Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 91, pp. 1–33. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND BLOW UP IN A SYSTEM OF WAVE EQUATIONS WITH NONSTANDARD NONLINEARITIES SALIM A. MESSAOUDI, OULIA BOUHOUFANI, ILHEM HAMCHI, MOHAMED ALAHYANE Abstract. In this article, we consider a coupled system of two nonlinear hyperbolic equations, where the exponents in the damping and source terms are variables. First, we prove a theorem of existence and uniqueness of weak solution, by using the Faedo Galerkin approximations and the Banach fixed point theorem. Then, using the energy method, we show that certain solutions with positive initial energy blow up in finite time. We also give some numerical applications to illustrate our theoretical results. 1. Introduction In this work, we study the following initial-boundary-value problem for the un- knowns u and v: utt − div(A∇u) + |ut|m(x)−2ut = f1(x, u, v) in Ω× (0, T ), vtt − div(B∇v) + |vt|r(x)−2vt = f2(x, u, v) in Ω× (0, T ), u = v = 0 on ∂Ω× (0, T ), u(0) = u0, ut(0) = u1 in Ω, v(0) = v0 and vt(0) = v1 in Ω, (1.1) where T > 0 and Ω is a bounded domain of Rn (n = 1, 2, 3) with a smooth boundary ∂Ω, m and r are continuous functions on Ω such that, for all x ∈ Ω, 2 ≤ m(x), if n = 1, 2, 2 ≤ m− ≤ m(x) ≤ m+ ≤ 6, if n = 3 (1.2) and 2 ≤ r(x), if n = 1, 2, 2 ≤ r− ≤ r(x) ≤ r+ ≤ 6, if n = 3, (1.3) where m− = inf x∈Ω m(x), m+ = sup x∈Ω m(x), r− = inf x∈Ω r(x), r+ = sup x∈Ω r(x). 2010 Mathematics Subject Classification. 35D30, 35B40, 35B44, 35L70. Key words and phrases. Hyperbolic system; existence; blow up, variable exponents; nonlinear. ©2021. This work is licensed under a CC BY 4.0 license. Submitted March 16, 2021. Published November 16, 2021. 1 2 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 The coupling terms f1 and f2 are as follows: for all x ∈ Ω and (u, v) ∈ R2, f1(x, u, v) = ∂ ∂u F (x, u, v), f2(x, u, v) = ∂ ∂v F (x, u, v), (1.4) with F (x, u, v) = a|u+ v|p(x)+1 + 2b|uv| p(x)+1 2 , (1.5) where a, b > 0 two positive constants, p is a given continuous function on Ω such that, for all x ∈ Ω, 3 ≤ p− ≤ p(x) ≤ p+, if n = 1, 2, p(x) = 3, if n = 3, (1.6) with max{m+, r+} ≤ p− = inf x∈Ω p(x). A and B are symmetric matrices of class C1(Ω× [0,∞)) such that for constants a0, b0 > 0 and all ξ ∈ Rn, Aξ · ξ ≥ a0|ξ|2, Bξ · ξ ≥ b0|ξ|2, (1.7) A′ξ · ξ ≤ 0, B′ξ.ξ ≤ 0, (1.8) where A′ = ∂A ∂t (·, t) and B′ = ∂B ∂t (·, t). The study of system (1.1) is motivated by the description of several models in physical phenomena, such as viscoelastic fluids, filtration processes through a porous media, fluids with temperature dependent viscosity, image processing, or robotics, etc. See for example [7] for an application of such functional spaces in the image recovery. Our system can be regarded as a model for interaction between two fields describing the motion of two nonlinear “smart” materials. For more details, see [1, 7]. A considerable effort has been devoted to the study of single wave equations in the case of constant exponents. The equation utt −∆u+ a|ut|m−2ut = b|u|p−2u in Ω× (0, T ), with initial and Dirichlet boundary conditions, has been studied by many re- searchers. For example, Ball in [5] showed that if a = 0, then the source term b|u|p−2u, with b > 0, forces the negative-energy solutions to explode in finite time. Haraux and Zuazua [10] proved that in the absence of the source term, the damping term a|ut|m−2ut, with a > 0, assures the global existence for arbitrary initial data. In the presence of both terms, the problem was first considered by Levine [12]. He established the blow up for solutions with negative initial energy, when m = 2. Georgiev and Todorava [8] pushed Levine’s result to the case m > 2, by introducing a different method. Messaoudi [14] proved that any solution with negative initial energy only, blows up in finite time when m < p. For a wave equation with variable-exponent nonlinearity, we mention some works. In [3], Antontsev studied the equation utt − div(a|∇u|p(x,t)−2∇u)− α∆ut − bu|u|σ(x,t)−2 = f in Ω× (0, T ), where α > 0 is a constant and a, b, p, σ are given functions. Under specific condi- tions, he proved the local and global existence of some weak solutions and a blow-up result for certain solutions having arbitrary initial energy. Guo and Gao [9] took EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 3 σ(x, t) = r > 2 and established a finite-time blow-up result. Sun et al. [22] studied the equation utt − div(a(x, t)∇u) + c(x, t)ut|ut|q(x,t)−1 = b(x, t)u|u|p(x,t)−2 in Ω× (0, T ) and established a blow-up result. Also, under some conditions on the initial data, the lower and upper bounds for the blow-up time are obtained. In addition, they provided numerical illustrations for their result. After that, Messaoudi and Talah- meh [17] studied the equation utt − div(|∇u|m(x)−2∇u) + µut = u|u|p(x)−2 in Ω× (0, T ), for µ ≥ 0 supplemented with Dirichlet-boundary conditions. They proved a blow- up result for certain solutions with arbitrary positive initial energy. In [18], the same authors considered the equation utt − div(|∇u|r(x)−2∇u) + aut|ut|m(x)−2 = bu|u|p(x)−2 in Ω× (0, T ), where a, b > 0 are two constants and m, r, p are given functions. They established a finite-time blow-up result for negative-initial-energy solutions and for certain so- lutions with positive initial energy. Recently, Messaoudi et al. [19] considered the equation utt −∆u+ aut|ut|m(x)−2 = bu|u|p(x)−2 in Ω× (0, T ). Using the Faedo-Galerkin method, they established the existence and uniqueness of a weak local solution and proved the finite-time blow up for solutions with negative initial-energy. Concerning the coupled systems of two nonlinear wave equations with constant exponents, Messaoudi and Said-Houari [16] proved a nonexistence theorem for positive-initial-energy solutions of a system of viscoelastic wave equations. Un- der some restrictions on the nonlinearity of the damping and source terms and the initial data, Said-Houari et al. [21] proved that the rate of decay of the total energy depends on those of the relaxation functions. Agre and Rammaha [2] obtained several results on the local and global existence, uniqueness, and the finite-time blow up of solutions, under appropriate conditions on the parameters in the sys- tem. Very recently, Messaoudi and Hassan [15] established a general decay result for a certain system of viscoelastic wave equations. In the case of coupled systems of two nonlinear hyperbolic equations with vari- able exponents, there is only the work of Bouhoufani and Hamchi [6], where they proved the global existence of a weak solution and established decay estimates of the solution energy. However, the existence of a local solution was not discussed. In this work, we push the local existence result of Agre and Rammaha [2], which was established for the case of constant-exponent nonlinearities, to our system which deals with variable-exponent nonlinearities. To the best of our knowledge, this is the first result of this kind and the generalization was not trivial at all. In addition to the local existence, we establish the blow up in finite time for certain solutions with positive initial energy and give some numerical illustrations. This paper consists of four Sections, in addition to the introduction. In Section 2, we give some preliminary results. The existence and uniqueness of weak solution is discussed in Section 3. Section 4 is devoted to the statement and the proof of the finite-time blow up result. In section 5, we present two numerical examples to illustrate our theoretical findings. 4 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 2. Preliminaries In this section, we define the Lebesgue and Sobolev spaces with variable expo- nents and present some facts and results related to this important class of spaces. See [4, 11] for more details. Let q : Ω→ [1,∞) be a measurable function. We define the Lebesgue space with a variable exponent by Lq(·)(Ω) = { f : Ω→ R measurable in Ω : %q(·)(λf) < +∞ for some λ > 0 } , where %q(·)(f) = ∫ Ω |f(x)|q(x)dx. Lq(·)(Ω) is a Banach space with respect to the following Luxembourg-type norm ‖f‖q(·) := inf { λ > 0 : ∫ Ω |f(x) λ |q(x)dx ≤ 1 } . We also define the variable exponent Sobolev space W 1,q(·)(Ω) = {f ∈ Lq(·)(Ω) : ∇f exists and |∇f | ∈ Lq(·)(Ω)}. Equipped with the norm ‖f‖W 1,q(·)(Ω) = ‖f‖q(·) + ‖∇f‖q(·), W 1,q(·)(Ω) is a Banach space. Furthermore, we denote by H 1,q(·) 0 (Ω) the closure of C∞0 (Ω) in W 1,q(·)(Ω) and by W−1,q′(·)(Ω) the dual space of W 1,q(·) 0 (Ω) (in the same way as the usual Sobolev spaces), where W k,p(·) 0 (Ω) = {u ∈W k,p(·)(Ω) : u = uχK for a compact K ⊂ Ω} and 1 p(·) + 1 p′(·) = 1. Lemma 2.1 (Young’s Inequality [11]). Let p, q, s : Ω → [1,∞) be measurable functions, such that 1 s(y) = 1 p(y) + 1 q(y) , for a.e y ∈ Ω. Then, for all a, b ≥ 0, we have (ab)s(·) s(·) ≤ ap(·) p(·) + bq(·) q(·) . By taking s = 1 and 1 < p, q < +∞, it follows that for any ε > 0, ab ≤ εap + Cεb q, where Cε = 1/q(εp)q/p. (2.1) Lemma 2.2 (Hölder’s Inequality [11]). Let p, q, s : Ω → [1,∞) be measurable functions, such that 1 s(y) = 1 p(y) + 1 q(y) , for a.e. y ∈ Ω. If f ∈ Lp(·)(Ω) and h ∈ Lq(·)(Ω), then fh ∈ Ls(·)(Ω), with ‖fh‖s(·) ≤ 2‖f‖p(·)‖h‖q(·). Lemma 2.3 ([11]). If 1 < q− ≤ q(x) ≤ q+ < +∞ holds, then for any f ∈ Lq(·)(Ω), min{‖f‖q − q(·), ‖f‖ q+ q(·)} ≤ %q(·)(f) ≤ max{‖f‖q − q(·), ‖f‖ q+ q(·)}. EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 5 Lemma 2.4 ([11]). If q+ < +∞, then C∞0 (Ω) is dense in Lq(·)(Ω). Lemma 2.5 (Embedding Property [11]). Let Ω ⊂ Rn be a bounded domain with a smooth boundary ∂Ω. If r, q ∈ C(Ω) such that r, q ≥ 1 and infx∈Ω(r∗(x)− q(x)) > 0 with r∗(x) = { nr(x) supx∈Ω(n−r(x)) , if r+ < n, ∞, if r+ ≥ n. Then, the embedding W 1,r(·) 0 (Ω) ↪→ Lq(·)(Ω) is continuous and compact. 3. Existence of weak solutions We begin this section by giving the definition of a weak solution for the system (1.1). Definition 3.1. Let (u0, u1), (v0, v1) ∈ H1 0 (Ω)×L2(Ω). Any pair of functions (u, v) such that u, v ∈ L∞([0, T ), H1 0 (Ω)), ut ∈ L∞([0, T ), L2(Ω)) ∩ Lm(·)(Ω× (0, T )), vt ∈ L∞([0, T ), L2(Ω)) ∩ Lr(·)(Ω× (0, T )), is called a weak solution of (1.1) on [0, T ), if d dt [ ∫ Ω utΦdx+ ∫ Ω A∇u.∇Φdx] + ∫ Ω |ut|m(x)−2utΦdx− ∫ Ω Φf1dx = 0, d dt [ ∫ Ω vtΨdx+ ∫ Ω B∇v.∇Ψdx] + ∫ Ω |vt|r(x)−2utΨdx− ∫ Ω Ψf2dx = 0, u(0) = u0, ut(0) = u1, v(0) = v0, vt(0) = v1, for a.e. t ∈ (0, T ) and all test functions Φ,Ψ ∈ H1 0 (Ω). To prove the existence of a local weak solution of problem (1.1), we first consider, as in [13], the initial-boundary-value problem utt − div(A∇u) + |ut|m(x)−2ut = f(x, t) in Ω× (0, T ), vtt − div(B∇v) + |vt|r(x)−2vt = g(x, t) in Ω× (0, T ), u = v = 0 on ∂Ω× (0, T ), u(0) = u0, ut(0) = u1 in Ω, v(0) = v0, vt(0) = v1 in Ω, (3.1) where f, g ∈ L2(Ω× (0, T )). Theorem 3.2. Under the above conditions, on m, r,A and B, and for the initial values (u0, u1), (v0, v1) ∈ H1 0 (Ω) × L2(Ω), problem (3.1) has a unique local weak solution (u, v) on [0, T ), in the sense of Definition 3.1. Proof. (Uniqueness.) Suppose that (3.1) has two weak solutions (u1, v1) and (u2, v2), in the sense of Definition 3.1. Then, (u, v) = (u1 − u2, v1 − v2) solves the problem utt − div(A∇u) + |u1t|m(x)−2u1t − |u2t|m(x)−2u2t = 0 in Ω× (0, T ), vtt − div(B∇v) + |v1t|r(x)−2v1t − |v2t|r(x)−2v2t = 0 in Ω× (0, T ), 6 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 u = v = 0 on ∂Ω× (0, T ), u(0) = ut(0) = 0, v(0) = vt(0) = 0 in Ω, in the sense of Definition 3.1. Taking Φ = ut, we obtain d dt [ ∫ Ω (u2 t +A∇u.∇u)dx] + 2 ∫ Ω (|u1t|m(x)−2u1t − |u2t|m(x)−2u2t)(u1t − u2t)dx ≤ 0, (3.2) by (1.8) and d dt (∫ Ω A∇u · ∇udx ) = ∫ Ω A′∇u · ∇u dx+ 2 ∫ Ω A∇u · ∇ut dx. (3.3) Since, for all x ∈ Ω, Y, Z ∈ R and q(x) ≥ 2, we have (|Y |q(x)−2Y − |Z|q(x)−2Z)(Y − Z) ≥ 0, q(x) ≥ 2, (3.4) inequality (3.2) leads to d dt ∫ Ω (u2 t +A∇u · ∇u)dx ≤ 0. Integrating over (0, t), t ≤ T , and using (1.7), we find that ‖ut‖22 + a0‖∇u‖22 = 0. Similarly, we obtain ‖vt‖22 + b0‖∇v‖22 = 0. Therefore, ut(·, t) = vt(·, t) = 0 and ∇u(·, t) = ∇v(·, t) = 0 for allt ∈ (0, T ). Which implies u = v = 0 on Ω × (0, T ), since u = v = 0 on ∂Ω × (0, T ). This proves the uniqueness. Existence. To prove the existence of a weak solution of (3.1), we proceed in four steps: Step 1. Approximate problem. We consider an orthonormal basis {ωj}∞j=1 of H1 0 (Ω) and define, for all k ≥ 1, a sequence (uk, vk) in the finite-dimensional subspace Vk = span{ω1, ω2, . . . , ωk}, as follows uk(x, t) = Σkj=1aj(t)ωj(x), vk(t) = Σkj=1bj(t)ωj(x), for x ∈ Ω and t ∈ (0, T ), satisfying the approximate problems∫ Ω uktt(x, t)ωjdx+ ∫ Ω A∇uk(x, t) · ∇ωjdx + ∫ Ω |ukt (x, t)|m(x)−2ukt (x, t)ωjdx = ∫ Ω f(x, t)ωjdx,∫ Ω vktt(x, t)ωjdx+ ∫ Ω B∇vk(x, t) · ∇ωjdx + ∫ Ω |vkt (x, t)|r(x)−2vkt (x, t)ωjdx = ∫ Ω g(x, t)ωjdx, (3.5) for j = 1, 2, . . . , k, and with the initial data uk(0) = uk0 = Σki=1〈u0, ωi〉ωi, ukt (0) = uk1 = Σki=1〈u1, ωi〉ωi vk(0) = vk0 = Σki=1〈v0, ωi〉ωi, vkt (0) = vk1 = Σki=1〈v1, ωi〉ωi, EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 7 where (uk0) and (vk0 ) are two sequences such that uk0 → u0, vk0 → v0 in H1 0 (Ω), uk1 → u1, vk1 → v1 in L2(Ω). This generates a system of k nonlinear ordinary differential equations, which admits a unique local solution (uk, vk) in [0, Tk), Tk ≤ T , by the standard ODE theory. Step 2. A priori Estimates. Now, we show, by a priory estimates, that Tk = T , for all k ≥ 1. For this, we multiply (3.5)1 and (3.5)2 by a′j(t) and b′j(t), respectively. Then sum each result over j, from 1 to k, and integrate each equation over (0, t), with t ≤ Tk. We obtain ‖ukt ‖22 − ‖uk1‖22 + ∫ Ω A∇uk · ∇ukdx− ∫ Ω A(x, 0)∇uk0 · ∇uk0dx + 2 ∫ t 0 ∫ Ω |ukt (x, t)|m(x) dx dt ≤ 2 ∫ t 0 ∫ Ω f(x, t)ukt (x, t) dx dt (3.6) and ‖vkt ‖22 − ‖vk1‖22 + ∫ Ω B∇vk · ∇vkdx− ∫ Ω B(x, 0)∇vk0 · ∇vk0dx + 2 ∫ t 0 ∫ Ω |vkt (x, t)|r(x) dx dt ≤ 2 ∫ t 0 ∫ Ω g(x, t)vkt (x, t) dx ds, (3.7) by (3.3) and (1.8). Under the assumptions on A and B, the addition of (3.6) and (3.7), Young’s inequality (2.1) gives ‖ukt ‖22 + ‖vkt ‖22 + a0‖∇uk‖22 + b0‖∇vk‖22 + 2 ∫ Tk 0 ∫ Ω (|ukt (x, t)|m(x) + |vkt (x, t)|r(x)) dx ds ≤ 2ε ∫ Tk 0 (‖ukt ‖22 + ‖vkt ‖22)ds+ 2Cε ∫ T 0 ∫ Ω (|f(x, t)|2 + |g(x, t)|2)) dx ds + ‖uk1‖22 + ‖vk1‖22 + α‖∇uk0‖22 + β‖∇vk0‖22, (3.8) where α = sup Ω×(0,T ) A(x, t), β = sup Ω×(0,T ) B(x, t). Since f, g ∈ L2(Ω× (0, T )) and uk0 → u0, vk0 → v0 in H1 0 (Ω), uk1 → u1, vk1 → v1 in L2(Ω), invoking Gronwall’s lemma, estimate (3.8) leads to sup (0,Tk) [‖ukt ‖22 + ‖vkt ‖22 + ‖∇uk‖22 + ‖∇vk‖22] + ∫ Tk 0 ∫ Ω (|ukt (x, t)|m(x) + |vkt (x, t)|r(x)) dx ds ≤ C, where C > 0, for all Tk ≤ T and k ≥ 1. Therefore, the local solution (uk, vk) of system (3.5) can be extended to (0, T ), for all k ≥ 1. Furthermore, (uk)k, (v k)k are bounded in L∞((0, T ), H1 0 (Ω)), (ukt )k is bounded in L∞((0, T ), L2(Ω)) ∩ Lm(·)(Ω× (0, T )), 8 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 (vkt )k is bounded in L∞((0, T ), L2(Ω)) ∩ Lr(·)(Ω× (0, T )). Consequently, we can extract two subsequences of (uk)k and (vk)k, which we denote by (ul)l and (vl)l, respectively, such that ul → u and vl → v weakly * in L∞((0, T ), H1 0 (Ω)), ult → ut weakly * in L∞((0, T ), L2(Ω)) and weakly in Lm(·)(Ω× (0, T )), vlt → vt weakly * in L∞((0, T ), L2(Ω)) and weakly in Lr(·)(Ω× (0, T )), as l→ +∞ Step 3. Nonlinear terms. In this step, we show that |ult|m(·)−2ult → |ut|m(·)−2ut weakly in L m(·) m(·)−1 (Ω× (0, T )), |vlt|r(·)−2vlt → |vt|r(·)−2vt weakly in L r(·) r(·)−1 (Ω× (0, T )) and then, we establish that (u, v) satisfies the differential equations (3.1) on Ω × (0, T ). So, by exploiting Hölder’s inequality (Lemma 2.2), one easily deduce that (|ult|m(·)−2ult)l is bounded in L m(·) m(·)−1 (Ω× (0, T )). Then, there exists a subsequence, still denoted by (|ult|m(·)−2ult)l, such that |ult|m(·)−2ult → Φ weakly in L m(·) m(·)−1 (Ω× (0, T )). To prove that Φ = |ut|m(·)−2ut, we set h(z) = |z|m(·)−2z and define, as in [13], the sequence Sl = ∫ T 0 ∫ Ω (h(ult)− h(z))(ult − z), for z ∈ Lm(·)((0, T ), H1 0 (Ω)) and l ≥ 1. Replacing uk by ul in (3.6), integrating the result over (0, T ), and letting l→∞, by (3.3), we obtain 0 ≤ lim sup l Sl ≤ 1 2 [ ‖u1‖22 − ‖ut(T )‖22 + ∫ Ω A(x, 0)∇u0 · ∇u0 ] − 1 2 ∫ Ω A(x, T )∇u(T ) · ∇u(T )]− ∫ T 0 ∫ Ω Φz − ∫ T 0 ∫ Ω h(z)(ut − z) + ∫ T 0 ∫ Ω fut. (3.9) On the other hand, if we use ul instead of uk in (3.5)1 and integrate the result over (0, t), we arrive at∫ Ω utω − ∫ Ω u1ω + ∫ t 0 ∫ Ω A∇u · ∇ω + ∫ t 0 ∫ Ω Φω = ∫ t 0 ∫ Ω fω, ∀ω ∈ H1 0 (Ω), since {ωj}∞j=1 is a basis of H1 0 (Ω). Therefore,∫ Ω uttω + ∫ Ω (A∇u · ∇ω + Φω) = ∫ Ω fω, ∀ω ∈ H1 0 (Ω), (3.10) EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 9 for all t ∈ (0, T ). Using the denseness of H1 0 (Ω) in L2(Ω), we use ut instead of ω in (3.10) and integrate the result over (0, T ) to obtain∫ T 0 ∫ Ω fut = 1 2 [ ‖ut(T )‖22 − ‖u1‖22 + ∫ Ω A(x, T )∇u(T ) · ∇u(T ) ] − 1 2 ∫ Ω A(x, 0)∇u0.∇u0] + ∫ T 0 ∫ Ω Φut. (3.11) Combining (3.9) and (3.11), we infer that∫ T 0 ∫ Ω [Φ− h(z)](ut − z) ≥ lim sup l Sl ≥ 0, ∀z ∈ Lm(·)(Ω× (0, T )), (3.12) since H1 0 (Ω) is dense in Lm(·)(Ω) (see Lemma 2.4). Now, let z = λω + ut, ω ∈ Lm(·)(Ω × (0, T )). Hence, inequality ((3.12) can be rewritten as −λ ∫ T 0 ∫ Ω [Φ− h(λω + ut)]ω ≥ 0, ∀λ 6= 0. The continuity of h with respect to λ yields∫ T 0 ∫ Ω (Φ− h(ut))ω = 0, ∀ω ∈ Lm(·)(Ω× (0, T )). Thus, Φ = h(ut) = |ut|m(x)−2ut. Therefore, inequality ((3.10) becomes∫ Ω uttω + ∫ Ω A∇u · ∇ω + ∫ Ω |ut|m(x)−2utω = ∫ Ω fω, ∀ω ∈ H1 0 (Ω). Consequently, utt − div(A∇u) + |ut|m(x)−2ut = f in D′(Ω× (0, T )). (3.13) Likewise and since H1 0 (Ω) is dense in Lr(·)(Ω) (Lemma 2.4), we obtain |vlt|r(·)−2vlt → |vt|r(·)−2vt weakly in L r(·) r(·)−1 (Ω× (0, T )) and vtt − div(B∇v) + |vt|r(x)−2vt = g in D′(Ω× (0, T )). (3.14) Step 4. Initial conditions. First, by Lions’ lemma [13, Lemma 1.2, page 7] and since ul ⇀ u weakly * in L∞((0, T ), H1 0 (Ω)), ult ⇀ ut weakly * in L∞((0, T ), L2(Ω)), we deduce that ul → u in C([0, T ], L2(Ω)). Therefore, ul(·, 0) is defined and ul(·, 0)→ u(·, 0) in L2(Ω). But, ul(·, 0) = ul0 → u0, in H1 0 (Ω). Then u(·, 0) = u0. Similarly, we obtain v(·, 0) = v0. 10 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 Second, for any φ ∈ C∞0 (0, T ) and j ≤ l, we obtain from (3.4) that∫ T 0 ∫ Ω ultt(x, t)ωj(x)φ(t) + ∫ T 0 ∫ Ω A∇ul(x, t) · ∇ωj(x)φ(t) = − ∫ T 0 ∫ Ω |ult(x, t)|m(x)−2ult(x, t)ωj(x)φ(t) + ∫ T 0 ∫ Ω f(x, t)ωj(x)φ(t). (3.15) By routine computations and taking l→ +∞, we find that for all ω ∈ H1 0 (Ω),∫ T 0 ∫ Ω utt(x, t)ω(x)φ(t) = ∫ T 0 ∫ Ω [div(A∇u(x, t))− |ut(x, t)|m(x)−2ut(x, t) + f(x, t)]ω(x)φ(t), which means utt ∈ L m(·) m(·)−1 ([0, T ), H−1(Ω)) and that u solves the equation utt − div(A∇u) + |ut|m(x)−2ut = f, in D′(Ω× (0, T )). So, we have ut ∈ L∞((0, T ), L2(Ω)), utt ∈ L m(·) m(·)−1 ([0, T ), H−1(Ω)). By Lions’ lemma [13], ut ∈ C([0, T ), H−1(Ω)) and consequently, ut(·, 0) has a mean- ing and, in addition, ult(·, 0)→ ut(·, 0), in H−1(Ω). Since, ult(·, 0) = ul1 → u1 in L2(Ω), this implies that ut(·, 0) = u1. Similarly, one has vt(·, 0) = v1. Therefore, (u, v) is the unique local solution of (3.1). � To state and prove the existence of a solution for problem (1.1), we recall the following elementary inequalities:∣∣|X|k − |Y |k∣∣ ≤ C|X − Y |(|X|k−1 + |Y |k−1), (3.16) for some constant C > 0, all k ≥ 1 and all X,Y ∈ R. Also∣∣|X|k′X − |Y |k′Y ∣∣ ≤ C|X − Y |(|X|k′ + |Y |k ′ ), (3.17) for some constant C > 0, all k′ ≥ 0 and all X,Y ∈ R. Theorem 3.3. Let (u0, u1), (v0, v1) ∈ H1 0 (Ω) × L2(Ω) be given. Assume that the conditions on p(·), r(·), m(·), A and B, given in Section 1, hold. Then, problem (1.1) has a unique weak local solution (u, v) on [0, T ), for some T > 0. Proof. (Existence.) Recall that the source terms are defined for all x ∈ Ω and (y, z) ∈ R2 by f1(x, y, z) = ∂ ∂y F (x, y, z), f2(x, y, z) = ∂ ∂z F (x, y, z), where F (x, y, z) = a|y + z|p(x)+1 + 2b|yz| p(x)+1 2 , a, b > 0. So, f1(x, y, z) = (p(x) + 1)[a|y + z|p(x)−1(y + z) + by|y| p(x)−3 2 |z| p(x)+1 2 ], EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 11 f2(x, y, z) = (p(x) + 1)[a|y + z|p(x)−1(y + z) + bz|z| p(x)−3 2 |y| p(x)+1 2 ]. Let y, z ∈ L∞((0, T ), H1 0 (Ω)). Using Young’s inequality (2.1) and the Sobolev embedding (Lemma 2.5), f1(y, z) and f2(y, z) are in L2(Ω × (0, T )). Indeed, we have ∫ Ω |f1(x, y, z)|2dx ≤ 2 ∫ Ω (p(x) + 1)2[a2|y + z|2p(x) + b2|y|p(x)−1|z|p(x)+1]dx ≤ 2(p+ + 1)2 [ a2 ∫ Ω |y + z|2p(x)dx+ b2 ∫ Ω |y|p(x)−1|z|p(x)+1dx ] ≤ C0 [ ∫ Ω |y + z|2p + dx+ ∫ Ω |y + z|2p − dx+ ∫ Ω |y|3(p+−1)dx ] + C0 [ ∫ Ω |y|3(p−−1)dx+ ∫ Ω |z| 32 (p++1)dx+ ∫ Ω |z| 32 (p−+1)dx ] , (3.18) where C0 = 2(p+ + 1)2 max{a2, 3b2} > 0. By the embeddings, one can obtain the following results • If n = 1, 2, then 1 < 3 2 (p− + 1) ≤ 3 2 (p+ + 1) ≤ 2p+ ≤ 3(p+ − 1) <∞, since 3 ≤ p− ≤ p(x) ≤ p+ <∞. Therefore, estimate (3.18) leads to∫ Ω |f1(x, y, z)|2dx ≤ C1 [ ‖∇(y + z)‖2p + 2 + ‖∇(y + z)‖2p − 2 + ‖∇y‖3(p+−1) 2 + ‖∇y‖3(p−−1) 2 ] + C1 [ ‖∇z‖ 3 2 (p++1) 2 + ‖∇z‖ 3 2 (p−+1) 2 ] < +∞, C1 = C0Ce (3.19) • If n = 3, then the Sobolev embeddings used in (3.19) are also satisfied, since p ≡ 3 on Ω. Consequently, under assumption (1.6), for all t ∈ (0, T ), we have∫ Ω |f1(x, y, z)|2dx <∞ and similarly, ∫ Ω |f2(x, y, z)|2dx <∞. Therefore, f1(y, z), f2(y, z) ∈ L2(Ω× (0, T )). By Theorem 3.2, there exists a unique weak solution (u, v) for the problem utt − div(A∇u) + |ut|m(x)−2ut = f1(y, z) in Ω× (0, T ), vtt − div(B∇v) + |vt|r(x)−2vt = f2(y, z) in Ω× (0, T ), u = v = 0 on ∂Ω× (0, T ), u(0) = u0, ut(0) = u1 in Ω, v(0) = v0, vt(0) = v1 in Ω. (3.20) 12 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 Now, let G : WT ×WT :→WT ×WT be a map defined by G(y, z) = (u, v), where WT = {w ∈ L∞((0, T ), H1 0 (Ω))/wt ∈ L∞((0, T ), L2(Ω))}. Note that WT is a Banach space with respect to the norm ‖w‖2WT = sup (0,T ) ∫ Ω |∇w|2dx+ sup (0,T ) ∫ Ω |wt|2dx. Our task is to prove that G is a contraction mapping from a bounded ball B(0, d) into itself, where B(0, d) = { (y, z) ∈WT ×WT /‖(y, z)‖WT0 ×WT0 ≤ d } , for d > 0 sufficiently large and T0 ≥ T to be fixed later. To do so, let (y, z) be in B(0, d) and (u, v) be the corresponding solution of system (3.20). Taking (Φ,Ψ) = (ut, vt) in Definition 3.1 and by (3.3) and (1.8), and then integrating each identity over (0, t), for all t ≤ T , we obtain 1 2 ‖ut‖22 − 1 2 ‖u1‖22 + 1 2 ∫ Ω A∇u · ∇udx− 1 2 ∫ Ω A(x, 0)∇u0 · ∇u0dx + ∫ t 0 ∫ Ω |ut(x, t)|m(x) ≤ ∫ t 0 ∫ Ω utf1(y, z) dx ds (3.21) and 1 2 ‖vt‖22 − 1 2 ‖v1‖22 + 1 2 ∫ Ω B∇v · ∇v dx− 1 2 ∫ Ω B(x, 0)∇vk0 · ∇v0 dx + ∫ t 0 ∫ Ω |vt(x, t)|r(x) ≤ ∫ t 0 ∫ Ω vtf2(y, z) dx ds. (3.22) Recalling the assumptions on A and B, inequalities (3.21), (3.22) lead to 1 2 ( ‖ut‖22 + a0‖∇u‖22 ) ≤ 1 2 ( ‖u1‖22 + α‖∇u0‖22 ) + ∫ t 0 ∫ Ω utf1(y, z) dx ds, 1 2 ( ‖vt‖22 + b0‖∇v‖22 ) ≤ 1 2 ( ‖v1‖22 + β‖∇v0‖22 ) + ∫ t 0 ∫ Ω vtf2(y, z) dx ds. Consequently, we arrive at 1 2 ‖u‖2WT ≤ λ0 + 1 min {1, a0} sup (0,T ) ∫ t 0 ∫ Ω utf1(y, z) dx ds, 1 2 ‖v‖2WT ≤ β0 + 1 min {1, b0} sup (0,T ) ∫ t 0 ∫ Ω vtf2(y, z) dx ds, where, λ0 = ‖u1‖22 + α‖∇u0‖22 2 min {1, a0} , β0 = ‖v1‖22 + β‖∇v0‖22 2 min {1, b0} . EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 13 The addition the last two inequalities yield 1 2 ‖(u, v)‖2WT×WT ≤ γ0 + C2 sup (0,T ) ∫ t 0 ( | ∫ Ω utf1(y, z)dx|+ | ∫ Ω vtf2(y, z)dx| ) ds, (3.23) where γ0 = λ0 + β0, C2 = 1 min {1, a0} + 1 min {1, b0} . Under assumption (1.6), we apply Young’s inequality (2.1) and the Sobolev embed- dings (Lemma 2.5) to obtain, for a.e. t ∈ (0, T ), | ∫ Ω utf1(y, z)dx| ≤ (p+ + 1) [ a ∫ Ω |ut||y + z|p(x) + b ∫ Ω |ut| |y| p(x)−1 2 |z| p(x)+1 2 ] ≤ (p+ + 1) [ε(a+ b) 2 ∫ Ω |ut|2 + 2a ε ∫ Ω |y + z|2p(x) + 2b ε ∫ Ω |y|p(x)−1|z|p(x)+1 ] ≤ c1 [ε 2 ‖ut‖22 + Cε (∫ Ω |y + z|2p + dx+ ∫ Ω |y + z|2p − dx )] + c1Cε (∫ Ω |y|3(p(x)−1)dx+ ∫ Ω |z| 32 (p(x)+1)dx ) ≤ c2 [ ε‖ut‖22 + ‖∇y‖2p − 2 + ‖∇z‖2p − 2 + ‖∇y‖2p + 2 + ‖∇z‖2p + 2 ] + c2 [ ‖∇y‖3(p−−1) 2 + ‖∇y‖3(p+−1) 2 + ‖∇z‖ 3 2 (p−+1) 2 + ‖∇z‖ 3 2 (p++1) 2 ] , (3.24) where ε, c1, c2 are positive constants. Likewise, we obtain∣∣ ∫ Ω vtf2(y, z)dx ∣∣ ≤ (p+ + 1) [ a ∫ Ω |vt||y + z|p(x) + b ∫ Ω |vt|.|z| p(x)−1 2 |y| p(x)+1 2 ] ≤ c2 [ ε‖vt‖22 + ‖∇y‖2p − 2 + ‖∇z‖2p − 2 + ‖∇y‖2p + 2 + ‖∇z‖2p + 2 ] + c2 [ ‖∇z‖3(p−−1) 2 + ‖∇z‖3(p+−1) 2 + ‖∇y‖ 3 2 (p−+1) 2 + ‖∇y‖ 3 2 (p++1) 2 ] . (3.25) Combining (3.24) and (3.25), we find sup (0,T ) ∫ t 0 (∣∣ ∫ Ω utf1(y, z)dx ∣∣+ ∣∣ ∫ Ω vtf2(y, z)dx ∣∣)ds ≤ εc2T‖(u, v)‖2WT×WT + c2T ( ‖(y, z)‖2p − WT×WT + ‖(y, z)‖2p + WT×WT + ‖(y, z)‖3(p−−1) WT×WT ) + c2T ( ‖(y, z)‖3(p+−1) WT×WT + ‖(y, z)‖ 3 2 (p−+1) WT×WT + ‖(y, z)‖ 3 2 (p++1) WT×WT ) . (3.26) 14 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 By substituting (3.26) into (3.23), we obtain, for some c3 > 0, 1 2 ‖(u, v)‖2WT×WT ≤ γ0 + εc3T‖(u, v)‖2WT×WT + c3T ( ‖(y, z)‖2p − WT×WT + ‖(y, z)‖2p + WT×WT + ‖(y, z)‖3(p−−1) WT×WT ) + c3T ( ‖(y, z)‖3(p+−1) WT×WT + ‖(y, z)‖ 3 2 (p−+1) WT×WT + ‖(y, z)‖ 3 2 (p++1) WT×WT ) . (3.27) Choosing ε such that εc3T = 1/4 and recalling that ‖(y, z)‖WT×WT ≤ d, for some d > 1, inequality (3.27) implies, for some c4 > 0, ‖(u, v)‖2WT×WT ≤ 4γ0 + 4c4T ( ‖(y, z)‖2p − WT×WT + ‖(y, z)‖2p + WT×WT + ‖(y, z)‖3(p−−1) WT×WT ) + 4c4T ( ‖(y, z)‖3(p+−1) WT×WT + ‖(y, z)‖ 3 2 (p−+1) WT×WT + ‖(y, z)‖ 3 2 (p++1) WT×WT ) ≤ 4γ0 + c4d 3(p+−1)T ≤ 4γ0 + c4d 3(p+−1)T0. Hence, if we take (d, T0) such that d2 >> 4γ0 and T0 < d2−4γ0 c4d3(p+−1) , we obtain ‖(u, v)‖2WT×WT ≤ ‖(u, v)‖2WT0 ×WT0 ≤ d2, which implies (u, v) ∈ B(0, d). Thus, G : B(0, d)→ B(0, d). Next, we show that G : B(0, d) → B(0, d) is a contraction. For this, let (y1, z1) and (y2, z2) be in B(0, d) and set (u1, v1) = G(y1, z1) and (u2, v2) = G(y2, z2). Then (u, v) = (u1 − u2, v1 − v2) is a weak solution of the problem utt − div(A∇u) + (|u1t|m(x)−2u1t − |u2t|m(x)−2u2t) = f1(y1, z1)− f1(y2, z2) in Ω× (0, T ), vtt − div(B∇v) + (|v1t|r(x)−2v1t − |v2t|r(x)−2v2t) = f2(y1, z1)− f2(y2, z2) in Ω× (0, T ), u = v = 0 on ∂Ω× (0, T ), (u(0), v(0)) = (ut(0), vt(0)) = (0, 0) in Ω, (3.28) in the sense of Definition 3.1. Therefore, by taking Φ = ut (in Definition 3.1) and integrating the result over (0, t), we obtain, for a.e. t ≤ T , d dt [ ‖ut‖22 + ∫ Ω A∇u · ∇u ] − ∫ Ω A′∇u · ∇u+ 2 ∫ Ω ut ( |u1t|m(x)−2u1t − |u2t|m(x)−2u2t ) = 2 ∫ Ω ut(f1(y1, z1)− f1(y2, z2))dx. (3.29) Integrating over (0, t) and using the initial conditions, the assumptions (1.7), (1.8) and inequality (3.3), we arrive at ‖ut‖22 + a0‖∇u‖22 ≤ 2 ∫ t 0 ∫ Ω ut(f1(y1, z1)− f1(y2, z2)) dx ds, EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 15 for all t ∈ (0, T ). Consequently, ‖u‖2WT ≤ C sup (0,T ) ∫ t 0 ∫ Ω |ut| |f1(y1, z1)− f1(y2, z2)| dx ds, (3.30) where C = 2/min{1, a0}. By repeating the same computations with Ψ = vt, in the second equation in Definition 3.1, we obtain ‖v‖2WT ≤ C sup (0,T ) ∫ t 0 ∫ Ω |vt| |f2(y1, z1)− f2(y2, z2)| dx ds, (3.31) where C = 2/min{1, b0}. Exploiting Young’s inequality (2.1), estimates (3.30) and (3.31), we arrive at ‖u‖2WT ≤ εCT‖u‖2WT + Cε sup (0,T ) ∫ t 0 ∫ Ω |f1(y1, z1)− f1(y2, z2)|2 dx ds, ‖v‖2WT ≤ εCT‖v‖2WT + Cε sup (0,T ) ∫ t 0 ∫ Ω |f2(y1, z1)− f2(y2, z2)|2 dx ds. By addition and choosing ε small enough, we obtain ‖(u, v)‖2WT×WT ≤ Cε sup (0,T ) ∫ t 0 ∫ Ω [|f1(y1, z1)− f1(y2, z2)|2 + |f2(y1, z1)− f2(y2, z2)|2] dx ds. (3.32) Now, we set Y = y1 − y2, Z = z1 − z2 and estimate∫ Ω |f1(y1, z1)− f1(y2, z2)|2dx and ∫ Ω |f2(y1, z1)− f2(y2, z2)|2dx. For this purpose, we recall inequalities (3.16) and (3.17) to obtain the following estimates satisfied by f1 and f2, respectively (as in [2]). |f1(y1, z1)− f1(y2, z2)| ≤ C4(|y1 − y2|+ |z1 − z2|) ( |y1|p(x)−1 + |z1|p(x)−1 + |y2|p(x)−1 + |z2|p(x)−1 ) + C5|z1 − z2|.|y1| p(x)−1 2 ( |z1| p(x)−1 2 + |z2| p(x)−1 2 ) + C5|y1 − y2|.|z2| p(x)+1 2 ( |y1| p(x)−3 2 + |y2| p(x)−3 2 ) , (3.33) and |f2(y1, z1)− f2(y2, z2)| ≤ C4(|y1 − y2|+ |z1 − z2|) ( |y1|p(x)−1 + |z1|p(x)−1 + |y2|p(x)−1 + |z2|p(x)−1 ) + C5|y1 − y2|.|z1| p(x)−1 2 ( |y1| p(x)−1 2 + |y2| p(x)−1 2 ) + C5|z1 − z2|.|y2| p(x)+1 2 ( |z1| p(x)−3 2 + |z2| p(x)−3 2 ) , (3.34) for some constants C4, C5 > 0 and for almost all x ∈ Ω and t ∈ (0, T ). So,∫ Ω |f1(y1, z1)− f1(y2, z2)|2dx ≤ I1 + I2 + I3 + I4, (3.35) 16 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 where I1 = C4 ∫ Ω |y1 − y2|2(|y1|2(p(x)−1) + |z1|2(p(x)−1))dx + C4 ∫ Ω |y1 − y2|2(|y2|2(p(x)−1) + |z2|2(p(x)−1))dx, I2 = C4 ∫ Ω |z1 − z2|2(|y1|2(p(x)−1) + |z1|2(p(x)−1))dx + C4 ∫ Ω |z1 − z2|2(|y2|2(p(x)−1) + |z2|2(p(x)−1))dx, I3 = C5 ∫ Ω |z1 − z2|2|y1|p(x)−1 ( |z1|p(x)−1 + |z2|p(x)−1 ) dx, I4 = C5 ∫ Ω |y1 − y2|2|z2|p(x)+1 ( |y1|p(x)−3 + |y2|p(x)−3 ) dx. By using Hölder’s and Young’s inequalities (Lemmas 2.1 and 2.2) and the Sobolev embeddings (Lemma 2.5), we obtain the following estimate for a typical term in I1 and I2,∫ Ω |y1 − y2|2|y1|2(p(x)−1)dx ≤ 2 (∫ Ω |y1 − y2|6dx )1/3(∫ Ω |y1|3(p(x)−1) )2/3 ≤ C‖y1 − y2‖26 [( ∫ Ω |y1|3(p+−1)dx )2/3 + (∫ Ω |y1|3(p−−1)dx )2/3] ≤ C‖∇(y1 − y2)‖22 ( ‖y1‖2(p+−1) 3(p+−1) + ‖y1‖2(p−−1) 3(p−−1) ) ≤ C‖∇Y ‖22 ( ‖∇y1‖2(p+−1) 2 + ‖∇y1‖2(p−−1) 2 ) ≤ C‖∇Y ‖22 ( ‖(y1, z1)‖2(p+−1) WT×WT + ‖(y1, z1)‖2(p−−1) WT×WT ) , (3.36) since 1 < 3(p− − 1) ≤ 3(p+ − 1) < ∞, when n = 1, 2; and 1 < 3(p− − 1) = 3(p+ − 1) = 6 = 2n n−2 , when n = 3. Similarly, we obtain∫ Ω |z1 − z2|2|y2|2(p(x)−1)dx ≤ C‖∇Z‖22 ( ‖(y2, z2)‖2(p+−1) WT×WT + ‖(y2, z2)‖2(p−−1) WT×WT ) . (3.37) Since (y1, z1), (y2, z2) ∈ B(0, d) and d > 1, estimates (3.36) and (3.37) lead to I1 ≤ Cd2(p+−1)‖∇Y ‖22 and I2 ≤ Cd2(p+−1)‖∇Z‖22. Hence, I1 + I2 ≤ Cd2(p+−1)(‖∇Y ‖22 + ‖∇Z‖22). (3.38) Similarly, a typical term in I3 can be handled as follows:∫ Ω |z1 − z2|2|y1|p(x)−1|z1|p(x)−1dx ≤ 2 (∫ Ω |z1 − z2|6dx )1/3(∫ Ω |y1| 3 2 (p(x)−1)|z1| 3 2 (p(x)−1) )2/3 EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 17 ≤ C‖z1 − z2‖26 [( ∫ Ω |y1|3(p(x)−1)dx )2/3 + (∫ Ω |z1|3(p(x)−1)dx )2/3] ≤ C‖∇(z1 − z2)‖22 ( ‖y1‖2(p+−1) 3(p+−1) + ‖y1‖2(p−−1) 3(p−−1) + ‖z1‖2(p+−1) 3(p+−1) + ‖z1‖2(p−−1) 3(p−−1) ) ≤ C‖∇(z1 − z2)‖22 ( ‖∇y1‖2(p+−1) 2 + ‖∇y1‖2(p−−1) 2 ) + C‖∇(z1 − z2)‖22 ( ‖∇z1‖2(p+−1) 2 + ‖∇z1‖2(p−−1) 2 ) ≤ 2C‖∇Z‖22 ( ‖(y1, z1)‖2(p+−1) WT×WT + ‖(y1, z1)‖2(p−−1) WT×WT ) . since 1 < 3(p− − 1) ≤ 3(p+ − 1) < ∞, when n = 1, 2; and 1 < 3(p− − 1) = 3(p+ − 1) = 6 = 2n n−2 , when n = 3. Therefore, I3 ≤ Cd2(p+−1)‖∇Z‖22. (3.39) Using the same arguments, we estimate a typical term in I4, as follows: Case 1. If n = 1, 2, we have 3 ≤ p− ≤ p+ <∞. So∫ Ω |y1 − y2|2|z2|p(x)+1|y1|p(x)−3dx ≤ 2 (∫ Ω |y1 − y2|3dx )2/3(∫ Ω |z2|3(p(x)+1)|y1|3(p(x)−3) )1/3 ≤ C‖y1 − y2‖23 [( ∫ Ω |z2|6(p(x)+1)dx )1/3 + (∫ Ω |y1|6(p(x)−3)dx )1/3] ≤ C‖∇Y ‖22 ( ‖∇z2‖2(p++1) 2 + ‖∇z2‖2(p−+1) 2 + ‖∇y1‖2(p+−3) 2 + ‖∇y1‖2(p−−3) 2 ) ≤ 4Cd2(p++1)‖∇Y ‖22, since (y1, z1), (y2, z2) ∈ B(0, d) and d > 1. Case 2. If n = 3, then p ≡ 3 on Ω and, hence,∫ Ω |y1 − y2|2|z2|p(x)+1|y1|p(x)−3dx = ∫ Ω |y1 − y2|2|z2|4dx ≤ C (∫ Ω |y1 − y2|6dx )1/3(∫ Ω |z2|6dx )2/3 ≤ C‖y1 − y2‖26.‖z2‖46 ≤ C‖∇Y ‖22.‖(y2, z2)‖4WT×WT . We deduce that I4 ≤ Cd2(p++1)‖∇Y ‖22. (3.40) Finally, by substituting (3.40), (3.39) and ((3.38) in (3.35), we arrive at∫ Ω |f1(y1, z1)− f1(y2, z2)|2dx ≤ Cd2(p++1)(‖∇Y ‖22 + ‖∇Z‖22), (3.41) for all t ∈ (0, T ). Similarly, we obtain∫ Ω |f2(y1, z1)− f2(y2, z2)|2dx ≤ Cd2(p++1)(‖∇Y ‖22 + ‖∇Z‖22). (3.42) 18 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 Now, we replace (3.41) and (3.42) in (3.32) to obtain ‖(u, v)‖2WT×WT ≤ Cεd2(p++1) sup (0,T ) ∫ t 0 ( ‖∇Y (s)‖22 + ‖∇Z(s)‖22 ) ds ≤ Cεd2(p++1)T‖(Y,Z)‖2WT×WT ≤ γT0‖(Y,Z)‖2WT×WT , where γ = Cεd 2(p++1). So, if we take T0 small enough, we obtain, for 0 < k < 1, ‖(u, v)‖2WT×WT ≤ k‖(Y,Z)‖2WT×WT . Thus, ‖G(y1, z1)−G(y2, z2)‖2WT×WT ≤ k‖(y1, z1)− (y2, z2)‖2WT×WT . This proves that G : B(0, d) → B(0, d) is a contraction. The Banach-fixed-point theorem guarantees the existence of a unique (u, v) ∈ B(0, d), such that G(u, v) = (u, v), which is obviously a local weak solution of (1.1). Uniqueness. Suppose that (1.1) has two solutions (u1, v1) and (u2, v2), in the sense of Definition 3.1. Therefore, (u, v) = (u1 − u2, v1 − v2) satisfies the problem utt − div(A∇u) + (|u1t|m(x)−2u1t − |u2t|m(x)−2u2t) = f1(u1, v1)− f1(u2, v2) in Ω× (0, T ), vtt − div(B∇v) + (|v1t|r(x)−2v1t − |v2t|r(x)−2v2t) = f2(u1, v1)− f2(u2, v2) in Ω× (0, T ), u = v = 0 on ∂Ω× (0, T ), (u(0), v(0)) = (ut(0), vt(0)) = (0, 0) in Ω. By taking (Φ,Ψ) = (ut, vt) in this definition, integrating each equation over (0, t) (t ≤ T ) and adding the two results, we obtain (as in (3.30) and (3.31)) the following ‖(ut, vt)‖22 + ‖(∇u,∇v)‖22 ≤ C ∫ t 0 ∫ Ω |ut||f1(u1, v1)− f1(u2, v2)| dx dt + C ∫ t 0 ∫ Ω |vt||f2(u1, v1)− f2(u2, v2)| dx dt. Under assumption (1.6) and applying similar arguments as in above, we arrive at ‖(ut, vt)‖22 + ‖(∇u,∇v)‖22 ≤ Cε ∫ t 0 (‖(ut(s), vt(s))‖22 + ‖(∇u(s),∇v(s))‖22)ds, for all t ∈ (0, T ). Gronwall’s lemma leads to ‖(ut, vt)‖22 + ‖(∇u,∇v)‖22 = 0, for all t ∈ (0, T ). Thanks to the boundary conditions, we obtain u = v = 0 on Ω×(0, T ). This proves the uniqueness of the solution of (1.1). � Theorem 3.3 is a generalization of the local existence of Agre and Rammaha [2], which dealt with constant exponents, to the situation of variable exponents. EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 19 4. Blow up result To prove our main blow-up result, we give some Lemmas. First we derive the energy functional associated with the problem. E(t) = 1 2 (‖ut‖22 + ‖vt‖22) + 1 2 ∫ Ω (A∇u · ∇u+B∇v · ∇v)dx − ∫ Ω F (x, u, v)dx, (4.1) for all t ∈ [0, T ), and α1 = (k(p− + 1)) 1 1−p− , E1 = ( 1 2 − 1 p− + 1 )α2 1, (4.2) where k = ( a2 p−+1 2 + 2b )( B̃2 c0 ) p−+1 2 , c0 = min{a0, b0} > 0 and B̃ is the best constant of the Sobolev embedding H1 0 (Ω) ↪→ Lp(·)+1(Ω). 4.1. Lemmas. Lemma 4.1 ([6]). The energy functional E is decreasing function and, for a.e. t ∈ (0, T ), we have E′(t) = − ∫ Ω |ut|m(x)dx− ∫ Ω |vt|r(x)dx+ 1 2 ∫ Ω (A′∇u · ∇u+B′∇v · ∇v)dx. (4.3) Lemma 4.2. [16] (1) 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). (4.4) (2) For all x ∈ Ω and (u, v) ∈ R2, we have u f1(x, u, v) + vf2(x, u, v) = (p(x) + 1)F (x, u, v). (4.5) Lemma 4.3. For any solution (u, v) of the system (1.1), with initial energy E(0) < E1 (4.6) and α1 < (∫ Ω (A∇u0 · ∇u0 +B∇v0 · ∇v0)dx )1/2 ≤ ( c0 2B̃2 )1/2, there exists α2 > α1 such that α2 ≤ (∫ Ω (A∇u · ∇u+B∇v · ∇v)dx )1/2 , ∀t ∈ [0, T ). (4.7) Proof. From the definition of the energy, it results that E(t) ≥ 1 2 ∫ Ω (A∇u · ∇u+B∇v · ∇v)dx− ∫ Ω F (x, u, v)dx. If we set α = (∫ Ω (A∇u · ∇u+B∇v · ∇v)dx )1/2 (4.8) 20 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 then E(t) ≥ 1 2 α2 − ∫ Ω F (x, u, v)dx. (4.9) From (1.7), we have ‖∇u‖22 + ‖∇v‖22 ≤ 1 c0 ∫ Ω (A∇u · ∇u+B∇v · ∇v)dx. So ‖∇u‖22 + ‖∇v‖22 ≤ α2 c0 . (4.10) On the other hand, by the definition of F, we have∫ Ω F (x, u, v) = a ∫ Ω |u+ v|p(x)+1dx+ 2b ∫ Ω |uv| p(x)+1 2 dx. Invoking Lemma 2.3, this leads to∫ Ω F (x, u, v) ≤ amax { ‖u+ v‖p −+1 p(·)+1, ‖u+ v‖p ++1 p(·)+1 } + 2bmax { ‖uv‖ p−+1 2 p(·)+1 2 , ‖uv‖ p++1 2 p(·)+1 2 } . (4.11) First, by the embedding (Lemma 2.5), we have ‖u+ v‖p(·)+1 ≤ B̃‖∇(u+ v)‖ 2 ≤ B̃[(‖∇u‖2 + ‖∇v‖2)2]1/2. Since (X + Y )δ ≤ 2δ−1(Xδ + Y δ), for all X,Y ≥ 0 and δ ≥ 1, (4.12) it follows that ‖u+ v‖p(·)+1 ≤ B̃[2(‖∇u‖ 2 2 + ‖∇v‖22)]1/2. By (4.10), ‖u+ v‖p(·)+1 (2B̃2α2 c0 )1/2 . Hence, ‖u+ v‖p −+1 p(·)+1 ≤ (2B̃2α2 c0 ) p−+1 2 , ‖u+ v‖p ++1 p(·)+1 ≤ ( 2B̃2α2 c0 ) p++1 2 . Therefore, max { ‖u+ v‖p −+1 p(·)+1, ‖u+ v‖p ++1 p(·)+1 } ≤ max {(2B̃2α2 c0 ) p−+1 2 , (2B̃2α2 c0 ) p++1 2 } . (4.13) Likewise, Hölder’s and Young’s inequalities (Lemmas 2.1 and 2.2) give ‖uv‖ p(·)+1 2 ≤ 2‖u‖p(·)+1‖v‖p(·)+1 ≤ ‖u‖2p(·)+1 + ‖v‖2p(·)+1 ≤ B̃ 2(‖∇u‖22 + ‖∇v‖22). Again, by (4.10), we find that ‖uv‖ p(·)+1 2 ≤ B̃2α2 c0 . So, we have ‖uv‖ p−+1 2 p(·)+1 2 ≤ ( B̃2α2 c0 ) p−+1 2 , ‖uv‖ p++1 2 p(·)+1 2 ≤ ( B̃2α2 c0 ) p++1 2 . EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 21 Therefore, max { ‖uv‖ p−+1 2 p(·)+1 2 , ‖uv‖ p++1 2 p(·)+1 2 } ≤ max {( B̃2α2 c0 ) p−+1 2 , ( B̃2α2 c0 ) p++1 2 } . (4.14) Substituting (4.13) and (4.14) in (4.11), we infer that∫ Ω F (x, u, v) ≤ amax {(2B̃2α2 c0 ) p−+1 2 , (2B̃2α2 c0 ) p++1 2 } + 2bmax {( B̃2α2 c0 ) p−+1 2 , ( B̃2α2 c0 ) p++1 2 } . (4.15) By inserting (4.15) into (4.9), we obtain E(t) ≥ h(α), for all α ≥ 0, (4.16) where h(α) := 1 2 α2 − amax {(2B̃2α2 c0 ) p−+1 2 , (2B̃2α2 c0 ) p++1 2 } − 2bmax {( B̃2α2 c0 ) p−+1 2 , ( B̃2α2 c0 ) p++1 2 } . For α in [0, ( c0 2B̃2 )1/2], one can easily check that B̃2α2 c0 ≤ 2B̃2α2 c0 ≤ 1. Consequently,(2B̃2α2 c0 ) p−+1 2 ≥ (2B̃2α2 c0 ) p++1 2 , ( B̃2α2 c0 ) p−+1 2 ≥ ( B̃2α2 c0 ) p++1 2 . Thus, (4.16) leads to E(t) ≥ 1 2 α2 − ( a2 p−+1 2 + 2b )( B̃2 c0 ) p−+1 2 αp −+1. That is, E(t) ≥ g(α), for all α ∈ [ 0, ( c0 2B̃2 )1/2] , (4.17) where g(α) = 1 2 α2 − kαp −+1. It is easy to verify that g is strictly increasing on [0, α1) and strictly decreasing on [α1,+∞). Since E(0) < E1 and E1 = g(α1), then, we can find α2 > α1 such that g(α2) = E(0). But α0 = (∫ Ω (A∇u0 · ∇u0 +B∇v0 · ∇v0)dx )1/2 ∈ [ α1, ( c0 2B̃2 )1/2] , therefore, by (4.17), we obtain g(α2) = E(0) ≥ g(α0). This implies α0 ≥ α2. Consequently α2 ∈ (α1, ( c0 2B̃2 )1/2]. To establish (4.7), we suppose on the contrary that(∫ Ω (A∇u(., t∗).∇u(., t∗) +B∇v(., t∗).∇v(., t∗))dx )1/2 < α2, 22 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 for some t∗ ∈ [0, T ). By the continuity of ( ∫ Ω A∇u · ∇u + B∇v · ∇vdx )1/2 and since α2 > α1, we can choose t∗ such that[ ∫ Ω (A∇u(., t∗).∇u(., t∗) +B∇v(., t∗).∇v(., t∗))dx ]1/2 > α1. The g being decreasing of on [α1, ( c0 2B̃2 )1/2] and (4.17) imply that E(t∗) ≥ g ([ ∫ Ω (A∇u(., t∗).∇u(., t∗) +B∇v(., t∗).∇v(., t∗))dx ]1/2) > g(α2) = E(0). This is impossible since E(t) ≤ E(0), for all t ∈ [0.T ). Thus, (4.17) is established. � Now, we set H(t) = E1 − E(t), for all t ∈ [0, T ). (4.18) Lemma 4.4. We have 0 < H(0) ≤ H(t) ≤ ∫ Ω F (x, u, v)dx, for all t ∈ [0, T ), (4.19)∫ Ω F (x, u, v)dx ≥ kαp −+1 2 . (4.20) Proof. From Lemma 4.1 and assumption (4.6), we have 0 < E1 − E(0) = H(0) ≤ H(t) (4.21) and by (4.9), we obtain H(t) ≤ E1 − 1 2 α2 + ∫ Ω F (x, u, v)dx. Since E1 = g(α1) and α ≥ α2 > α1, we obtain H(t) ≤ ( g(α1)− 1 2 α2 1 ) + ∫ Ω F (x, u, v)dx ≤ −kαp −+1 1 + ∫ Ω F (x, u, v)dx ≤ ∫ Ω F (x, u, v)dx. Thus, (4.19) is established. To prove (4.20), we note that E is nonincreasing. Hence, E(0) ≥ E(t) ≥ 1 2 α2 − ∫ Ω F (x, u, v)dx. Consequently, ∫ Ω F (x, u, v)dx ≥ 1 2 α2 − E(0). But E(0) = g(α2) and α ≥ α2, so∫ Ω F (x, u, v)dx > 1 2 α2 2 − g(α2) = kαp −+1 2 . � EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 23 In what follows and for simplicity, we denote ρ(u) = ∫ Ω |u |p(x)+1dx, ρ(v) = ∫ Ω |v |p(x)+1dx and we define Ω+ = {x ∈ Ω : |u(x, t)| ≥ 1}, Ω− = {x ∈ Ω : |u(x, t)| < 1}. Lemma 4.5. There exists C3 > 0 such that any solution of (1.1) satisfies ‖u‖p −+1 p−+1 + ‖v‖p −+1 p−+1 ≤ C3(ρ(u) + ρ(v)). (4.22) Proof. Since p− ≤ p(·) ≤ p+, one easily sees that ρ(u) = ∫ Ω+ |u|p(x)+1dx + ∫ Ω− |u|p(x)+1dx ≥ ∫ Ω+ |u|p −+1dx + ∫ Ω− |u|p ++1dx ≥ ∫ Ω+ |u|p −+1dx + c1( ∫ Ω− |u|p −+1dx) p++1 p−+1 , for some c1 > 0. Thus, ρ(u) ≥ ∫ Ω+ |u|p −+1dx and (ρ(u) c1 ) p−+1 p++1 ≥ ∫ Ω− |u|p −+1dx. By addition, for some c2 > 0, we obtain ‖u‖p −+1 p−+1 ≤ ρ(u) + c2(ρ(u)) p−+1 p++1 ≤ ρ(u) + ρ(v) + c2(ρ(u) + ρ(v)) p−+1 p++1 = (ρ(u) + ρ(v)) [ 1 + c2(ρ(u) + ρ(v)) p−− p+ p++1 ] . Recalling (4.19) and (4.4), we infer that 0 < H(0) ≤ H(t) ≤ C2(ρ(u) + ρ(v)), (4.23) then ρ(u) + ρ(v) ≥ H(0)/C2. Therefore, ‖u‖p −+1 p−+1 ≤ (ρ(u) + ρ(v)) [ 1 + c2(H(0)/C2) p−− p+ p++1 ] . Hence ‖u‖p −+1 p−+1 ≤ c3(ρ(u) + ρ(v)), where c3 = 1 + c2(H(0)/C2) p−− p+ p++1 > 0. Similarly, we arrive at ‖v‖p −+1 p−+1 ≤ c3(ρ(u) + ρ(v)). Therefore, (4.22) is satisfied with C3 = 2c3. � Corollary 4.6. There exist constants C4, C5 > 0 such that∫ Ω |u|m(x)dx ≤ C4 [ (ρ(u) + ρ(v)) m+ p−+1 + (ρ(u) + ρ(v)) m− p−+1 ] , (4.24)∫ Ω |v|r(x)dx ≤ C5 [ (ρ(u) + ρ(v)) r+ p−+1 + (ρ(u) + ρ(v)) r− p−+1 ] . (4.25) 24 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 Proof. Since p− ≥ max{m+, r+}, it follows that∫ Ω |u|m(x)dx ≤ ∫ Ω+ |u|m + dx + ∫ Ω− |u|m − dx ≤ c1 ( ∫ Ω+ |u|p −+1dx ) m+ p−+1 + c1 (∫ Ω− |u|p −+1dx ) m− p−+1 ≤ c1 ( ‖u‖m + p−+1 + ‖u‖m − p−+1 ) , c1 > 0. Recalling Lemma 4.5, we obtain, for a constant C4 > 0,∫ Ω |u|m(x)dx ≤ C4 [ (ρ(u) + ρ(v)) m+ p−+1 + (ρ(u) + ρ(v)) m− p−+1 ] . Similarly, we obtain, for some C5 > 0,∫ Ω |v|r(x)dx ≤ C5 [ (ρ(u) + ρ(v)) r+ p−+1 + (ρ(u) + ρ(v)) r− p−+1 ] . Thus, (4.23) and (4.24) are proved. � 4.2. Main result. Now, we state and prove our main blow-up result. Theorem 4.7. Let the assumptions given in Subsection 4.1 hold. Then any solution of the system (1.1) blows up in finite time. Proof. We assume that the solution exists for any t > 0 and reach to a contradiction. This will be established in 4 steps. Step 1. For small ε > 0 to be fixed later, we define the auxiliary functional G(t) = H1−σ(t) + ε ∫ Ω (uut + vvt)dx, t > 0, where 0 < σ ≤ min { p− −m+ + 1 (p− + 1)(m+ − 1) , p− − r+ + 1 (p− + 1)(r+ − 1) , p− − 1 2(p− + 1) } . (4.26) Our goal is to show that G satisfies a differential inequality which leads to a blow up in finite time. Now, we have G′(t) = (1− σ)H−σ(t)H ′(t) + ε(‖ut‖22 + ‖vt‖22) + ε ∫ Ω (uf1(x, u, v) + vf2(x, u, v))dx − ε ∫ Ω (A∇u · ∇u+B∇v · ∇v)dx − ε ∫ Ω (|ut|m(x)−2utu+ |vt|r(x)−2vtv)dx. (4.27) From Lemma 4.2, it follows that∫ Ω (uf1(x, u, v) + vf2(x, u, v))dx = ∫ Ω (p(x) + 1)F (x, u, v)dx ≥ (p− + 1) ∫ Ω F (x, u, v)dx. (4.28) EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 25 By the definition of H and E, we obtain∫ Ω (A∇u · ∇u+B∇v · ∇v)dx = 2 ∫ Ω F (x, u, v)dx− ‖ut‖22 − ‖vt‖22 + 2E1 − 2H(t). (4.29) If we insert (4.28) and (4.29) into (4.27), we then obtain G′(t) ≥ (1− σ)H−σ(t)H ′(t) + 2ε(‖ut‖22 + ‖vt‖22) + 2εH(t) − 2εE1 + ε(p− − 1) ∫ Ω F (x, u, v)dx − ε ∫ Ω (|u||ut|m(x)−1 + |v||vt|r(x)−1)dx. (4.30) Using (4.20), we have E1 ≤ (kαp −+1 2 )−1E1 ∫ Ω F (x, u, v)dx. Hence, (4.30) becomes G′(t) ≥ (1− σ)H−σ(t)H ′(t) + 2ε(‖ut‖22 + ‖vt‖22) + εc1 ∫ Ω F (x, u, v)dx + 2εH(t)− ε ∫ Ω (|u||ut|m(x)−1 + |v| |vt|r(x)−1)dx, (4.31) where c1 = p− − 1− 2(kαp −+1 2 )−1E1 > 0, since α2 > α1. Step 2. In this step, we estimate the last two terms in the right-hand side of (4.31). We set I1 := ∫ Ω |u||ut|m(x)−1dx, I2 := ∫ Ω |v||vt|r(x)−1dx and apply the Young inequality XY ≤ δλ λ Xλ + δ−β β Y β , for all X, Y ≥ 0, δ > 0 and 1 λ + 1 β = 1, with X = |u|, Y = |ut|m(x)−1, λ = m(x), β = m(x) m(x)− 1 , δ > 0, to obtain I1 ≤ ∫ Ω δm(x) m(x) |u|m(x)dx+ ∫ Ω m(x)− 1 m(x) δ−m(x)/(m(x)−1)|ut|m(x)dx. (4.32) By taking δ = [KH−σ(t)] 1−m(x) m(x) , where K is a large constant to be chosen later, we obtain I1 ≤ K1−m− m− ∫ Ω [H(t)]σ(m(x)−1)|u|m(x)dx + m+ − 1 m− KH−σ(t) ∫ Ω |ut|m(x)dx. (4.33) 26 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 From Lemma 4.1, we have H ′(t) = ∫ Ω |ut|m(x)dx+ ∫ Ω |vt|r(x)dx− 1 2 ∫ Ω (A′∇u · ∇u+B′∇v · ∇v)dx and by (1.8)), we obtain ∫ Ω |ut|m(x)dx ≤ H ′(t). (4.34) On the other hand, since m(x) ≤ m+ and H(t) ≥ H(0) > 0, one has∫ 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 x ∈ Ω, where c3 > 0. So, for a constant c4 > 0, we obtain∫ Ω [H(t)]σ(m(x)−1)|u|m(x)dx ≤ c4[H(t)]σ(m+−1) ∫ Ω |u|m(x)dx. (4.35) Replacing (4.35) and (4.34)) in (4.33), we infer that I1 ≤ K1−m− m− c4[H(t)]σ(m+−1) ∫ Ω |u|m(x)dx + m+ − 1 m− KH−σ(t)H ′(t). (4.36) Likewise, we obtain, for some c5 > 0, I2 ≤ K1−r− r− c5[H(t)]σ(r+−1) ∫ Ω |v|r(x)dx + r+ − 1 r− KH−σ(t)H ′(t). (4.37) Also, from (4.19), we have, for some c6 > 0, [H(t)]σ(m+−1) ≤ c6(ρ(u) + ρ(v))σ(m+−1). This inequality and estimate (4.24) imply that for some c7 > 0, [H(t)]σ(m+−1) ∫ Ω |u|m(x)dx ≤ c7(ρ(u) + ρ(v)) σ(m+−1)+ m+ p−+1 + c7(ρ(u) + ρ(v)) σ(m+−1)+ m− p−+1 . (4.38) Now, if we use (4.26) and the algebraic inequality zτ ≤ z + 1 ≤ (1 + 1 a )(z + a), for all z ≥ 0, 0 < τ ≤ 1 and a > 0, (4.39) with z = ρ(u) + ρ(v), a = H(0), τ = σ(m+ − 1) + m+ p− + 1 and then with τ = σ(m+ − 1) + m− p−+1 , respectively, we obtain (ρ(u) + ρ(v)) σ(m+−1)+ m+ p−+1 ≤ [1 + 1 H(0) ](ρ(u) + ρ(v) +H(0)) ≤ γ(ρ(u) + ρ(v) +H(t)) (4.40) EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 27 and (ρ(u) + ρ(v)) σ(m+−1)+ m− p−+1 ≤ γ(ρ(u) + ρ(v) +H(t)) (4.41) where γ = 1 + 1 H(0) . Combining (4.41) and (4.40) with (4.38), we obtain that for some c8 > 0, [H(t)]σ(m+−1) ∫ Ω |u|m(x)dx ≤ c8(ρ(u) + ρ(v) +H(t)). (4.42) Similarly, we have for some c9 > 0, [H(t)]σ(r+−1) ∫ Ω |v|r(x)dx ≤ c9(ρ(u) + ρ(v) +H(t)). (4.43) Substituting (4.42) into (4.36), we find that I1 ≤ K1−m− m− c10(ρ(u) + ρ(v) +H(t)) + m+ − 1 m− KH−σ(t)H ′(t), (4.44) and substituting (4.43) into (4.37), we obtain I2 ≤ K1−r− r− c11(ρ(u) + ρ(v) +H(t)) + r+ − 1 r− KH−σ(t)H ′(t), (4.45) where c10 and c11 are two positive constants. Step 3. Now, we estimate G′. By inserting (4.44) and (4.45) into (4.31), we arrive at G′(t) ≥ (1− σ − εM)H−σ(t)H ′(t) + 2ε(‖ut‖22 + ‖vt‖22) + 2εH(t) + c12(ρ(u) + ρ(v))− εK 1−m− m− c10(ρ(u) + ρ(v) +H(t)) − εK 1−r− r− c11(ρ(v) + ρ(u) +H(t)), where c12 > 0 and M = K(m +−1 m− + r+−1 r− ). Therefore, G′(t) ≥ (1− σ − εM)H−σ(t)H ′(t) + 2ε(‖ut‖22 + ‖vt‖22) + ε ( 2− K1−m− m− c10 − K1−r− r− c11 ) H(t) + ε ( c12 − K1−m− m− c10 − K1−r− r− c11 ) (ρ(u) + ρ(v)). For a large value of K, we can find c13 > 0 such that G′(t) ≥ (1− σ − εM)H−σ(t)H ′(t) + εc13(‖ut‖22 + ‖vt‖22 +H(t) + ρ(u) + ρ(v)). Once K is fixed, we select ε small enough so that 1− σ − εM ≥ 0 and G(0) = H1−σ(0) + ε ∫ Ω (u0u1 + v0v1)dx > 0. From Lemma 4.1, we have H ′(t) ≥ 0. Therefore, there exists h > 0 satisfying G′(t) ≥ εh(H(t) + ‖ut‖22 + ‖vt‖22 + ρ(u) + ρ(v)). (4.46) Consequently, G(t) ≥ G(0) > 0, for t > 0. 28 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 Step 4. Finally, we complete the proof of the blow up result. By the definition of G and using (4.12), it follows that G1/(1−σ)(t) ≤ ( H1−σ(t) + ε ∫ Ω |uut + vvt|dx )1/(1−σ) ≤ 2σ/(1−σ) ( H(t) + (ε ∫ Ω (|uut|+ |vvt|)dx)1/(1−σ) ) ≤ c14 ( H(t) + ( ∫ Ω (|u||ut|+ |v||vt|)dx)1/(1−σ) ) , where c14 = 2σ/(1−σ) max{1, ε1/(1−σ)}. Also, we have(∫ Ω (|u||ut|+ |v||vt|)dx )1/(1−σ) ≤ 2σ/(1−σ) (∫ Ω |u‖ut|dx )1/(1−σ) + 2σ/(1−σ) (∫ Ω |v||vt|dx )1/(1−σ) . (4.47) Since p− ≥ 2, Hölder’s and Young’s inequalities yield, for c15, c16 > 0,(∫ Ω |u‖ut|dx )1/(1−σ) ≤ ‖u‖1/(1−σ) 2 ‖ut‖1/(1−σ) 2 ≤ c15‖u‖1/(1−σ) p−+1 ‖ut‖1/(1−σ) 2 ≤ c16(‖u‖µ/(1−σ) p−+1 + ‖ut‖β/(1−σ) 2 ), where 1 µ + 1 β = 1. If we set β = 2(1−σ), we obtain µ/(1−σ) = 2/(1− 2σ). Hence,(∫ Ω |u‖ut|dx )1/(1−σ) ≤ c16(‖u‖2/(1−2σ) p−+1 + ‖ut‖ 2 2). (4.48) By Lemma 4.4, estimate (4.48) becomes(∫ Ω |u‖ut|dx )1/(1−σ) ≤ c17((ρ(u) + ρ(v))τ + ‖ut‖22), where c17 > 0 and τ = 2/(p−+1)(1−2σ). Again, by (4.26), (4.39) and since τ ≤ 1, we obtain, for some c18 > 0,(∫ Ω |u||ut|dx )1/(1−σ) ≤ c18(ρ(u) + ρ(v) +H(t) + ‖ut‖ 2 2). (4.49) Similar computations lead to( ∫ Ω |v||vt|dx )1/(1−σ) ≤ c18(ρ(u) + ρ(v) +H(t) + ‖vt‖ 2 2). (4.50) By adding (4.49) and (4.50), estimate (4.47) yields, for some c19 > 0,(∫ Ω (|u||ut|+ |v||vt|)dx )1/(1−σ) ≤ c19 ( ρ(u) + ρ(v) + ‖ut‖ 2 2 + ‖vt‖ 2 2 +H(t) ) . Therefore, for some c20 > 0, we arrive at G1/(1−σ)(t) ≤ c20(ρ(u) + ρ(v) +H(t) + ‖ut‖ 2 2 + ‖vt‖ 2 2). (4.51) Combining (4.51) and (4.46), we deduce that G′(t) ≥ ΓG1/(1−σ)(t), for all t > 0, EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 29 where Γ = εh c20 . A simple integration over (0, t) yields Gσ/(1−σ)(t) ≥ 1 G −σ 1−σ (0)− σΓt 1−σ , which implies that G(t) → +∞, as t → T ∗, where T ∗ ≤ 1−σ σΓ[G σ (1−σ) (0)] . Conse- quently, the solution of problem (1.1) blows up in finite time. � 5. Numerical tests In this section, we show some numerical experiments to illustrate the theoretical results in Theorem 4.7. We solve the system (1.1) under specific initial data and Dirichlet boundary conditions. We use a numerical scheme based on the finite element method in space and the Newmark method in time [24, 23]. We consider problem (1.1) in two space-dimensions and take the functions m, r and p fulfilling the assumptions (1.2), (1.3) and (1.6). Precisely, we have m(x, y) = 2 + 1 1 + x2 , r(x, y) = 2 + 1 1 + y2 , p(x, y) = 3 + 2 1 + x2 + y2 , and the source terms are given by (1.4) and (1.5) with a = b = 1. Whereas, the matrices A and B are given as follows A = (1 + e−t) ( 2 1 0 1 ) , B = (1 + 1 1 + t ) ( 3 0 1 2 ) . Test 1. We consider the circular domain Ω1 = {(x, y) : x2 + y2 < 1} with a triangulation discretization (see the mesh-grid in Figure 1) which consists of 281 triangles and 162 degrees of freedoms [20] and use the initial conditions u0(x, y) = 2(1− x2 − y2), v0(x, y) = 3(1− x2 − y2), u1 = v1 = 0. We run our code with a time step ∆t = 10−3, which is small enough to catch the below-up behavior. Figure 1. Uniform mesh grid of Ω1. Figure 2 shows the approximate numerical results of the solution (u, v) at differ- ent time iterations t = 0, t = 0.02, t = 0.023, and t = 0.024, where the left column shows the approximate values of u and the right column shows the approximate values of v. Note that the blow-up is occurring at instant t = 0.024. 30 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 (A) t = 0 (B) t = 0.02 (C) t = 0.023 (D) t = 0.024 Figure 2. Numerical results of Test 1 at different times. Figure 3. Test 1: Blow-up of H in finite time. Figure 3 presents the numerical values of the functional H(t) defined by (4.18) during the time iterations. It shows the blow-up of the energy of system (1.1). EJDE-2021/91 EXISTENCE AND BLOW UP FOR WAVE EQUATIONS 31 Figure 4. Uniform mesh grid of Ω2. (A) t = 0 (B) t = 0.02 (C) t = 0.0205 (D) t = 0.021 Figure 5. Numerical results of Test 2 at different times. Test 2. We consider the elliptical domain Ω2 = {(x, y) : x2 4 + y2 < 1} with a triangulation discretization (see the mesh-grid in Figure 4) which consists of 311 triangles and 180 degrees of freedom [20] and take the initial conditions u0(x, y) = 2(1− x2 4 − y2), v0(x, y) = 3(1− x2 4 − y2), u1 = v1 = 0. 32 S. A. MESSAOUDI, O. BOUHOUFANI, I. HAMCHI, M. ALAHYANE EJDE-2021/91 Figure 6. Test 2: Blow-up of H in finite time. We run our code with a time step ∆t = 5 · 10−4, which is small enough to catch the below-up behavior. In Figure 5, we show the approximate numerical results of the solution (u, v) at different time iterations t = 0, t = 0.02, t = 0.0205 and t = 0.021, where the left column shows the approximate values of u and the right column shows the approximate values of v. Note that the blow-up takes place at instant t = 0.021. For Test 2, the numerical values of the functional H(t) are presented in Figure 6. Observe the blow-up of the energy from t = 0.02. Acknowledgment. S. A. Messaoudi was supported by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah. Grant No. 2002144089, 2019-2020. References [1] Aboulaich, A.; Meskin, D.; Suissi, A.; New diffusion models in image processing, Comput. Math. App., 56 (4) (2008), 874–882. [2] Agre, K.; Rammaha, M.; Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235–1270. 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Box 27272 Sharjah, United Arab Emirates Email address: smessaoudi@sharjah.ac.ae Oulia Bouhoufani Department of Mathematics, University Batna-2, 05000 Batna, Algeria Email address: o.bouhoufani@univ-batna2.dz Ilhem Hamchi Department of Mathematics, University Batna-2, 05000 Batna, Algeria Email address: i.hamchi@univ-batna2.dz Mohamed Alahyane Department of Mathematics, RISE, University of Sharjah, P. O. Box 27272 Sharjah, United Arab Emirates Email address: malahyane@sharjah.ac.ae 1. Introduction 2. Preliminaries 3. Existence of weak solutions 4. Blow up result 4.1. Lemmas 4.2. Main result 5. Numerical tests Acknowledgment References