΩEη(t)u(t)f1(x,u,v)dxdt

    and

    TsΩEη(t)[v(t)vtt(t)v(t)Δv(t)+v(t)|vt(t)|r(x)2vt(t)]dxdt=TsΩEη(t)v(t)f2(x,u,v)dxdt.

    Green's formula and the boundary conditions lead to

    TsΩEη(t)[(u(t)ut(t))t|ut(t)|2+|Δu(t)|2+u(t)ut(t)|ut(t)|m(x)2]dxdt=TsΩEη(t)u(t)f1(x,u,v)dxdt, (5.15)

    and

    TsΩEη(t)[(v(t)vt(t))t|vt(t)|2+|v(t)|2+v(t)vt(t)|vt(t)|r(x)2]dxdt=TsΩEη(t)v(t)f2(x,u,v)dxdt. (5.16)

    Adding and subtracting the following two terms

    |TsΩEη(t)[β|Δu(t)|2+(1+β)|ut(t)|2]dxdtTsΩEη(t)[β|v(t)|2+(1+β)|vt(t)|2]dxdt,

    to (5.15) and (5.16), respectively, and recalling (5.11), we arrive at

    (1β)TsEη(t)Ω(|Δu(t)|2+|v(t)|2+|ut(t)|2+|vt(t)|2)dxdt+TsEη(t)Ω[(u(t)ut(t)+v(t)vt(t))t(2β)(|ut(t)|2+|vt(t)|2)]dxdt+TsEη(t)Ω(u(t)ut(t)|ut(t)|m(x)2+v(t)vt(t)|vt(t)|r(x)2)dxdt=TsEη(t)Ω[β(|Δu(t)|2+|v(t)|2)(p(x)+1)F(x,u,v)]dxdt0. (5.17)

    Now, by exploiting the formula:

    Eη(t)Ω(u(t)ut(t)+v(t)vt(t))tdx=ddt(Eη(t)Ω(u(t)ut(t)+v(t)vt(t))dx)ηEη1(t)E(t)Ω(u(t)ut(t)+v(t)vt(t))dx,

    estimate (5.17) yields

    2(1β)TsEη+1(t)dtηTsEη1(t)E(t)Ω(u(t)ut(t)+v(t)vt(t))dxdtTsddt(Eη(t)Ω(u(t)ut(t)+v(t)vt(t))dx)dtTsEη(t)Ω(u(t)ut(t)|ut(t)|m(x)2+v(t)vt(t)|vt(t)|r(x)2)dxdt+(2β)TsEη(t)Ω(|ut(t)|2+|vt(t)|2)dxdt=I1+I2+I3+I4. (5.18)

    Next, we handle the terms Ii,i=¯1,4 and denote by C a positive generic constant.

    ● First, applying Young's and Poincaré's inequalities, we obtain

    I1=ηTsEη1(t)E(t)Ω(u(t)ut(t)+v(t)vt(t))dxdtη2TsEη1(t)(E(t))[u(t)22+ut(t)22+v(t)22+vt(t)22]dtCTsEη1(t)(E(t))[Δu(t)22+v(t)22+ut(t)22+vt(t)22]dt,

    By (5.12), this gives

    I1CTsEη(t)(E(t))dtCEη+1(s)CEη+1(T)CEη(0)E(s)CE(s). (5.19)

    ● Concerning the second term, we have

    I2=Tsddt(Eη(t)Ω(u(t)ut(t)+v(t)vt(t))dx)dt=Eη(s)(Ω(u(x,s)ut(x,s)+v(x,s)vt(x,s))dx)Eη(T)(Ω(u(x,T)ut(x,T)+v(x,T)vt(x,T))dx)

    Again, by (5.12) and the inequalities of Young and Poincaré, we get

    |Ωu(x,s)ut(x,s)dx|C(Δu(s)22+ut(s)22)CE(s),|Ωu(x,T)ut(x,T)dx|C(Δu(T)22+ut(T)22)CE(T)

    and likewise

    |Ωv(x,s)vt(x,s)dx|C(v(s)22+vt(s)22)CE(s)|Ωv(x,T)vt(x,T)dx|C(v(T)22+vt(T)22)CE(T).

    Therefore,

    I2CEη+1(s)CEη(0)E(s)CE(s). (5.20)

    ● For the third term, we apply Young's inequality (as in (4.30)) to obtain, for some ε>0,

    I3=TsEη(t)Ω(u(t)ut(t)|ut(t)|m(x)2+v(t)vt(t)|vt(t)|r(x)2)dxdtTsEη(t)(ε2Ω|u(t)|m(x)dx+1εΩ|ut(t)|m(x)dx)dt+TsEη(t)(ε2Ω|v(t)|r(x)dx+1εΩ|vt(t)|r(x)dx)dt.

    Invoking Lemma 5.2 and recalling (3.40), yields

    I3ε2TsEη(t)Ω(|u(t)|m(x)+|v(t)|r(x))dxdt+1εTsEη(t)(E(t))dtεCTsEη+1(t)dt+CεE(s). (5.21)

    ● Now, we handle I4, as follows:

    I4=(2β)TsEη(t)Ω(|ut(t)|2+|vt(t)|2)dxdt=(2β)[TsEη(t)Ω|ut(t)|2dxdt+TsEη(t)Ω|vt(t)|2dxdt]=(2β)(J1+J2).

    We claim that

    J1,J2εCTsEη+1(t)dt+CεE(s). (5.22)

    Since 2˜αm(.)α on Ω, we obtain

    J1=TsEη(t)Ω|ut(t)|2dxdt=TsEη(t)[Ω|ut(t)|2dx+Ω+|ut(t)|2dx]dtCTsEη(t)[(Ω|ut(t)|αdx)2/α+(Ω+|ut(t)|˜αdx)2/˜α]dtCTsEη(t)[(Ω|ut(t)|m(x)dx)2/α+(Ω+|ut(t)|m(x)dx)2/˜α]dt,

    where

    ˜α= min{m,r}, α= max{m+,r+},
    Ω+={xΩ:|u(x,t)|1} and Ω={xΩ:|u(x,t)|<1}.

    Therefore,

    J1CTsEη(t)(E(t))2/αdt+CTsEη(t)(E(t))2/˜αdt=C(Jα+J˜α). (5.23)

    Three cases are possible:

    (1) if α=˜α=2 (m(x)=r(x)=2, on Ω), then

    J1CTsEη(t)(E(t))dtCE(s)εCTsEη+1(t)dt+CE(s).

    (2) if α>2 and ˜α=2, we exploit Young's inequality with

    δ=(η+1)/η and  δ=η+1

    to find

    Jα=TsEη(t)(E(t))2/αdtεCTsEη+1(t)dt+CεTs(E(t))2(η+1)/αdt.

    So, for η=α21, we get

    JαεCTsEη+1(t)dt+CεTs(E(t))dtεCTsEη+1(t)dt+CεE(s). (5.24)

    Also, in this case, we have

    J˜α=TsEη(t)(E(t))dtCE(s). (5.25)

    By inserting (5.24) and (5.25) into (5.23), we infer that J1 (and similarly J2) satisfies (5.22).

    (3) if α>˜α>2, we apply Young's inequality with

    δ=˜α/(˜α2) and  δ=˜α/2

    to obtain

    J˜α=TsEη(t)(E(t))2/˜αdtεCTsE(t)η˜α/(˜α2)dt+CεE(s).

    But η˜α/(˜α2)=η+1+(α˜α)/(˜α2), then

    J˜αεC(E(s))(α˜α)/(α2)TsEη+1(t)dt+CεE(s)εCTsEη+1(t)dt+CεE(s). (5.26)

    The addition of (5.24) and (5.26) leads to (5.22).

    We conclude that the claim is true for any α˜α2. Therefore,

    I4εCTsEη+1(t)dt+CεE(s). (5.27)

    Now, substituting (4.22)–(5.21) and (5.27) into (5.18), we get

    2(1β)TsEη+1(t)dtεCTsEη+1(t)dt+CεE(s),

    with η=α21. So,

    2(1β)TsEα2(t)dtεCTsEα2(t)dt+CεE(s).

    Choosing ε small enough, we obtain

    TsEα2(t)dtCE(s).

    Letting T, it yields

    sEα2(t)dtCE(s),s>0.

    Applying Komornik's lemma [23], we get the desired decay estimates.

    We considered a coupled system of Laplacian and bi-Laplacian equations with nonlinear damping and source terms of variable-exponents nonlinearities. We gave a detailed proof of the local existence using Faedo-Galerkin method and Banach-fixed-point theorem. We also showed that the solutions with positive-initial energy blow-up in a finite time. Furthermore, we proved a global existence theorem, using the Stable-set method and established a decay estimate of the solution energy, by Komornik's integral approach.

    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. INCB2205, is also greatly acknowledged.

    The authors declare that there is no conflict of interest.



    [1] S. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells, New York: McGraw-hill, 1959.
    [2] P. Shankar, The eddy structure in stokes flow in a cavity, J. Fluid mech., 250 (1993), 371–383. https://doi.org/10.1017/S0022112093001491 doi: 10.1017/S0022112093001491
    [3] R. Srinivasan, Accurate solutions for steady plane flow in the driven cavity. i. stokes flow, Z. angew. Math. Phys., 46 (1995), 524–545. https://doi.org/10.1007/BF00917442 doi: 10.1007/BF00917442
    [4] V. Meleshko, Steady stokes flow in a rectangular cavity, P. Roy. Soc. A-Math. Phy., 452 (1996), 1999–2022. https://doi.org/10.1098/rspa.1996.0106 doi: 10.1098/rspa.1996.0106
    [5] P. Shankar, M. Deshpande, Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech., 32 (2000), 93–136. https://doi.org/10.1146/annurev.fluid.32.1.93 doi: 10.1146/annurev.fluid.32.1.93
    [6] N. E. Sevant, M. I. Bloor, M. J. Wilson, Aerodynamic design of a flying wing using response surface methodology, J. Aircraft, 37 (2000), 562–569. https://doi.org/10.2514/2.2665 doi: 10.2514/2.2665
    [7] M. I. Bloor, M. J. Wilson, Method for efficient shape parametrization of fluid membranes and vesicles, Phys. Rev. E, 61 (2000), 4218–4229. https://doi.org/10.1103/PhysRevE.61.4218 doi: 10.1103/PhysRevE.61.4218
    [8] V. Meleshko, Biharmonic problem in a rectangle, In: In fascination of fluid dynamics, Springer, 1998,217–249. https://doi.org/10.1007/978-94-011-4986-0_14
    [9] V. Meleshko, Selected topics in the history of the two-dimensional biharmonic problem, Appl. Mech. Rev., 56 (2003), 33–85. https://doi.org/10.1115/1.1521166 doi: 10.1115/1.1521166
    [10] V. Meleshko, Bending of an elastic rectangular clamped plate: Exact versus 'engineering'solutions, J. Elasticity, 48 (1997), 1–50. https://doi.org/10.1023/A:1007472709175 doi: 10.1023/A:1007472709175
    [11] V. Meleshko, A. Gomilko, Infinite systems for a biharmonic problem in a rectangle, P. Roy. Soc. A Math. Phy., 453 (1997), 2139–2160. https://doi.org/10.1098/rspa.1997.0115 doi: 10.1098/rspa.1997.0115
    [12] 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
    [13] S. Antontsev, S. Shmarev, Evolution pdes with nonstandard growth conditions, In: Atlantis studies in differential equations, 2015. https://doi.org/10.2991/978-94-6239-112-3
    [14] B. Guo, W. Gao, Blow-up of solutions to quasilinear hyperbolic equations with p(x, t)-Laplacian and positive initial energy, C. R. Mecanique, 342 (2014), 513–519. https://doi.org/10.1016/j.crme.2014.06.001 doi: 10.1016/j.crme.2014.06.001
    [15] S. A. Messaoudi, O. Bouhoufani, I. Hamchi, M. Alahyane, Existence and blow up in a system of wave equations with nonstandard nonlinearities, Electron. J. Differ. Eq., 2021 (2021), 1–33.
    [16] 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 (2022), 211–238. https://doi.org/10.3233/ASY-211704 doi: 10.3233/ASY-211704
    [17] S. H. Park, J. R. Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Method. Appl. Sci., 42 (2019), 2083–2097. https://doi.org/10.1002/mma.5501 doi: 10.1002/mma.5501
    [18] O. Bouhoufani, I. Hamchi, Coupled system of nonlinear hyperbolic equations with variable-exponents: Global existence and stability, Mediterr. J. Math., 17 (2020), 166. https://doi.org/10.1007/s00009-020-01589-1 doi: 10.1007/s00009-020-01589-1
    [19] L. Diening, P. Harjulehto, P. Hästö, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer, 2011.
    [20] D. V. Cruz-Uribe, A. Fiorenza, Variable Lebesgue spaces: Foundations and harmonic analysis, Springer Science & Business Media, 2013.
    [21] 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
    [22] C. O. Alves, M. M. Cavalcanti, V. N. D. Cavalcanti, M. A. Rammaha, D. Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete Cont. Dyn. S, 2 (2009), 583–608. https://doi.org/10.3934/dcdss.2009.2.583 doi: 10.3934/dcdss.2009.2.583
    [23] V. Komornik, Decay estimates for the wave equation with internal damping, In: Control and estimation of distributed parameter systems: Nonlinear phenomena, 1994. https://doi.org/10.1007/978-3-0348-8530-0_14
  • This article has been cited by:

    1. 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, 2023, 8, 2473-6988, 19971, 10.3934/math.20231018
    2. Muhammad I. Mustafa, Viscoelastic Wave Equation with Variable-Exponent Nonlinear Boundary Feedback, 2024, 30, 1079-2724, 10.1007/s10883-024-09714-z
    3. Muhammad I. Mustafa, On the interaction between the viscoelasticity and the boundary variable-exponent nonlinearity in plate systems, 2024, 103, 0003-6811, 2923, 10.1080/00036811.2024.2327436
    4. Adel M. Al-Mahdi, Long-time behavior for a nonlinear Timoshenko system: Thermal damping versus weak damping of variable-exponents type, 2023, 8, 2473-6988, 29577, 10.3934/math.20231515
    5. Oulia Bouhoufani, Well-posedness and decay in a system of hyperbolic and biharmonic-wave equations with variable exponents and weak dampings, 2023, 12, 2193-5343, 513, 10.1007/s40065-023-00431-2
    6. Mohammad Kafini, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Existence and stability results of nonlinear swelling equations with logarithmic source terms, 2024, 9, 2473-6988, 12825, 10.3934/math.2024627
    7. Mohammad Kafini, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Existence and Blow-up Study of a Quasilinear Wave Equation with Damping and Source Terms of Variable Exponents-type Acting on the Boundary, 2024, 30, 1079-2724, 10.1007/s10883-024-09695-z
  • Reader Comments
  • © 2023 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(1756) PDF downloads(112) Cited by(7)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog