On wave equation: review and recent results

Arabian Journal of Mathematics, Nov 2017

The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs.

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On wave equation: review and recent results

On wave equation: review and recent results Salim A. Messaoudi 0 Ala A. Talahmeh 0 0 S. A. Messaoudi ( The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs. Mathematics Subject Classification 35L05 · 35L70 · 35B44 · 35B35 1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Decay in the case of constant exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Blowup in the case of constant exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Blowup in the case of variable-exponent nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 History of variable-exponent Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Variable-exponent Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Variable-exponent Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - Contents 4 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Literature review A considerable and great effort has been devoted to the study of linear and nonlinear wave equations in the case of constant and variable-exponent nonlinearities. Our aim here is to give an overview of the existing results and introduce some other ones. 1.1 Decay in the case of constant exponents There is an extensive literature on the stabilization of the wave equation by internal or boundary feedbacks. Zuazua [95] proved the exponential stability of the energy for the wave equation by a locally distributed internal feedback depending linearly on the velocity. Komornik [37] and Nakoa [72] extended the result of Zuazua [95] by considering the case of a nonlinear damping term with a polynomial growth near the origin. Martinez [53,54] studied a damped wave equation and used the piecewise multiplier technique combined with some nonlinear integral inequalities to establish explicit decay rate estimates. These decay estimates are not optimal for some cases including the case of the polynomial growth. The following initial-boundary value problem of the Kirchhoff equation with a general dissipation of the form ⎧ utt − φ ⎪⎨ u(x , t ) = 0, ⎪⎩ u(x , 0) = u0(x ), ut (x , 0) = u1(x ), |∇u|2 u + σ (t )g(ut ) = 0, in where is a bounded domain Rn (n ≥ 1), with a smooth boundary ∂ and φ, σ and g are given functions and the functions (u0, u1) are the given initial data, was considered by many authors in the literature. For instance, in the case when g = σ = 0, the one-dimensional equation of (1.1) was first introduced by Kirchhoff [35] in 1876, and then was called the Kirchhoff string after his name. When σ = 1, φ (r ) = r α (α ≥ 1) and g(x ) = τ x (τ > 0), problem (1.1) was treated by Nishihara and Yamada [73]. They proved the existence and uniqueness of a global solution and the polynomial decay for small data (u0, u1) ∈ (H01( ) ∩ H 2( )) × H 1( ) with 0 u0 = 0 . In [76], Ono extended the work of [73] to the case where φ (r ) = r and σ (t ) ≡ (1 + t )−δ, δ < 13 using the decay lemma of Nakao [69]. Benaissa and Guesmia [ 18 ] extended the results obtained by Ono [76] and proved an existence and uniqueness theorem of a global solution in Sobolev spaces to the problem (1.1) when φ (r ) = r, g(v) = v and general functions σ. Also, they obtained an explicit and general decay rate, depending on σ, g and φ, for the energy of solutions of (1.1) without any growth assumption on g and φ at the origin, and on σ at infinity. Also, the following problem ⎧ utt − u + g(ut ) + f (u) = 0, ⎪ ⎨ u(x , t ) = 0, ⎪⎩ u(x , 0) = u0(x ), ut (x , 0) = u1(x ), For instanicsea, binouthnedecdasreegwihoenninf R(un) (=n |≥u|1p)−,2wu,itgh(auts)m=oo|tuht |bmo−u2nudta,rmy,∂ p ,>w2a,sNcoanksaiode[7re0d] sbhyomwaendythaautth(1o.r2s). where has a unique global weak solution if 0 ≤ p − 2 ≤ 2/(n − 2), n ≥ 3 and a global unique strong solution if p −2 > 2/(n −2), n ≥ 3. In addition to global existence, the issue of the decay rate was also addressed. In both cases it has been shown that the energy of the solution decays algebraically if m > 2 and decays exponentially if m = 2. This improved an earlier result in [68], where Nakao studied the problem in an abstract setting and established a theorem concerning decay of the solution energy only for the case m − 2 ≤ 2/(n − 2), n ≥ 3. Also in a joint work, Nakao and Ono [71] extended this result to the Cauchy problem utt − u(x , 0) = u0(x ), ut (x , 0) = u1(x ), u + λ2(x )u + ρ(ut ) + f (u) = 0, in Rn, in Rn × (0, +∞) (1.1) (1.2) (1.3) where ρ(ut ) behaves like |ut |β ut and f (u) behaves like −bu|u|α. In this case the authors required that the initial data be small enough in the H 1 × L2 norm and with compact supports. In [56] Messaoudi considered problem (1.2) in the case f (u) = bu|u| p−2, g(ut ) = a(1 + |ut |m−2)ut , a, b > 0, p, m > 2, and showed that, for any initial data (u0, u1) ∈ H 1( ) × L2( ), the problem has a unique global solution with energy 0 decaying exponentially. Benaissa and Messaoudi [ 16 ] studied (1.2), for f (u) = −bu|u| p−2, and g(ut ) = a(1 + |ut |m−2)ut , and showed that, for suitably chosen initial data, the problem possesses a global weak solution which decays exponentially even if m ≥ 2. In [ 28 ], Guesmia looked into the following problem ⎧ utt − u + h(∇u) + g(ut ) + f (u) = 0, ⎪ ⎨ u(x , t ) = 0, ⎪⎩ u(x , 0) = u0(x ), ut (x , 0) = u1(x ), in where is a bounded open domain in Rn (n ≥ 1), with a smooth boundary ∂ and f, g : R → R and h : Rn → R are continuous nonlinear functions satisfying some general properties. He obtained uniform decay of strong and weak solutions under weak growth assumptions on the feedback function and without any control of the sign of the derivative of the energy related to the above equation. Guesmia and Messaoudi [ 29 ], considered (1.4) with h(∇u) = −∇φ · ∇u, where φ ∈ W 1,∞( ), and proved local and global existence results and showed that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. Pucci and Serrin [78] discussed the stability of the following problem ⎧ utt − u + Q(x , t, u, ut ) + f (x , u) = 0, ⎪ ⎨ u(x , t ) = 0, ⎪⎩ u(x , 0) = u0(x ), ut (x , 0) = u1(x ), in and proved that the energy of the solution is a Lyapunov function. Although they did not discuss the issue of the decay rate, they did show that in general the energy goes to zero as t approaches infinity. They also considered an important special case of (1.5), when Q(x , t, u, ut ) = a(t )t αut and f (x , u) = V (x )u, and showed that the behavior of the solution depends crucially on the parameter α. Precisely, they showed if |α| ≤ 1 then the rest field is asymptotically stable. On the other hand, when |α| > 1 there are solutions that do not approach zero or approach nonzero function φ (x ) as t → ∞. In [ 27 ], Guesmia studied the following elasticity system ⎧⎪ ∂tt ui − σi j, j + i (x , ∂t ui ) = 0, ⎨ u(x , t ) = 0, ⎪⎩ ui (0) = ui0, ∂t ui (0) = ui1, in where i (x , ∂t ui ) = bi (x )gi (∂t ui ), bi ’s ∈ L∞( ), are bounded nonnegative functions and gi ’s are nondecreasing continuous real-valued functions satisfying certain conditions. He proved precise decay estimates of the energy for the system (1.6) with some localized dissipations. Zuazua [96] considered the following damped semilinear wave equation utt − u + αu + f (u) + a(x )ut = 0 in Rn × (0, ∞), with α > 0. He proved the exponential decay of the energy of the solution under suitable conditions on the functions f and a. In [ 17 ], Benaissa and Mokeddem looked into the following equation utt − div |∇u| p−2∇u − σ (t )div |∇ut |m−2∇ut = 0, where σ is a positive function, p, m ≥ 2 and is a bounded domain in Rn (n ≥ 1) with a regular boundary. They gave an energy-decay estimate for the solutions and extended the results of Yang [92] and Messaoudi [59]. Cavalcanti and Guesmia [ 19 ] looked into the following problem ⎧ utt − u + F (x , t, u, ∇u) = 0, ⎪ ⎪⎪⎨ u(x , t ) = 0, t u + 0 g(t − s) ∂∂uν (s)ds = 0, ⎪ ⎪⎪⎩ u(x , 0) = u0(x ), ut (x , 0) = u1(x ), in × (0, +∞) on ∂ 0 × (0, +∞) on ∂ 1 × (0, +∞) in , (1.7) where is a bounded region in Rn whose boundary is partitioned into disjoint sets 0, 1, g is the relaxation function which satisfies some assumptions. They proved that the dissipation given by the memory term is strong enough to assure exponential (or polynomial) decay provided the relaxation function also decays exponentially (or polynomially). In both cases the solution decays with the same rate of the relaxation function. This result was later generalized by Messaoudi and Soufyane [60], where relaxation functions of general decay type were considered. Alabau-Boussouira [ 1 ] used some weighted integral inequalities and convexity arguments and proved a semi-explicit formula which leads to decay rates of the energy in terms of the behavior of the nonlinear feedback near the origin, for which the optimal exponential and polynomial decay rate estimates are only special cases. The following problem has been widely studied in the literature: utt − u(x , t ) = 0, u + α(t )g(ut ) = 0, in on ∂ and g, α are specific functions. For instance, where is a bounded domain of Rn with a smooth boundary ∂ when α ≡ 1 and g satisfies where c1, c2 > 0 are constants and q > 1, it was proved that c1 min{|s|, |s|q } ≤ |g(s)| ≤ c2 max{|s|, |s|1/q }, E (t ) ≤ C E (0) t −2/(q−1), ∀t > 0, and for q = 1, the decay rate is exponential (see [36,43]). In the presence of a weak frictional damping, Benaissa and Messaoudi [ 15 ] treated problem (1.8) for g having a polynomial growth near the origin, and established energy decay results depending on α and h. Decay results for arbitrary growth of the damping term have been considered for the first time in the work of Lasiecka and Tataru [44]. They showed that the energy decays as fast as the solution of an associated differential equation whose coefficients depend on the damping term. Mustafa and Messaoudi [65] considered (1.8) and established an explicit and general decay rate result, using some properties of convex functions. Their result was obtained without imposing any restrictive growth assumption on the frictional damping term. Wu and Xue [91] studied the following quasilinear hyperbolic equation utt − ψ (t ) div(|∇ut | p−2∇ut ) − σi (uxi ) + μ|ut |αut = 0, n i=1 ∂ ∂ xi where μ, α ≥ 0, and p ≥ 2 are constants, the functions σi (i = 1, 2, . . . , n) and ψ are nonlinear, the domain is bounded in Rn (n ≥ 1), with a regular boundary. They investigated, using the multiplier methods, the stability of weak solutions and obtained an explicit estimation for the rate of the decay. In 2015, Mokeddem and Mansour [64] revisited the problem considered in [ 17 ] with some modifications. Precisely they treated the equation utt − div(|∇u| p−2∇u) − σ (t )(ut − div(|∇ut |m−2∇ut )) = 0, and gave the same decay result. Recently, Cavalcanti et al. [ 20 ] treated the following damped wave problem ⎧ utt − u + a(x )ut − div(b(x )∇ut ) = 0, ⎪ ⎨ u(x , t ) = 0, ⎪⎩ u(x , 0) = u0(x ), ut (x , 0) = u1(x ), in where is a bounded open domain in Rn (n ≥ 1), with a smooth boundary ∂ and a, b : → R+ are nonnegative functions satisfying specific conditions. Under appropriate assumptions on the coefficients and the initial data (u0, u1), they proved stabilization results for problem (1.9). Taniguchi [83] studied the following problem with nonlinear boundary condition: ⎧ utt (t ) − ρ(t ) u(t ) + b(x )ut (t ) = f (u(t )), ⎪ ⎪⎪⎨ u(t ) = 0, ⎪ ∂u∂(νt) + γ (ut (t )) = 0, ⎪⎪⎩ u(0) = u0, ut (0) = u1, on on on in × (0, T ), 0 × (0, T ), 1 × (0, T ) , (1.9) (1.10) where is a bounded domain of Rn with a smooth boundary ∂ = 0 ∪ 1 and 0 ∩ 1 = φ. Under some conditions on u0 and E(0), the global existence and exponential decay of the energy E(t) of weak solution to (1.10) were established. 1.2 Blowup in the case of constant exponents The first study of finite-time blowup of solutions of hyperbolic PDEs of the form utt − u = f (u) utt − u + aut = f (u). goes back to early 70s in the work of Levine [46] and Ball [ 10 ]. The interaction between the damping and the source terms was considered by Levine for an equation of the form He introduced the concavity method and showed that solutions with negative initial energy blow up at finite time. This method was later improved by Kalantarov and Ladyzhenskaya [ 34 ] to accommodate more general cases. After 20 years, Georgiev and Todorova [ 26 ] extended Levine’s result to the nonlinearly damped equation utt − u + a|ut |m ut = b|u|pu, in × (0, ∞), for a, b, m, p > 0. In their work, Georgiev and Todorova introduced a different method and determined appropriate relations between the nonlinearities in the damping and the source, for which there is either global existence or alternatively finite-time blowup. Precisely, they showed that solutions with negative energy exist globally ’in time’ if m ≥ p and blow up in finite time if p > m and initial energy is ’sufficiently’ negative. This result was later generalized to an abstract setting and to unbounded domains by Levine et al. [47], Levine and Serrin [48], Levine and Park [49], and Messaoudi [55,57]. In all these papers, the authors showed that no solution with negative or sufficiently negative energy can be extended on [0, ∞), if the nonlinearity dominates the damping effect ( p > m). Vitillaro [87] combined the arguments in [45] and [ 26 ] to extend these results to situations where the damping is nonlinear and the solution has positive initial energy. For more results concerning blowup and nonexistence, we mention here the work of Vitillaro [88], Todorova [84], Todorova and Vitillaro [85], Wang [89], Liu [51], Wu [90], and the very recent book of Al’shin et al. [ 2 ]. For the nonlinear Kirchhoff-type problem of the form ⎧ utt − ( ⎪ ⎨ u(x, t) = 0, ⎪⎩ u(x, 0) = u0(x), ut (x, 0) = u1(x), |Dm u|2dx)q u + ut |ut |r = |u|pu, in where p, q, r ≥ 0, is a bounded domain of Rn, with a smooth boundary ∂ and a unit outer normal ν, several results concerning global existence and blowup have been established; see in this regard [ 11–13,74,75 ], and the references therein. Messaoudi and Said-Houari [58] considered the nonlinear wave equation utt − ut − div |∇u|α−2∇u − div |∇ut |β−2∇ut + a|ut |m−2ut = b|u|p−2u, where a, b > 0, α, β, m, p > 2 and is a bounded domain in Rn (n ≥ 1), with a regular boundary. They proved, under appropriate conditions on α, β, p, m > 2, a global nonexistence result for solutions associated with negative initial energy. Chen et al. [ 21 ] considered the following nonlinear p-Laplacian-wave equation utt − div(|∇u|p−2∇u) − ut + q(x, u) = f (x) in a bounded domain ⊂ Rn, where 2 ≤ p < n and f, q are given functions. They established global existence and uniqueness under appropriate conditions on the initial data and the functions f, q. They also discussed the long-time behavior of the solution. Ibrahim and Lyaghfouri [ 32 ] considered the following equation utt − div(|∇u|p−2∇u) = |u|p∗−2u in Rn, n ≥ 3, and 2 < p < n, p∗ = np−np is the critical Sobolev exponent. Under appropriate assumptions on the initial data, they proved the finite-time blow up of solutions and, hence, extended a result by Galaktionov and Pohozaev [ 23 ]. Ye [93] investigated the blowup property of solutions of a quasilinear hyperbolic system of equations and proved that certain solutions with positive initial energy explode in finite time and he also gave estimation for the solution lifespan. Recently, Kafini and Messaoudi [ 33 ] studied a nonlinear wave equation with damping and delay terms and showed, under suitable hypotheses on the initial data, that the solution energy explodes in a finite time. For more results, we refer the reader to [ 14,24,31,81 ]. 1.3 Blowup in the case of variable-exponent nonlinearities In recent years, a great deal of attention has been given to the investigation of nonlinear models of hyperbolic, parabolic and elliptic equations with variable exponents of nonlinearity. For instance, some models from physical phenomena like flows of electro-rheological fluids or fluids with temperature-dependent viscosity, filtration processes in a porous media, nonlinear viscoelasticity, and image processing, give rise to such problems. More details on the subject can be found in [ 3 ] and [ 4 ]. Regarding hyperbolic problems with nonlinearities of variable-exponent type, only few works have appeared. For instance, Antontsev [ 6 ] considered the equation utt − div(a(x , t )|∇u| p(x,t)−2∇u) − α ut = b(x , t )u|u|σ (x,t)−2 in a bounded domain ⊂ Rn , where α > 0 is a constant and a, b, p, σ are given functions. For specific conditions on a, b, p, σ, he proved some blowup results, for certain solutions with non-positive initial energy. He also discussed the case when α = 0 and established a blowup result. Subsequently, Antontsev [ 5 ] discussed the same equation and proved a local and a global existence of some weak solutions under certain hypotheses on the functions a, b, p, σ. He also established some blowup results for certain solutions having non-positive initial energy. Guo and Gao [ 30 ] looked into the same problem of [ 6 ] and established several blowup results for certain solutions associated with negative initial energy. Precisely, they took σ (x , t ) = σ > 2, a constant, and established a result of blowup in finite time. For the case σ (x , t ) = σ (x ), they claimed the same blowup result but no proof has been given. This work is considered to be an improvement for that of [ 6 ]. In [82], Sun et al. looked into the following equation utt − div(a(x , t )∇u) + c(x , t )ut |ut |q(x,t)−1 = b(x , t )u|u| p(x,t)−1 in a bounded domain, with Dirichlet-boundary conditions, and established a blowup result for solutions with positive initial energy. They also gave lower and upper bounds for the blowup time and provided a numerical illustrations for their result. Recently, Messaoudi and Talahmeh [61] studied utt − div(|∇u|m(x)−2∇u) + μut = |u| p(x)−2u, (1.12) with Dirichlet-boundary conditions and for μ ≥ 0. They proved a blowup result for certain solutions with arbitrary positive initial energy. This result generalized that of Korpusov [39] established for (1.12), with m and p constants. This latter result was later extended by the same authors in [62] to an equation of the form utt − div(|∇u|r(·)−2∇u) + a|ut |m(·)−2ut = b|u| p(·)−2u, where a, b > 0 are constants and the exponents of nonlinearity m, p and r are given functions. They proved a finite-time blowup result for the solutions with negative initial energy and for certain solutions with positive energy. Very recently, Messaoudi et al. [63] studied utt − u + aut |ut |m(·)−2 = bu|u| p(·)−2, (1.13) where a, b are positive constants. They established the existence of a unique weak solution using the Faedo– Galerkin method under suitable assumptions on the variable exponents m and p and they also proved the finite-time blow up of solutions and gave a two-dimension numerical example to illustrate the blowup result. Yunzhu Gao and Wenjie Gao [ 25 ] studied a nonlinear viscoelastic equation with variable exponents and proved the existence of weak solutions using the Faedo–Galerkin method under suitable assumptions. Autuori et al. [ 9 ] looked into a nonlinear Kirchhoff system in the presence of the −→p (x , t )-Laplace operator, a nonlinear force f (t, x , u) and a nonlinear damping term Q = Q(t, x , u, ut ). They established a global nonexistence result under suitable conditions on f, Q, p. For more results concerning the blowup of hyperbolic problems, we refer the reader to Antontsev and Ferreira [ 7 ] and the book by Antontsev and Shmarev [ 8 ]. 2 Preliminaries 2.1 History of variable-exponent Lebesgue spaces Variable Lebesgue spaces were first introduced by Orlicz in 1931 in his article [77]. He started by looking for necessary and sufficient conditions on a real sequence (yi ) under which xi yi converges, for any other real sequence (xi ) such that xipi converges, where ( pi ) is a sequence of real numbers with pi > 1. Furthermore, he also considered the variable-exponent function space L p(·) on the real line. Orlicz later concentrated much on the theory of the function spaces that were named after him. In the theory of Orlicz spaces, the space Lϕ is defined as follows: Lϕ := u : → R such that (λu) = ϕ(λ|u(x)|)dx < +∞ , for some λ > 0, where ϕ is a real-valued function which may depend on x and satisfies some additional conditions. Putting certain properties of in an abstract setting, a more general class of function spaces, called modular spaces, was first studied by Nakano [66,67]. Following the work of Nakano, modular spaces were investigated by several people, most importantly by groups at Sapporo (Japan), Voronezh (U.S.S.R.), and Leiden (the Netherlands). An explicit version of modular function spaces was investigated by Polish Mathematicians, like Hudzik and Kaminska. The variable-exponent Lebesgue space L p(·)( ) is defined as the Orlicz space Lϕp(·) ( ), where t p(·) ϕp(·)(t) = t p(·) or ϕp(·)(t) = p(·) , with p(·) : → [1, ∞] as a measurable function. Variable-exponent Lebesgue spaces on the real line have been independently developed by Russian researchers. Their results originated in 1961 in a paper by Tsenov [86]. The Luxemburg norm was introduced by Sharapudinov for the Lebesgue space. He showed that this space is Banach if the exponent p(·) satisfies 1 < essinf p ≤ esssup p < +∞. In the mid-80s, Zhikov [94] started a new line of investigation of variable-exponent spaces, by considering variational integrals with non-standard growth conditions. The next major step in the study of variable-exponent spaces was by Kovacik and Rakosnik [40] in the early 90s. In their paper, they established many of the basic properties of Lebesgue and Sobolev spaces in Rn. 2.2 Variable-exponent Lebesgue Spaces In this section, we present some preliminary facts about the Lebesgue spaces with variable exponents. Definition 2.1 Let X be a K−vector space. A function : X −→ [0, ∞] is said to be left-continuous if the mapping λ −→ (λx) is left-continuous on [0, ∞), for every x ∈ X ; that is, lim λ→1− (λx) = (x), ∀ x ∈ X. Definition 2.2 Let X be a K−vector space. A function : X −→ [0, ∞] is called a semimodular on X if the following properties hold: (a) (0) = 0. (b) (λx) = (x), for all x ∈ X and λ ∈ K, with |λ| = 1. (c) is convex. (d) is left-continuous. (e) (λx) = 0, for all λ > 0 implies x = 0. A semimodular is called modular if (f) (x) = 0 implies x = 0 A semimodular is called continuous if (g) the mapping λ −→ (λx) is continuous on [0, ∞) for all x ∈ X Examples 2.3 Let L0( ) be the set of all Lebesgue-measurable functions defined on . If 1 ≤ p < +∞, then p( f ) := | f (x)|pdx defines a continuous modular on L0( ). Theorem 2.4 Let be a semimodular on X . Then, the mapping λ → every x ∈ X . Moreover, (λx) is non-decreasing on [0, ∞) for (λx) = (|λ|x) ≤ |λ| (x) for all |λ| ≤ 1, (λx) = (|λ|x) ≥ |λ| (x) for all |λ| ≥ 1. (2.1) Proof • Assume that 0 ≤ λ < μ, then 0 ≤ μλ < 1. So for a fixed x ∈ X we have, by convexity and non-negativeness of and the fact that (0) = 0, (λx) = λ μ (μx) + λ 1 − μ · 0 λ ≤ μ (μx) + λ 1 − μ λ (0) = μ (μx) ≤ (μx). Hence, for any fixed x ∈ X , we have • • • For λ = 0 we have For |λ| ≤ 1, we have Therefore, For |λ| ≥ 1, we have Thus, (λx) ≤ (μx), for 0 ≤ λ < μ. λ λ |λ|x | | (λx) = = (|λ|x) since Definition 2.5 Let ( A, , μ) be a σ -finite, complete measure space. We define P( A, μ) to be the set of all μ-measurable functions p : → [1, ∞]. The functions p ∈ P( A, μ) are called variable exponents on A. We set p1 := essinfy∈A p(y) and p2 := esssupy∈A p(y). If p2 < +∞, then we call p a bounded variable exponent. If p ∈ P( A, μ), then we define p ∈ P( A, μ) by 1 1 p(y) + p (y) = 1, where 1 ∞ := 0. The function p is called the dual variable exponent of p. In the special case when μ is the n-dimensional Lebesgue measure and is an open subset of Rn, we abbreviate P( ) := P( , μ). where Definition 2.6 We define the Lebesgue space with a variable-exponent p(·) by L p(·)( ) := {u : → R; measurable in : p(·)(λu) < ∞, for some λ > 0}, is a modular. We equip L p(·)( ) with the following Luxembourg-type norm p(·)(u) = |u(x)|p(x)dx u p(·) := inf λ > 0 : u(x) p(x)dx ≤ 1 . λ Examples 2.7 Let p(x) = x on = (1, 2). Then 1 p(·) = 1. Indeed, p(·)(1/λ) = 1 2 λ − 1 λ−x dx = λ2 ln λ . Since p(·)(1) = 1, then, by definition of 1 p(·), we have 1 p(·) ≤ 1. On the other hand, it is easy to check that p(·)(1/λ) > 1, for 0 < λ < 1. This gives 1 p(·) ≥ 1. Hence, we conclude that 1 p(·) = 1. Lemma 2.8 If p(x) ≡ p, where p is constant. Then, = |x − x0||x + x0| 4 log δ ≤ log δ A ≤ − log |(x, y) − (x0, y0)| Lemma 2.11 (Unit ball property) Let p ∈ P( A, μ) and f ∈ L p(·)( A, μ). Then (i) f p(·) ≤ 1 if and only if p(·)( f ) ≤ 1 Proof Since p(·)(u/λ0) = 1, then 1 Since, p(·)(u/λk ) = (λk)p u p | | ≤ 1, then we have Combining (2.3) and (2.4) gives (2.2). Definition 2.9 We say that a function q : 0 < δ < 1 such that u p(·) = λ0 = Next, using property of Inf, there exists a sequence {λk }k∞=1 such that λk ≥ u p(·), with → R is log-Hölder continuous on , if there exist A > 0 and A |q(x) − q(y)| ≤ − log |x − y| , for all x, y ∈ , with |x − y| < δ. Examples 2.10 Let p(x) = x2 + 1 be defined on = B(0, 1). Then p : → R is log-Hölder continuous on . Indeed, Let (x, y), (x0, y0) ∈ , with |(x, y) − (x0, y0)| < δ and 0 < δ < 1. Then, | p(x, y) − p(x0, y0)| = |x2 − x02| (2.2) (2.3) (2.4) (2.5) (ii) If f p(·) ≤ 1, then p(·)( f ) ≤ f p(·) (iii) Iff fp(·p)(≤·) ≥1 +1, thpe(·n)( f ) f p(·) ≤ p(·)( f ) (iv) Proof (i) If p(·)( f ) ≤ 1, then f p(·) ≤ 1 by definition of · p(·). On the other hand, if f p(·) ≤ 1, then p(·) λf ≤ 1 for all λ > 1. Since p(·) is left-continuous it follows that p(·)( f ) ≤ 1. (ii) The claim is obvious for f = 0, so assume that 0 < f p(·) ≤ 1. By (i) and f/ f p(·) = 1, it follows that p(·) f/ f p(·) ≤ 1. Since f p(·) ≤ 1, it follows from (2.1) that p(·)( f )/ f p(·) ≤ 1. This implies p(·)( f ) ≤ f p(·). (iii) Assumethat f p(·) > 1.Then p(·) λf > 1for1 < λ < f p(·) andby(2.1)itfollowsthat1 < p(·λ)(f). which implies λ < p(·)( f ), for 1 < λ < f p(·). Since λ is arbitrary, we have f p(·) ≤ p(·)( f ). (iv) This follows immediately from (ii) and (iii). We also state, without proof, some useful results from [41]. Lemma 2.12 If 1 < p1 ≤ p(x) ≤ p2 < +∞ holds, then min{ u pp1(·), u pp2(·)} ≤ p(·)(u) ≤ max{ u pp1(·), u pp2(·)}, for any u ∈ Lp(·)( ). Theorem 2.13 If p ∈ P(A,μ), then Lp(·)(A,μ) is a Banach space. Lemma 2.14 If p : → [1,∞) is a measurable function with p2 < +∞, then C0∞( ) is dense in Lp(·)( ). Lemma 2.15 (Young’s inequality) Let p,q,s ∈ P( ) such that (2.6) Then for all a,b ≥ 0, By taking s = 1, and 1 < p,q < +∞, then we have for any ε > 0, where Cε = 1 q . For p = q = 2, we have q(εp) p Lemma 2.16 (Hölder’s Inequality) Let p,q,s ∈ P( ) such that If f ∈ Lp(·)( ) and g ∈ Lq(·)( ), then fg ∈ Ls(·)( ) and By taking p = q = 2, we have the Cauchy–Schwarz inequality. 1 1 1 , for a.e y ∈ . s(y) = p(y) + q(y) 2 ab ≤ εa2 + 4bε. 1 1 1 , for a.e y ∈ . s(y) = p(y) + q(y) fg s(·) ≤ 2 f p(·) g q(·). 2.3 Variable-exponent Sobolev spaces In this section we study some functional analysis-type properties of Sobolev spaces with variable exponents. We start by recalling the definition of weak derivative. Definition 2.17 (Weak derivative) Let ⊂ Rn be a domain. Assume that u ∈ Ll1oc( ). Let α := (α1, . . . , αn) ∈ Nn be a multi-index and let |α| = α1 + · · · + αn. If there exists g ∈ Ll1oc( ) such that u ∂|α|ψ ∂α1 x1 · · · ∂αn xn dx = (−1)|α| ψg dx, ∂αu or ∂α1 x∂1|·α··|∂uαn xn . Definition 2.18 Let k ∈ N. We define the space W k,p(·)( ) by for all ψ ∈ C0∞( ), then g is called a weak partial derivative of u of order α. The function g is denoted by W k,p(·)( ) := {u ∈ L p(·)( ) such that ∂αu ∈ L p(·)( ), ∀ |α| ≤ k}. We define a semimodular on W k,p(·)( ) by W k,p(·)( )(u) = L p(·)( )(∂αu). This induces a norm given by For k ∈ N, the space W k,p(·)( ) is called Sobolev space and its elements are called Sobolev functions. Clearly W 0,p(·)( ) = L p(·)( ) and W 1,p(·)( ) = {u ∈ L p(·)( ) such that ∇u exists and |∇u| ∈ L p(·)( )}, equipped with the norm u W 1,p(·)( ) = u p(·) + ∇u p(·). Theorem 2.19 Let p ∈ P( ). The space W k,p(·)( ) is a Banach space, which is separable if p is bounded, and reflexive if 1 < p1 ≤ p2 < +∞. Definition 2.20 Let p ∈ P( ) and k ∈ N. The Sobolev space W k,p(·)( ) “with zero boundary trace” is the 0 closure in W k,p(·)( ) of the set of W k,p(·)( )−functions with compact support, i.e., W0k,p(·)( ) = {u ∈ W k,p(·)( ) : u = uχK for a compact K ⊂ }. Remark 2.21 [41] Let p ∈ P( ) and k ∈ N. Then (i) The space H k,p(·)( ) is defined as the closure of C0∞( ) in W k,p(·)( ). 0 (ii) H k,p(·)( ) ⊂ W k,p(·)( ). 0 0 (iii) If p is log-Hölder continuous on , then W k,p(·)( ) = H k,p(·)( ). 0 0 (iv) The dual of W 1,p(·)( ) is defined as W −1,p (·)( ), in the same way as the usual Sobolev spaces, where 0 1 1 p(·) + p (·) = 1. Theorem 2.22 Let p ∈ P( ). The space W k,p(·)( ) is a Banach space, which is separable if p is bounded, 0 and reflexive if 1 < p1 ≤ p2 < +∞. Theorem 2.23 (Poincaré’s inequality) Let continuity property, then be a bounded domain of Rn and p(·) satisfies the Log-Hölder u p(·) ≤ C ∇u p(·), for all u ∈ W 1,p(·)( ), 0 norm given by u W 1,p(·)( ) = ∇u p(·). 0 If p = 2, then we set H01( ) = W01,2( ). where the positive constant C depends on p(·) and only. In particular, the space W 1,p(·)( ) has an equivalent 0 Remark 2.24 Contrary to the constant-exponent case, there is no Poincaré inequality version for modular. The following example shows that the Poincaré inequality does not, in general, hold in a modular form. Examples 2.25 [41] Let p : (−2, 2) −→ [ 2, 3 ] be a Lipschitz continuous exponent defined by Let uμ be a Lipschitz function defined by Then as μ → 0+. p(x) = ⎧ 3, if x ∈ (−2, −1) ∪ (1, 2) ⎪⎪⎪⎨ 2, if x ∈ − 21 , 21 ⎪⎪⎪⎩ 2−x2+x+1,1, if x ∈ − 1, − 21 if x ∈ 21 , 1 . ⎧ μx + 2μ, uμ(x) = ⎨⎪ μ, We end this section with some essential embedding results. See [41]. Lemma 2.26 Let be a bounded domain in Rn with a smooth boundary ∂ . Assume that p : is a measurable function such that → (1, ∞) 1 < p1 ≤ p(x) ≤ p2 < +∞, for a.e. x ∈ . If p(x), q(x) ∈ C( ) and q(x) < p∗(x) in with p∗(x) = ∞, Then the embedding W 1,p(·)( ) → Lq(·)( ) is continuous and compact. nn−pp(x(x)) , if p2 < n if p2 ≥ n. As a special case we have Corollary 2.27 Let be a bounded domain in Rn with a smooth boundary ∂ . Assume that p : is a continuous function such that 2 ≤ p1 ≤ p(x) ≤ p2 < , n ≥ 3. 2n n − 2 Then the embedding H 1( ) → L p(·)( ) is continuous and compact. → (1, ∞) (2.7) 3 Existence In this section we discuss a wave problem in the presence of a nonlinear damping, where the nonlinearity is of variable-exponent type. We establish the well posedness, using the Galerkin method and adopting the steps of the book [50] used for standard linearities. Let be a bounded domain in Rn with a smooth boundary ∂ . We consider the following initial-boundary value problem: where m(·) is a given continuous function on and the log-Hölder continuity condition: ⎧ utt − ⎪ ⎨ u(x , t ) = 0, u + |ut |m(·)−2ut = 0, , m1 := ess infx∈ m(x ), Integrate over (0, t ), to get Using the inequality for all a,b ∈ R and a.e. x ∈ , we have which implies that w = C = 0, since w = 0 on ∂ . Hence, the uniqueness. A |m(x ) − m(y)| ≤ − log |x − y| , for a.e. x , y ∈ Theorem 3.1 Let u0 ∈ H01( ), u1 ∈ L2( ). Assume that the exponent m(·) satisfies conditions (3.1) and (3.2). Then problem (P) has a unique weak solution such that u ∈ L∞((0, T ), H01( )), ut ∈ L∞((0, T ), L2( )) ∩ Lm(·)( × (0, T )), utt ∈ L2((0, T ), H −1( )), ∀ T > 0. Proof Uniqueness Suppose that ( P) has two solutions u and v. Then, w = u − v satisfies ⎪⎩ w(x , 0) = wt (x , 0) = 0, ⎧ wtt − ⎪ ⎨ w(x , t ) = 0, w + ut |ut |m(·)−2 − vt |vt |m(·)−2 = 0, in on ∂ in Multiply the equation by wt and integrate over , to obtain wt2 + |∇w|2 + (ut |ut |m(x)−2 − vt |vt |m(x)−2)(ut − vt )dx = 0. 0 t (wt2 + |∇w|2) + 2 (ut |ut |m(x)−2 − vt |vt |m(x)−2)(ut − vt )dx ds = 0. (|a|m(x)−2a − |b|m(x)−2b)(a − b) ≥ 0, wt2 + |∇w|2 = 0; (P) (3.1) (3.2) Existence Let {v j } ∞j=1 be an orthonormal basis of H 1( ), with 0 − v j = λ j v j , in , v j = 0, on ∂ . look for functions We define the finite-dimensional subspace Vk = span{v1, . . . , vk }. By normalization, we have v j 2 = 1. We uk (x, t) = a j (t)v j k j=1 which satisfy the following approximate problems utkt (x, t)v j (x)dx + ∇uk (x, t)∇v j (x)dx + |utk (x, t)|m(x)−2utk (x, t)v j (x)dx = 0, k k ⎩⎪ uk (x, 0) = u0, utk (x, 0) = u1, ∀ j = 1, 2, . . . , k, k i=1(u0, vi )vi and uk1 = k i=1(u1, vi )vi are two sequences in H 1( ) and L2( ), respectively, 0 This generates the system of k ordinary differential equations uk0 → u0 in H01( ) and uk1 → u1 in L2( ). a j (t) + λ j a j (t) = G j (a1(t), . . . , ak (t)), a j (0) = (u0, v j ), a j (0) = (u1, v j ), ∀ j = 1, 2, . . . , k, | k i=1 G j (a1(t), . . . , ak (t)) = − ai (t)vi (x)|m(x)−2 ai (t)vi (x)v j (x)dx. This system can be solved by standard ODE theory. Hence, we obtain functions a j : [0, tk ) → R, 0 < tk < T . Next, we have to show that tk = T , ∀k ≥ 1. We multiply the equation in (3.3) by a j (t) and sum over j to get ⎧ ⎪⎨ where uk 0 = such that where 0 t k i=1 (3.3) (3.4) (3.5) (3.6) 1 d 2 dt 1 2 = ≤ 1 2 1 2 Integration over (0, t) gives So, we have (|utk (x, t)|2dx + |∇uk (x, t)|2)dx + |utk (x, t)|m(x)dx = 0. (|utk (x, t)|2dx + |∇uk (x, t)|2)dx + |utk (x, s)|m(x)dxds (|u1| + |∇uk0|2)dx k 2 (u12 + |∇u0|2)dx = C. sup(0,tk) |utk (x, t)|2dx + sup(0,tk) |∇uk (x, t)|2dx + 0tk |utk (x, s)|m(x)dxds ≤ C. Thus, the solution can be extended to [0, T ) and, in addition, we have (uk ) is a bounded sequence in L∞((0, T ), H01( )) (utk ) is a bounded sequence in L∞((0, T ), L2( )) ∩ Lm(·)( Therefore, we can extract a subsequence (u ) such that u → u weakly ∗ in L∞((0, T ), H01( )) ut → ut weakly ∗ in L∞((0, T ), L2( )) and weakly in Lm(·)( × (0, T )). We can conclude by Lion’s Lemma [50] that u ∈ C [0, T ], L2( ) so that u(x , 0) has a meaning. Since (ut ) is bounded in Lm(·)( × (0, T )) then (|ut |m(x)−2ut ) is bounded in L mm(·()·−) 1 ( × (0, T )); hence, up to a subsequence, |ut |m(·)−2ut → ψ weakly in L mm(·()·−) 1 utt − u + ψ = 0, in D ( A(ut ) − A(v))(ut − v)dt ≥ 0, ∀v ∈ Lm(·)((0, T )H01( )). |ut (x , T )|2 A(ut )v − 0 T ψ v − ψ − A(λw + ut ) w ≥ 0, ∀λ = 0, ∀w ∈ Lm(·) 0 ≤ lim sup X ≤ 0 ψ v − (3.12) (3.13) (3.14) (ψ − A(λw + ut ))w ≤ 0, ∀w ∈ Lm(·)( (ψ − A(ut ))w ≥ 0, ∀w ∈ Lm(·)( which gives u + |ut |m(x)−2ut = 0, in D ( utt − × (0, T )). To handle the initial conditions, we note that ul u weakly ∗ in L∞((0, T ), H01( )) ult ut weakly ∗ in L∞((0, T ), L2( )). Thus, using Lions’ Lemma [50], we obtain, up to a subsequence, ul → u in C ([0, T ], L2( )). Therefore, ul (x , 0) makes sense and ul (x , 0) → u(x , 0) in L2( ). Also we have that ul (x , 0) = ul0(x ) → u0(x ) in H01( ). u(x , 0) = u0(x ). As in [52], let φ ∈ C0∞(0, T ) and replacing (uk ) by (ul ), we obtain, from (3.3) and for any j ≤ l, 0 T 0 T ∇u (x , t )∇v j (x )φ (t )dx dt − |ut (x , t )|m(x)−2ut (x , t )v j (x )φ (t )dx dt. (3.15) ∇u(x , t )∇v j (x )φ (t )dx dt − |ut (x , t )|m(x)−2ut (x , t )v j (x )φ (t )dx dt, (3.16) − = 0 T 0 T ut (x , t )v(x )φ (t )dx dt u − |ut (x , t )|m(x)−2ut (x , t ) v(x )φ (t )dx dt, As l → +∞, we obtain that for all j ≥ 1. This implies − 0 T = − − 0 ult (x , t )v j (x )φ (t )dx dt ut (x , t )v j (x )φ (t )dx dt m(·) for all v ∈ H 1( ). This means utt ∈ L m(·)−1 ([0, T ), H −1( )) and u solves the equation 0 m(·) Thus, ut ∈ L∞([0, T ), L2( )), utt ∈ L m(·)−1 ([0, T ), H −1( )). Consequently, So, ult (x , 0) makes sense (see [52, p.116]). It follows that utt − u + |ut |m(·)−2ut = 0. ut ∈ C ([0, T ), H −1( )). ult (x , 0) → ut (x , 0) in H −1( ). ult (x , 0) = ul1(x ) → u1(x ) in L2( ). ut (x , 0) = u1(x ). But 4 Decay where (3.17) (3.18) (3.19) (3.20) This ends the proof of Theorem (3.1). For the best of our knowledge, there are not many stability results for hyperbolic problem involving nonstandard nonlinearities. The only works, we are aware of, are that by Ferreira and Messaoudi [ 22 ], where they studied a nonlinear viscoelastic plate equation with a lower order perturbation of a −→p (x , t )−Laplacian operator of the form 0 t utt + 2 u − −→p (x,t)u + h(t − s) u(s)ds − ut + f (u) = 0, −→p (x,t)u = n i=1 ∂ ∂ xi ∂u pi (x,t)−2 ∂u ∂ xi ∂ xi , −→p = ( p1, p2, . . . , pn)T and where and h ≥ 0 is a memory kernel that decays at a general rate and f is a nonlinear function. They proved a general decay result under appropriate assumptions on h, f, and the variable-exponent −→p (x , t )−Laplacian operator. Also, the work of Yunzhu Gao and Wenjie Gao [ 25 ], where they considered the following nonlinear viscoelastic hyperbolic problem: ⎪⎧ utt − u − ⎨ u(x , t ) = 0, ⎪⎩ u(x , 0) = u0(x ), ut (x , 0) = u1(x ), utt + 0t g(t − τ ) u(τ )dτ + |ut |m(x)−2ut = |u| p(x)−2u, in on ∂ in × (0, T ), × [0, T ) , (4.1) where m(x ), p(x ) are continuous functions in such that 1 < inf m(x ) ≤ m(x ) ≤ sup m(x ) < +∞, 1 < inf p(x ) ≤ p(x ) ≤ sup p(x ) < +∞ x∈ x∈ x∈ x∈ ∀z, ξ ∈ , |z − ξ | < 1, |m(z) − m(ξ )| + | p(z) − p(ξ )| ≤ ω |z − ξ | , They also assumed that (i) g : R+ → R+ is a C 1 function satisfying (ii) There exists η > 0 such that lim sup ω(τ ) ln t→0+ 1 τ = C < +∞. g(0) > 0, 1 − g(s)ds = > 0; 0 +∞ g (t ) ≤ −ηg(t ), t ≥ 0 and proved the existence of the weak solution to problem (4.1). To establish our decay result, we need the following well-known lemma. Lemma 4.1 [38] Let E : R+ → that R+ be a nonincreasing function. Assume that there exist σ > 0, ω > 0 such s ∞ 1 E 1+σ (t )dt ≤ ω E σ (0)E (s) = c E (s), ∀s > 0. which satisfies We define the energy of the solution by This means that E (t ) is a nonincreasing function. where which implies that It follows that Using the definition of E (t ) and the relation s T T T Equation (4.2) becomes Estimates · − s dt Theorem 4.2 Suppose the conditions of Theorem 3.1 hold. Then there exist two constants c, α > 0 such that the energy satisfies, ∀t ≥ 0, Proof Multiplying (P) by u E q (t ), for q > 0 to be specified later, and integrating over obtain that × (s, T ), T > s, we s T (uutt − u u + |ut |m(x)−2ut u) = 0, s T s T d dt = E q (s) ≤ E q (s) ≤ E q (s) + E q (T ) (ut u)t − ut2 + |∇u|2 + |ut |m(x)−2ut u = 0. uut dx + dt T s s T (4.2) (4.3) where c∗ is the embedding constant. Since E (t ) is nonincreasing, then we have dt For the third term of the right-hand side of (4.3), we set, as in [38], and exploit Hölder’s and Young’s inequalities and (3.1) as follows (4.4) (4.5) ut2 + + + |ut |m1 |ut |m(x) ut2 − 2/m1 2/m1 + + −E (t ) 2/m1 + −E (t ) 2/m2 − |ut |m2 |ut |m(x) 2/m2 ! 2/m2 ! s T E q (t ) −E (t ) 2/m2 dt + c E q (t ) −E (t ) 2/m1 dt E q+1(t )dt + cε (−E (t ))2(q+1)/m2 dt + cε (E (t ))qm1/(m1−2)dt + cε (−E (t ))dt. ≤ c ≤ c ≤ c ≤ c ≤ cε s s T T s s s s s T T T T s T T s T |ut |m(x)−1|u| ≤ ε|u|m(x) + cε(x )|ut |m(x), cε(x ) = ε1−m(x)(m(x ))−m(x)(m(x ) − 1)m(x)−1. s T 2 The case m1 = 2 is similar. m(x) , p (x ) = m(x ); so for a.e. For the last term of (4.3), we use Young’s inequality with p(x ) = m(x)−1 , we have at x ∈ where We choose q such that q = m22 − 1 and notice that mq1m−12 = q + 1 + mm21−−m21 , if m1 > 2. Hence, we arrive Therefore, s T T T T ≤ ε s + s T ≤ ε s T + s + s E q (t ) cε(x )|ut |m(x)dx . Therefore, a combination of (4.3)–(4.7) leads to 2 T E q+1(t )dt ≤ cε(1 + (E (0)) m21 −1 + (E (0)) m22 −1 T s s + cε E (s) + s E q (t ) cε(x )|ut |m(x)dx . We then pick ε > 0 so small that cε 1 + ( E (0) m21 −1 + E (0) m22 −1 < 1. Once ε is fixed, then cε(x ) ≤ M, where M is constant since m(x ) is bounded. Thus, we arrive at s T E q+1(t )dt ≤ c E (s) + M ≤ c E (s) + q M+s1 E q+1(s) − E q+1(T ) ≤ c E (s) + E q+1(s) ≤ c 1 + E q (0) E (s) = c E (s), ∀ T > s > 0. By taking T → ∞, we get s Hence, Komornik’s Lemma (with σ = m22 − 1) implies the desired result. ∞ E m22 (t )dt ≤ c E (s). Using the definition of E (t ), we get T s 5 Numerical study In this section, we present two numerical applications to illustrate the decay results in Theorem 4.2. We consider the initial-boundary value problem ( P) on the domain B(0, 1) × (0, T ), where B(0, 1) is the unit disk in R2. We take the initial data u0(x1, x2) = 1 − x1 − x22 and u1(x1, x2) = 0. We consider the following applications: 2 1. Polynomial decay Here we take the variable-exponent m(x1, x2) = numerically verify that x1 2 + 4, where m2 = 5, and for some c > 0. 2. Exponential decay We take m(x1, x2) = 2 and we numerically verify that E (t ) ≤ c E (0)(1 + t )−2/3, ∀t ≥ 0, E (t ) ≤ ce−αt , ∀t ≥ 0, for some c > 0 and α > 0. In the above applications, the exponent function m(·) satisfies conditions (3.1) and (3.2). Below we introduce a numerical scheme of ( P) using finite-element and finite-difference methods for the space and time discretization, respectively. Extensive details about these methods can be found in [42,79,80]. 5.1 Numerical method We discretize the wave equation utt − u + |ut |m(·)−2ut = 0, in B(0, 1) × (0, T ), (5.1) using finite differences for the time variable and a finite-element method for the space variable x = (x1, x2) ∈ B(0, 1). We first discretize the wave equation in time. The time interval [0, T ] is divided into N equal subintervals [t j , t j+1], j = 0, 1, . . . , N − 1, where t j = j τ and τ = T /N is the time step. Denoting we approximate the time derivatives of u(x , t ) at t = t j , for j = 1, 2, . . . , N − 1, using the following finite-difference formulas: U ( j) := u(x , y, t j ), U ( j) t Ut(tj) = U ( j) − U ( j−1) τ U ( j+1) − 2U ( j) + U ( j−1) τ 2 , . Then, a semi-discrete formulation of (5.1) at t = t j+1 reads Ut(tj+1) U ( j+1) + |Ut( j)|m−2Ut( j+1) = 0, (5.2) for j = 1, 2, ..., N − 1. Note that we have used the history data U ( j) and U ( j) to make the formulation linear t in U ( j+1), which will be essential for the space discretization. The above formulation is defined for j = 0 by taking U (0) = u0(0) and Ut(0) = u1(0). Now, we discretize the space variable. Let Bh be a triangulation of B(0, 1), with a maximal diameter size h. Let Ph := P1(Bh ) ∩ H0(Bh ), where H0(Bh ) denotes the linear Lagrangian finite-dimensional space. Multiplying (5.2) by a test function wh ∈ Ph , integrating by parts and using the boundary condition Uh( j+1) = 0 on ∂ Bh , we obtain the full discrete, weak formulation problem: given initial data Uh(0) := uh (0) and Uh(0,t) := u1,h (0), find Uh( j+1) ∈ Ph that satisfies (Uh(,jt+t1) + |Uh(,jt)|m−2Uh(,jt+1), wh ) + (∇Uh( j+1), ∇wh ) = 0, j = 0, 1, . . . , N − 1, (5.3) for all wh ∈ Ph . Here, (·, ·) denotes the L2 inner product. The above numerical problem is advanced, and at this stage we assume that the discrete solution of (5.3) is unique and converges to the exact solution of ( P), in the H 1-norm, as (h, τ ) → (0, 0). Fig. 1 h Fig. 2 Initial data: u(h0) 5.2 Numerical results In this section, we present the decay results of the full discrete model (5.3). The numerical results are obtained using Matlab. Figure 1 shows the triangulation h , which consists of 7808 degrees of freedom and 3024 elements. We take a small enough time step τ = 0.01, large enough time span T = 100 and α = 0.1. Figure 2 shows the initial data u(h0) = 1 − x12 − x22 projected in the finite-element space Ph . Now, we present the numerical results corresponding to the polynomial and exponential decay applications. 1. Polynomial decay Figure 3 lists the solutions u(n) corresponding to tn = 5, 20, 50, 100. h Figure 4 shows that energy function satisfies E (tn )(1 + tn )2/3 ≤ 4 for tn ∈ {0, 1, . . . , 100}. This implies that the numerical solution u(n) has a polynomial decay, which agrees with Theorem 4.2. h Fig. 4 Polynomial decay: y = E (t )(1 + t )2/3 2. Exponential decay In this case, the numerical solutions u(n) make the energy function E (tn ) satisfy h E (tn )e0.1t ≤ 4 for tn ∈ {0, 1, . . . , 100}. This implies that the numerical solution u(n) has an exponential decay, which verifies Theorem 4.2. h In Figs. 4 and 5, the corresponding curves are plotted using τ = 0.1 (the dashed curve) and using τ = 0.01 (the solid curve). In both applications the two curves are coinciding, which indicates that the corresponding numerical solutions are promising. 6 Blowup Let be a bounded domain in Rn with a smooth boundary ∂ . We consider the following initial-boundary value problem: where a > 0 and p is a given continuous function on satisfying ⎧ utt − ⎪ ⎨ u(x, t) = 0, u = a|u|p(·)−2u, in , p1 := ess infx∈ p(x), p2 := ess supx∈ p(x), with and the log-Hölder continuity condition: A | p(x) − p(y)| ≤ − log |x − y| , for a.e. x, y ∈ Therefore, (Q) can be rewritten as an initial-value problem: = (u, v)T , (0) = 0 = (u0, u1)T and J ( ) = 0, a|u|p(·)−2u T . where the linear operator A : D(A) → H is defined by A The state space of is the Hilbert space equipped with the inner product t + A (0) = = J ( ) 0, = (−υ, − u)T . H = H01( ) × L2( ), , H = (∇u · ∇u + υυ)dx, for all = (u, v)T and = (u, v)T in H. The domain of A is Then, we have the following local existence result. D(A) = { ∈ H : u ∈ H 2( ) ∩ H01( ), υ ∈ H01( )}. (Q) (6.1) (6.2) (Q1) Theorem 6.1 Assume that the exponent p satisfies conditions (6.1) and (6.2). Then for any (Q) has a unique local weak solution ∈ C ( ; H). That is, u ∈ L∞((0, T ), H01( )), ut ∈ L∞((0, T ), L2( )), utt ∈ L2((0, T ), H −1( )). Proof First, for all ∈ D(A), we have A , H = − ∇υ · ∇u dx − υ u = − ∇υ · ∇u dx + ∇υ · ∇u dx = 0. Therefore, A is a monotone operator. To show that A is maximal, we prove that for each there exists V = (u, υ)T ∈ D(A) such that (I + A)V = F. That is, We deduce from (6.5) that where Now we define, over H 1( ), the bilinear and linear forms 0 F = ( f, g)T ∈ H, u − υ = f υ − u = g. u − u = G, G = f + g ∈ L2( ). 0 ∈ H, problem (6.3) (6.4) (6.5) (6.6) (6.7) (6.8) B(u, w) = uw + ∇u · ∇w, L(w) = Gw. It is easy to verify that B is continuous and coercive and L is continuous on H 1( ). Then, Lax–Milgram 0 theorem implies that the equation B(u, w) = L(w), ∀w ∈ H01( ), has a unique solution u ∈ H 1( ). Hence, υ = u − f ∈ H 1( ). Thus, V ∈ H. 0 0 Using (6.6), we get uw + ∇u · ∇w = Gw, ∀ w ∈ H01( ). The elliptic regularity theory, then, implies that u ∈ H 2( ). Therefore, Consequently, I + A is surjective and then A is maximal. Finally, we show that J : H → H is locally Lipchitz. So, if we set V = (u, υ)T ∈ D(A). J ( ) − J ( ) 2H = = = (0, a|u| p(·)−2u − a|u| p(·)−2u) 2 H a|u| p(·)−2u − a|u| p(·)−2u) 2L2 f (u) − f (u) 2L2 . As a consequence of the mean value theorem, we have, for 0 ≤ θ ≤ 1, 2n As u, u ∈ H 1( ), we then use the Sobolev embedding H 1( ) → L n−2 ( ), Hölder’s inequality and (6.1), 0 0 to obtain |θ u + (1 − θ )u|2( p(x)−2)|u − u|2dx ≤ + u Ln(p1−2) + u Ln(p1−2) ≤ C u − u 2H01( ) + u H01( ) + u H01( ) 2( p1−2) u H01( ) + u H01( ) 2( p1−2) 2 n (6.9) (6.10) Therefore, J is locally Lipchitz. The proof of Theorem 6.1 is completed. See [38]. Next, we show that the solution (6.3) blows up in finite time if (6.1) holds and E (0) < 0, where 2 H J ( ) − J ( ) ≤ C (u, u) − 1 E (t ) := 2 ut2 + |∇u|2 dx − a 2 H . u p(x) | | p(x ) dx . We also denote by (u) = blowup result. |u(x )| p(x)dx . First, we establish several lemmas needed for the proof of our Lemma 6.2 Suppose the conditions of Corollary 2.27 hold. Then there exists a positive C > 1, depending on only, such that s p1 (u) ≤ C ∇u 22 + (u) , (u) , and use C to denote a generic positive constant depending on only. As a result of (6.9) and (6.10), we have Corollary 6.4 Let the assumptions of Lemma 6.2 hold. Then we have As a special case, we have Corollary 6.5 Let the assumptions of Lemma 6.2 hold. Then we have u sp1 ≤ C |H (t )| + 2 ut 2 + for any u ∈ H 1( ) and 2 ≤ s ≤ p1. 0 Lemma 6.6 Let the assumptions of Lemma 6.2 hold and let u be the solution of problem (Q), with E (0) < 0. Then, (u) ≥ C u pp11 . Therefore, (6.10) follows. As a special case, we have Corollary 6.3 Let the assumptions of Lemma 6.2 hold. Then we have where so we get This implies that This yields u p(x)dx = | | u p(x)dx + | | + − u p(x)dx , | | + u p1 + | | − u p2 ≥ | | + u p1 + c1 | | − u p1 | | p2 p1 . p1 c2 (u) p2 ≥ − u p1 and | | (u) ≥ + |u| p1 . p1 c2 (u) p2 + (u) ≥ u pp11 . (6.11) (6.12) (6.13) (6.14) (6.15) Since then (6.15) leads to Hence, (6.14) follows. Theorem 6.7 Let the conditions of Theorem 6.1 be fulfilled. Assume further that (6.1) holds and 0 < H (0) ≤ H (t ) ≤ (u), (u) 1 + c2 H (0) p1 u p1 . a Our main blowup result reads as follows: Then the solution (6.3) blows up in finite time. Proof We multiply Eq. (Q) by ut and integrate over to get hence H (t ) = 0 and E (0) < 0. E (t ) = 0, for every t in [0, T ), by virtue of (6.16). We then define for ε small to be chosen later and 0 < H (0) = H (t ) ≤ (u), L(t ) := H 1−α(t ) + ε uut (x , t )dx , 0 < α ≤ min 1, p1 − 2 2 p1 . By taking the derivative of (6.19) and using equation (Q), we obtain L (t ) = (1 − α)H −α(t )H (t ) + ε ut − |∇u|2 2 + εa u p(x). | | Add and subtract ε(1 − η) p1 H (t ), for 0 < η < 1, from the right-hand side of (6.21), to arrive at L (t ) ≥ ε(1 − η) p1 H (t ) + εaη u p(x) | | + ε + ε (1 − η) p1 2 − 1 2 ∇u 2. (1 − η) p1 2 + 1 2 ut 2 For η small enough, we see that where Therefore, using (6.14), we arrive at L (t ) ≥ εβ H (t ) + 2 2 ut 2 + ∇u 2 + (u) , β = min (1 − η) p1, aη, (1 − η) p1 2 + 1, (1 − η) p1 2 − 1 > 0. L (t ) ≥ γ ε H (t ) + 2 ut 2 + (u) ≥ γ ε H (t ) + 2 ut 2 + (6.16) (6.17) (6.18) (6.19) (6.20) (6.21) (6.22) (6.23) (6.24) Consequently, we have Next, we would like to show that L (t ) ≥ L (0) > 0, for all t ≥ 0. L (t ) ≥ where is a positive constant depending on εγ and C (the constant of Corollary 6.3). Once (6.25) is established, we obtain in a standard way the finite-time blowup of L (t ). To prove (6.25), we first note that L α/(1−α)(t ) ≥ 1 L −α/(1−α)(0) − t α/(1 − α) T ∗ ≤ 1 − α α[L (0)]α/(1−α) , Therefore, (6.28) shows that L (t ) blows up in finite time where and α are positive constant with α < 1 and L is given by (6.19) above. This completes the proof. Remark 6.8 The estimate (6.29) shows that the larger L (0) is, the quicker the blow up takes place. Acknowledgements The authors thank King Fahd University of Petroleum and Minerals for its support. This work is supported by KFUPM under Project no. FT 161004. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. which implies Again Young’s inequality gives uut (x , t )dx ≤ u 2 ut 2 ≤ C u p1 ut 2, uut (x , t )dx 1 1−α θ/(1−α) ut 2 ], Therefore, (6.26) becomes for μ1 + θ = 1. We take θ = 2(1 − α), to get μ/(1 − α) = 2/(1 − 2α) ≤ p1 by (6.20). 1 where s = 2/(1 − 2α) ≤ p1. Using Corollary 6.5, we obtain Finally, by noting that uut (x , t )dx 1/(1−α) 1/(1−α) ≤ C [ u sp1 + ut 22], uut (x , t )dx ≤ C [ H (t ) + p1 u p1 + ut 22], for all t ≥ 0. 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Salim A. Messaoudi, Ala A. Talahmeh. On wave equation: review and recent results, Arabian Journal of Mathematics, 2017, 1-33, DOI: 10.1007/s40065-017-0190-4