Abstract
In this paper, we investigate the existence of a local solution in time and discuss the exponential asymptotic behavior to a weakly damped wave equation involving the variable-exponents
with simply supported boundary condition, where Ω is a bounded domain of \(\mathbb {R}^{n}\), g > 0 is a memory kernel that decays exponentially, and M(s) is a locally Lipschitz function. This kind of problem without the memory term when k(.) and p(.) are constants models viscoelastic Kirchhoff equation.
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1 Introduction
Let \({\Omega } \subset \mathbb {R}^{n}\) \(\left (n\geq 2\right ) \) be an open bounded domain with smooth boundary ∂Ω. In this paper, we are concerned with the existence and the asymptotic behavior of weak solutions to a weakly damped wave equation of Kirchhoff type with nonlinear damping and source terms involving the variable-exponent nonlinearities
where \(M\left (s\right ) \) is a locally Lipschitz function in s, g > 0 is a memory kernel, \(\left (.\right )^{\prime }\) denotes the derivative with respect to time t thus \(u^{\prime }=u_{t}=\frac {\partial u}{\partial t},\) \( u^{\prime \prime }=u_{tt}=\frac {\partial ^{2}u}{\partial t^{2}},\) γ1 positive real number, and Δ stands for the Laplacian with respect to the spatial variables. The exponents k(.) and p(.) are given measurable functions on Ω satisfying
where
We also assume that k satisfies the following Zhikov-Fan uniform local continuity condition:
In recent years, according to the values of the function M, a considerable effort has been devoted to the study of problem (1)–(3) in case of constant and variable-exponent nonlinearities.
Constant Exponent
In (1)–(3), when g ≥ 0 and k, p are constants, this equation has its origin in the nonlinear vibration of an elastic string, where the source term \(\left \vert u\right \vert ^{p-1}u\) forces the negative-energy solutions to explode in finite time while the dissipation term \(\left \vert u_{t}\right \vert ^{k-1}u_{t}\) assures the existence of global solutions for any initial data. Local, global existence and long-time behavior have been considered by many authors (see, for example, [27, 36, 47, 56] and the references therein). It is well known that Kirchhoff first investigated the following nonlinear vibration of an elastic string for f = g = 0:
where 0 < x < L, t ≥ 0, and u(x,t) is the lateral displacement, E the Young modulus, ρ the mass density, h the cross-section area, L the length, p0 the initial axial tension, δ the resistance modulus, and f and g the external forces. The above equation is described by the second order hyperbolic (1) and it is seemed to be important and natural that the equation with external forces is considered for analyzing phenomena in real world.
Kirchhoff type problems are often referred to as being nonlocal because of the presence of the integral over the entire domain Ω. It is analogous to the stationary case of equations that arisen in the study of string or membrane vibrations, namely,
where Ω is a bounded domain in \(\mathbb {R}^{N},u\) denotes the displacement, f(x,u) is the external force, and b is the initial tension while a is related to the intrinsic properties of the string (such as Young’s modulus). Equations of this type were first proposed by Kirchhoff in 1883 to describe the transversal oscillations of a stretched string, particularly taking into account the subsequent change in string length caused by oscillations. The solvability of the Kirchhoff type (4) has been well-studied in general dimensions and domains by various authors (see, for examples, [18, 19, 28, 30, 37, 45, 58, 60] and the references therein).
For the initial boundary value problem of Kirchhoff type equation
K. Nishihara and Y. Yamada [46] have proved the global existence of a unique solution under the assumptions that the initial data u0,u1 are sufficiently small and u0≠ 0 for the case a > 0, b = 0. For the problem with linear damping aut, there are the works of E. H. Brito [14], R. Ikehata [29], and the references therein. In the case of φ(s) = s, M. D. Silva Alves [3] has proved the existence of weak solutions of the unilateral problem using the Galerkin method. In [56], Taniguchi considered the following wave equation of Kirchhoff type with the damping term and the source term
where M(r) is a nonnegative locally Lipschitz function in r ≥ 0 and \( {\Delta } =\sum \partial ^{2}/\partial {x_{i}^{2}}\) is a Laplace operator, using the approximation of Faedo-Galerkin, he established the global existence and exponential asymptotic behavior of solutions. The initial boundary value problem for the following nonlinear Kirchhoff systems with memory term was considered in [9]
the authors established the nonexistence result of global solutions with the initial energy controlled above by a critical value, using energy method. Liu et al. [41], considered the initial boundary value problem for the following nonlinear Kirchhoff systems with memory term
using the energy method, the nonexistence result of global solutions with the initial energy controlled above by a critical value, was established. Li et al. in [36] considered the following initial boundary value problem for nonlinear wave equations of Kirchhoff systems with memory type in a bounded domain
where Ω is a bounded domain in \(\mathbb {R}^{n}\) (n ≥ 1) with a smooth boundary ∂Ω, p > 1, M(s) is a nonnegative C1 function like M(s) = a + bsγ for s ≥ 0, a ≥ 0, b ≥ 0, a + b > 0, γ > 0 and g(t) represents the kernel of memory term. Using the energy method, under some suitable assumptions on g and the initial data, they established a global nonexistence result for certain solutions with arbitrarily high energy. In the study of Yang et al. [61], they looked into the following equation
with suitable initial data and boundary conditions. Under certain assumptions on the kernel g and the initial data, by using the energy method together with simple analysis techniques, they established a new blow-up result for arbitrary positive initial energy. Wu et al. [59] considered the following initial boundary value problem for an integro-differential equation with strong damping in a bounded domain,
where h represents the friction or damping, and f the source. Using the energy method they established a decay estimates of the energy function and the estimates of the lifespan of blow-up solutions under specific conditions on the initial data.
The equations in (1)–(3) with M ≡ 1 form a class of nonlinear viscoelastic equations used to investigate the motion of viscoelastic materials. As these materials have a wide application in the natural sciences, their dynamics are interesting and of great importance.
Hence, questions related to the behavior of the solutions for the wave equation with Dirichlet’s boundary condition have attracted considerable attention from many authors. In particular, there are many results of proving the nonexistence and blow-up of solutions with negative initial energy (see [26, 33, 34, 48, 49] and a list of the references therein), also these results were obtained with the convexity method. However, much less is known when the initial energy is positive (cf. [2, 35, 50]), and these results used several other methods, for example, contradiction method, decomposition method, and so on.
The equations in (1) with \(M\left (s\right ) =a+bs\) and a > 0, b > 0 is the model to describe the motion of deformable solids as hereditary effect is incorporated, which was first studied by Torrejón and Yong [57]. They proved the existence of a weakly asymptotic stable solution for the large analytical datum. Later, Munoz Rivera [44] showed the existence of global solutions for small datum, and the total energy decays to zero exponentially under some restrictions.
Variable-Exponent Nonlinearity
Recently, a great deal of attention has been given to the investigation of nonlinear models of physical phenomena such as flows of electro-rheological fluids or fluids with temperature-dependent viscosity, nonlinear viscoelasticity, filteration processes through a porous media and image processing which give rise to equations with nonstandard growth conditions, that is, equations with variable exponents of nonlinearities. These models include hyperbolic, parabolic or elliptic equations that are nonlinear in gradient of the unknown solution and with variable exponents of nonlinearity. More details on these problems can be found in previous studies [1, 6,7,8, 17, 25, 38, 52, 53]. For instance, starting from the celebrated paper by Stanislav Antontsev [5], where he discussed the Dirichlet problem for following equation
with negative initial energy under suitable conditions on the functions a,b,f,p,σ, using the Galerkin method and energy method the local, global, and blow-up solutions have been established. The same problem (5) is considered in [4], where the authors proved the existence of solutions using the energy method. Lili Sun et al. [55] considered the following nonlinear hyperbolic equation,
Using the potential well, under some conditions on the initial data, the author obtained the lower and upper bounds for blowing up time. Autuori et al. in [9], thanks to approach of the classical potential well (energy) and concavity methods, investigated the question of the nonexistence of global solutions of dissipative nonlinear Kirchhoff systems, involving the p(x)-Laplacian operator and governed by nonlinear driving forces f
where Δp(x) denotes the vectorial p(x)-Laplacian operator defined as div(|∇u|p(x)− 2∇u), while the associated p(x)-Dirichlet energy integral is \({\mathscr{I}} u(t)={\int \limits }_{\Omega }\frac {1}{p(x)}|\nabla u(t,x)|^{p(x)}\mathrm {d}x.\) The functions f, M and Q represent a source force, a Kirchhoff dissipative term and an external damping term, respectively. Y. Gao and W. Gao [24] studied a nonlinear viscoelastic equation with variable exponents
and proved the existence of weak solutions by using the Faedo-Galerkin method under suitable assumptions. In related work, recently, Liao et al. [39] have regarded on the following viscoelastic hyperbolic equation of Kirchhoff type with initial-boundary value condition
By means of energy estimates, the bounds for blow-up time with positive initial energy were discussed, lower bounds for blow-up time in different range of exponent were obtained. A. Rahmoune [52], considered the following semilinear hyperbolic boundary value problem associated to the nonlinear generalized viscoelastic equations with the Direchlet-Neumann boundary conditions
where, F and G are nonlinear continuous functions, using the Faedo-Galerkin with fixed point theorem, he established the well-posedness of a weak solution. For the wave equation (g = 0), in the case of M ≡ 1, and p := p(.), k := k(.), A. Rahmoune [53] considered the following initial-boundary value problem
where the exponents k(.) and p(.) are given measurable functions on Ω satisfying
Based on the combination of the Faedo-Galerkin approximation and the compactness method, the author, under suitable assumptions on the variable exponents, established the existence of a unique weak solution. They also proved the uniform decay rates of solution.
The Faedo-Galerkin approach may be used within a variational formulation for studying more regular solutions of the equations under possibly weaker assumptions on the data, and also to prove a very useful tool for the numerical approximation of the equations [54]. In [43], the author has considered the abstract Cauchy problem and discussed the existence and Faedo-Galerkin approximation of the mild solution of the problem. In [12], authors have extended the results of [43] and considered the Faedo-Galerkin approximations of the solutions to a class of functional integrodifferential equation. In [37], the existence and Faedo-Galerkin approximation of the solution have been established, with the help of the Banach fixed point theorem and analytic semigroup theory. In [13], authors have considered a class of stochastic evolution integrodifferential equation and established the existence of the mild solution by using analytic semigroup. Utilizing the Faedo-Galerkin approximation, authors have also proven some convergence results in [13]. The Faedo-Galerkin approximations of the solutions of a stochastic differential equation are similar to the Faedo-Galerkin approximations of solutions of a deterministic problem. For more details on the existence of an approximate solution and Faedo-Galerkin approximations, we refer to papers [4, 5, 10,11,12,13, 15, 16, 32, 42, 51,52,53] and the references cited therein.
Problem (1)–(2) is the extension of those problems in which the variable-exponent are constants and g ≥ 0, concentrating our attention on difficulties caused by variable exponents p(x) and k(x).
In this paper, a class of a weakly damped wave equation of generalized Kirchhoff type with nonlinear damping and source terms involving the variable-exponent nonlinearity is considered. Hence, by using the Faedo-Galerkin arguments and compactness method as in [40], together with the Banach fixed point theorem, we will show the local existence of problem (1)–(3).
The purpose of this paper is to generalize the existence theorems of local solutions due to constant-exponents. In other words, we prove the existence of local solutions to weakly damped degenerate wave equations of Kirchhoff type (1)–(3) with nonlinear damping and source terms.
The contents of this paper are as follows. In Section 2, we give preliminaries and function spaces. In Section 3, we consider the existence of a local solution in time. In Section 4, we discuss the existence of a global solution to (1)–(3). In Section 5, we consider the exponential asymptotic behavior of the energy E(t) of a solution with a small initial value (u0,u1).
2 Preliminaries, Function Spaces
In this section, we list and recall some well-known results and facts from the theory of the Sobolev spaces with variable exponent (for the basic properties of the spaces W1,p(x)(Ω)and Lp(x)(Γ) we refer to [20,21,22,23, 31]).
Throughout the rest of the paper we assume that Ω is a bounded domain of \(\mathbb {R}^{n}\), n ≥ 1 with smooth boundary Γ and assume that p(.) is a measurable function on \(\overline {\Omega }\) such that
where
We also assume that p satisfies the following Zhikov–Fan uniform local continuity condition
Given a function \(p:\overline {\Omega }\rightarrow \left [ p^{-},\text {\ }p^{+} \right ] \subset \left (1,\infty \right ) ,\) p± = const, we define the set
The variable-exponent space Lp(.)(Ω) equipped with the Luxemburg norm
becomes a Banach space.
In general, variable-exponent Lebesgue spaces are similar to classical Lebesgue spaces in many aspects, see the first discussed the Lp(x) spaces and \(W^{k,p\left (x\right ) }\) spaces by Kovácik and Rákosnik in [31].
Let us list some properties of the spaces Lp(.)(Ω) which will be used in the study of the problem (1)–(3).
∙ If p(x) is measurable and \(1<p^{-}\leq p(x)\leq p^{+}<\infty \) in Ω, then Lp(.)(Ω) is a reflexive and separable Banach space and \(C_{0}^{\infty }({\Omega } )\) is dense in Lp(.)(Ω).
∙ If condition (6) is fulfilled, and Ω has a finite measure and p, q are variable exponents so that p(x) ≤ q(x) almost everywhere in Ω, the inclusion Lq(.)(Ω) ⊂ Lp(.)(Ω) is continuous and
∙ The variable Sobolev space \(W^{1,p\left (.\right ) }\left ({\Omega } \right ) \) is defined as the closure of \(C_{0}^{\infty }({\Omega } )\) with respect to the norm
It is known that for the elements of \(W_{0}^{1,p(.)}({\Omega } )\) the Poincaré inequality holds,
and an equivalent norm of \(W_{0}^{1,p(.)}({\Omega } )\) can be defined by
According to (6) \(W_{0}^{1,p(.)}({\Omega } )\subset W_{0}^{1,p^{-}}({\Omega } )\). If \(p^{-}>\frac {2n}{n+2},\) then the embedding \( W_{0}^{1,p^{-}}({\Omega } )\subset L^{2}({\Omega } )\) is compact.
∙ It is known that \(C_{0}^{\infty }({\Omega } )\) is dense in \( W_{0}^{1,p(.)}({\Omega } )\) according to (6) if \(p(x)\in C_{\log }(\overline {\Omega })\), that is, the variable exponent p(x) is continuous in \(\overline {\Omega }\) with the logarithmic module of continuity.
∙It follows directly from the definition of the norm that
∙The following generalized Hölder inequality
holds for all u ∈ Lp(.)(Ω), v ∈ \(L^{p^{\prime }(.)}({\Omega } )\) with \(p\left (x\right ) \in \left (1,\infty \right ) ,\) \( p^{\prime }\left (x\right ) =\frac {p\left (x\right ) }{p\left (x\right ) -1}.\) If \(p:{\Omega } \rightarrow \left [ p^{-},\text {\ }p^{+}\right ] \subset \left [ 1,+\infty \right ) \) is a measurable function and \(p_{\ast }>\underset { \left \{ x\in {\Omega } \right \} }{ess\text { }\sup }p\left (x\right ) \) with \( p_{\ast }\leq \frac {2n}{n-2}\), then the embedding \({H_{0}^{1}}\left ({\Omega } \right ) =W_{0}^{1,2}({\Omega } )\hookrightarrow L^{p(.)}({\Omega } )\) is continuous and compact.
Lemma 1 (Poincaré inequality)
[20] Let Ω be a bounded domain of \(\mathbb {R}^{n}\) and p(.) satisfies (6), then
D0 is the optimal constant of the Poincaré inequality and depends on p± and HCode \(\left \vert {\Omega } \right \vert \).
Lemma 2
[20] Let Ω be a bounded domain in \(\mathbb {R}^{n}\), n ≥ 1 with a smooth boundary Γ = ∂Ω, p(.) is a given measurable function on \(\overline {\Omega }\) satisfying condition (6) and q = const ≥ 1. If q ≤ p(x) a.e. in Ω, then
There are many proprieties of Lebesgue and Sobolev spaces with variable exponent, see the detailed exposition given in the monograph [6, Chapter 1].
Lemma 3 (Modified Gronwall inequality)
Let ϕ and h be nonnegative functions on \([0,+\infty )\) satisfying
with K > 0 and r > 0. Then,
as long as the right-hand side exists.
2.1 Mathematical Hypotheses
We begin this section by introducing some hypotheses for our main result. Throughout this paper, we use standard functional spaces and denote by (.,.), \(\left \Vert .\right \Vert \) the inner products and norms in L2(Ω) and \({H_{0}^{1}}\left ({\Omega } \right ) \), respectively, and they are given by
Next, we state sufficient assumptions for problem (1)–(3).
(H1) Hypotheses on M. Let \(M\in C\left (\left [ 0,+\infty \right ) , \mathbb {R}_{+}\right ) \) be a nonnegative locally Lipschitz function and for some positive constant m3 > 0,
(H2) Hypotheses on g. The memory kernel g has typical properties as follows \(g:[0,\infty )\rightarrow (0,\infty )\) is a nondecreasing C1 function such that
(H3) Hypotheses on p(.), k(.). Let k(.) and p(.) be given measurable functions on \(\overline {\Omega }\) satisfying the following condition
According to (9), we have
where
endowed with the Hilbert structure induced by \(H^{1}\left ({\Omega } \right ) \), is a Hilbert space and B > 0, B∗ > 0 are the optimal constants of the Sobolev embedding \({H_{0}^{1}}\left ({\Omega } \right ) \hookrightarrow L^{p^{+}+1}({\Omega } )\) and \({H_{0}^{1}}\left ({\Omega } \right ) \hookrightarrow L^{k^{+}+1}({\Omega } ),\) respectively.
3 Main Result
This section first presents the local existence and uniqueness of the solution for problem (1)–(3). Our method of proof is based on the combination of the Faedo-Galerkin approximation and the compactness method together with the Banach fixed point theorem (see [4, 5, 10,11,12, 42, 52, 53, 56]).
3.1 Existence of Local Solutions
In this section, we prove the existence of the local solution to the wave equation of Kirchhoff type (1)–(3) for any initial value \(\left (u_{0},u_{1}\right ) \in {H_{0}^{1}}\left ({\Omega } \right ) \cap H^{2}\left ({\Omega } \right ) \times {H_{0}^{1}}\left ({\Omega } \right ) \). For proving the main theorem we need the local existence and uniqueness of the solution to the following wave equation
where \(\varphi :\left [ 0,T\right ] \rightarrow {H_{0}^{1}}\left ({\Omega } \right ) \) is a continuous function. So we first prove the existence and uniqueness of the local solution to (P4). Let \(\left (w_{j}\right ) ,\) \(j=1,2,\dots ,\) be a completely orthonormal system in L2(Ω) having the following properties
∙\(\forall j;w_{j}\in {H_{0}^{1}}\left ({\Omega } \right ) \);
∙The family \(\left \{ w_{1},w_{2},\dots ,w_{m}\right \} \) is linearly independent;
∙Vm is the space generated by \(\left \{ w_{1},w_{2},\dots ,w_{m}\right \} ,\) ∪mVm is dense in \( {H_{0}^{1}}\left ({\Omega } \right ) \). We construct approximate solutions, um \(\left (m=1,2,3,\dots \right ) \) in Vm in the form
where Kjm(t) are determined by the following ordinary differential equation
and will be completed by the following initial conditions um(0), \({u_{t}^{m}}(0)\) which satisfy
Then, it holds that for any given \(v\in \text {Span}\left \{ w_{1},w_{2},\dots ,w_{m}\right \} ,\)
By virtue of the theory of ordinary differential equations, system (11), (12) and (13) has a unique local solution which is extended to a maximal interval \(\left [ 0,t_{m}\right [ .\)
A solution u to the problem (1)–(3) on some interval \(\left [ 0,t_{m}\right [ \) will be obtained as the limit of um as \(m\rightarrow \infty \). Then, this solution can be extended to the whole interval \(\left [ 0,T\right ] \), for all T > 0, as a consequence of the priori estimates that shall be proven in the next step. In this section, c > 0, c∗ > 0 and c∗ > 0 denote a various positive constants which change from line to line and are independent of natural number m and depend only (possibly) on the initial value.
Let us first recall a useful identity for the memory term which plays an important role in the sequel. By denoting
it is easy, by differentiating the term \(\left (\upbeta \diamond v\right ) \left (t\right ) \) with respect to t, to show that
We prove by the Galerkin method the following lemma on the existence and uniqueness of local solution to (P4) in time.
Lemma 4
Let M(.) be a nonnegative locally Lipschitz function. Let \(\left (u_{0},u_{1}\right ) \in {H_{0}^{1}}\left ({\Omega } \right ) \cap H^{2}\left ({\Omega } \right ) \times {H_{0}^{1}}\left ({\Omega } \right ) \). Assume that the differentiable function φ(t) satisfies φ(0) = u0, \(\varphi ^{\prime }(0)=u_{1}\) and consider the hypotheses (H1)– (H3). Then, there exists a time T0 = T0(u0,u1,m1,m2, m3) > 0 such that if there exist m1, m2, m3 > 0 and T > 0 satisfying \(\left \vert \nabla \varphi \left (t\right ) \right \vert \leq m_{1}\), \(\left \vert \nabla \varphi ^{\prime }\left (t\right ) \right \vert \leq m_{2},\) \(M\left (\left \vert \nabla \varphi \left (t\right ) \right \vert ^{2}\right ) \geq m_{3}>0\) for all t ∈ [0,T], then there exists a unique local weak solution in time u(t) to (P4) with the initial value (u0,u1) on [0,T0], where T0 ≤ T satisfying
Proof
By taking \(v={u_{t}^{m}}(t)\) in (13), we have that
Since it holds that
where L = L(m1) is a local Lipschitz constant for M(r), we have for \(t\in \left (0,t_{m}\right ) \)
Young’s inequality gives
consequently, taking (7) and (8) into account
Combining above results, and observing that g > 0 and \(g^{\prime }\leq 0\), we deduce
Thus, there exist B0 > 0, β0 > 0 and r0 > 0 such that
where we note that B0 and β0 are independent of m and r0. Since r0 > 0, there exists an enough small time T0:= T0(u0,u1,m3) ∈ (0,T1) satisfying
Thus, we have by the modified Gronwall lemma (Lemma 3)
Therefore, there exist constants ci = ci(u0,u1,m3) > 0 (i = 1,2,3) such that for any t ∈ [0,T0]
Furthermore, by (14) it follows that
where Ci are positive constants which are independent of m, γ1, and t. Thus, the solution can be extended to [0,T) and, in addition, we have
The second estimate \(\left (\text {Estimates on }u_{tt}^{_{m}}\right )\):
First of all, we are going to estimate \(u_{tt}^{_{m}}\left (0\right ) \). By taking t = 0 in (13), we get
Employing Green’s formula, (12) and Hölder’s inequality, we have
Therefore,
Now, by differentiating (13) with respect to t and substituting \( v=u_{tt}^{m}(t)\), we have
We need to do an analysis of the term \(2M^{\prime }(\left \vert \nabla \varphi \left (t\right ) \right \vert ^{2}) \left (\nabla \varphi ,\nabla \varphi ^{\prime }\right ) (\nabla u^{m},\nabla u_{tt}^{_{m}}(t))\). By multiplying both sides of (13) by \(f\left (t\right ) =\frac { 2M^{\prime }\left (\left \vert \nabla \varphi \left (t\right ) \right \vert ^{2}\right ) \left (\nabla \varphi ,\nabla \varphi ^{\prime }\right ) }{ M\left (\left \vert \nabla \varphi \left (t\right ) \right \vert ^{2}\right ) } \left (\leq \frac {2Lm_{1}m_{2}}{m_{3}}\right ) \) and replacing \( v=u_{tt}^{_{m}}(t)\), we have
By replacing above equality in (18), we have
Next, we are going to analyze the term on the right-hand side of (19), taking into account the estimates (15) and (16). Estimate for I1:
By using (7), (9) and the imbedding (10), we deduce that
Estimate for I2:
From the generalized Hölder’s inequality, it holds that
Estimate for I3:
Estimate for I4:
Estimate for I5:
Estimate for I6:
Estimate for I7:
Estimate for I8:
By replacing (20)–(21) in (19), we obtain
Employing Hölder’s inequality, Young’s inequality, integrating by parts on \(\left (0,t\right ) \), the first and the second terms on the right-hand side and the first term on the left-hand side of (22) can be estimated as follows, for
since from estimate (15) we have
and
and
Then,
where
Thus, there exist B1 > 0, β1 > 0 and r1 > 0 such that
where we note that B1 and β1 are independent of m and r1. Since r1 > 0, there exists an enough small time \(T_{2}\in \left (0,T_{1}\right ) \) satisfying
Thus, encore by the modified Gronwall lemma (Lemma 3), we have
Furthermore, by (23)
Here, let T1 ≥ T0 ≥ T2 without loss of generality, thanks to (17), (24), and (25), we obtain
From (26)–(28), there exists a subsequence of \(\left (u^{m}\right ) ,\) still denoted by \(\left (u^{m}\right ) \), such that
Since \({H_{0}^{1}}({\Omega } )\hookrightarrow L^{2}({\Omega } )\) is compact and from the Aubin-Lions theorem, we deduce that
and consequently, by making use of Lion’s Lemma [40, Lemma 1.3], we have
The convergences (29)–(33) permit us to pass to the limit in (13). Since \(\left (w_{j}\right ) \ \)is a basis of \({H_{0}^{1}}\left ({\Omega } \right ) \) and Vm is dense in \({H_{0}^{1}}\left ({\Omega } \right ) \), after passing to the limit, we obtain
for all \(\theta \in D\left (0,T_{0}\right ) \), and for all \(v\in {H_{0}^{1}}\left ({\Omega } \right ) .\)
By taking \(v\in D\left ({\Omega } \right ) \), we get that
Therefore,
Finally, we prove the uniqueness of the local solution. To this end, let u(t) and v(t) be two local solutions to (13) with the same initial value. Let w(t) = u(t) − v(t). Then, w(0) = 0, wt(0) = 0 for all t ∈ [0,T0] and
for all \(\psi \in {H_{0}^{1}}\left ({\Omega } \right ) .\) By replacing \(\psi =w_{t}\left (t\right ) \) in (34) and observing that the function \( h\left (v\right ) =\left \vert v\left (t\right ) \right \vert ^{k\left (.\right ) -1}v\left (t\right ) \) is monotone, it holds that
From generalized Hölder’s and Young’s inequalities and taking estimates (26)–(29) into account, it holds that
Substituting the last two inequalities in (35) and integrating the results over (0,t), since \(\left (g\diamond \nabla w\right ) \left (t\right ) -\left (g^{\prime }\diamond \nabla w\right ) \left (t\right ) +g\left (t\right ) \left \vert \nabla w\left (t\right ) \right \vert ^{2}\geq 0,\) it holds
Thus, employing Gronwall’s lemma, we conclude that \(\left \vert w^{\prime }(t)\right \vert ^{2}=\left \vert \nabla w\left (t\right ) \right \vert ^{2}=0.\) Consequently, this completes the proof of the lemma.
We are concerned with the existence and uniqueness of local solution in time to degenerate wave equation (1)–(3). So by using Lemma 4, we prove the existence and uniqueness of local solution in time to (1)–(3) by the Banach fixed point theorem. □
Theorem 1
Assume that M(r) is a nonnegative locally Lipschitz function and assume that the following condition is satisfied
Let \(\left (u_{0},u_{1}\right ) \in {H_{0}^{1}}\left ({\Omega } \right ) \cap H^{2}\left ({\Omega } \right ) \times {H_{0}^{1}}\left ({\Omega } \right ) \) with \( \left \vert \nabla u_{1}\right \vert \neq 0\) or \(\left \vert \nabla u_{0}\right \vert \neq 0\). Assume that \(M(\left \vert \nabla u_{0}\right \vert ^{2}) >0\). Then, there exist a time T0 > 0 and a unique local weak solution u(t) to (1)–(3) with the initial value \(\left (u_{0},u_{1}\right ) \) satisfying
Proof
Since M(|∇u0|2) > 0, we have a positive real number m3 such that 0 < m3 < M(|∇u0|2). We may assume that \(m_{3}-{\int \limits }_{0}^{+\infty }g\left (t\right ) \mathrm {d}t<1\). Let R0 be a positive real number such that
Since \(M(\left \vert \nabla u_{0}\right \vert ^{2}) >0\), for sufficiently small time T > 0, we define the space BT(R0) by
We introduce the metric d on the space BT(R0) by
Then, the space BT(R0) is a complete metric space. Let ϕ ∈ BT(R0). Then, \(\left \vert \nabla \phi (t)\right \vert \leq R_{0}\), \( \left \vert \nabla \phi ^{\prime }(t)\right \vert \leq R_{0}\) and M(|∇ϕ(t)|2) ≥ m3 for all t ∈ [0,T]. Thus, thanks to Lemma 4 we obtain a unique local weak solution u(t) on [0,T1] with T1 ≤ T to the following wave equation:
Here, let T = T1 without loss of generality. Define the mapping Φ by
Then, we have that
For showing (37), posing v = ut in (36), we have that
So, we have the estimates for I1 and I2 as follows
And by taking estimates (29) into account
Thus, we have
where
and \(C_{1}=\frac {L{R_{0}^{2}}}{l}\). The simple calculations Gronwall inequality yields
for sufficiently small 0 < T3 ≤ T1. Thus,
We have that
Let T = T3. Thus, (37) is satisfied. Next, we show (38). Let w = u1 − u2, where \(u_{1}={\Phi } \left (\varphi _{1}\right ) ,\) \( u_{2}={\Phi } \left (\varphi _{2}\right ) \) with φ1, φ2 ∈ BT(R0). Then, we have that
Set
Since \(0<l=m_{3}-{\int \limits }_{0}^{\infty }g\left (s\right ) \mathrm {d}s<1\), we have that
By replacing v in (39) by wt, we have that
Then,
and
Since
It follows that
by the Gronwall lemma we have that
Choose a 0 < T3 ≤ T small enough which satisfies that
Thus, by the Banach contraction mapping theorem, there exists a fixed point \(u={\Phi } (u)\in B_{T_{3}}(R_{0})\), which is a unique local weak solution in time to (1)–(3). This completes the proof of the theorem. □
4 Global Existence
In this section, we consider the asymptotic behavior of solutions to (1)–(3). In this section, we assume that
In order to formulate our results, it is convenient to introduce the energy of the system
where
then we have
so the energy E(t) is a nonincreasing function.
In this note, we shall extend the above exponential rate of decay to the general case, which is similar to that of g. We use the following assumption which is weaker than (8).
-
(H4)
Hypotheses on g, ξ. The differentiable function ξ(t) satisfies
$$ \begin{array}{@{}rcl@{}} g^{\prime }\left( t\right) &\leq &-\zeta \left( t\right) g\left( t\right) \text{ for all }t\geq 0, \\ \zeta \left( t\right) &\geq &0,\text{ }\zeta^{\prime }\left( t\right) \leq 0,\text{ \ }\forall t>0,\text{ \ }{\int}_{0}^{\infty }\xi \left( t\right) \mathrm{d}t=+\infty . \end{array} $$
Remark 1
There are many functions satisfying (H4). Examples of such functions are
for β, b > 0 to be chosen properly.
Before we state and prove our global existence result, we need the following lemma.
Lemma 5
Let u be the solution of (1)–(3). If the initial data satisfies that u0 ∈ Hγ and
then \(u\left (t\right ) \) ∈ Hγ, for each t ∈ [0,T]. Here, B(p,Ω) is the Sobolev-Poincaré constant, and
Proof
Let u0 ∈ Hγ, then Iγ(u0) > 0. By continuity, this implies the existence of Tm ≤ T such that Iγ(u(t)) ≥ 0 for all t ∈ [0,Tm]. Therefore, from (41), (42), and (43), we have
Thus, it follows that
This relation together with (10), implies that for t ∈ [0,Tm],
That is, by our assumption on α
Hence,
which means that u(t) ∈ Hγ, \(\forall t\in \left [ 0,T_{m}\right ] \).
Since the energy E is decreasing along trajectories, we have the following inequality
By repeating the procedure, Tm extends in an increasing manner until it reaches the bound T. □
Theorem 2
Suppose that (H1)–(H3) and assumptions of Lemma 5 hold. If \(\left (u_{0},u_{1}\right ) \in {H_{0}^{1}}\left ({\Omega } \right )\) \( \cap H^{2}\left ({\Omega } \right ) \times {H_{0}^{1}}\left ({\Omega } \right ) \). Then, the solution is global and bounded.
Proof
From (46), we have
since Iγ(t), \(\left (\frac {\alpha }{2\left (\gamma +1\right ) }- \frac {\alpha }{p^{-}+1}\right ) \) and \(\left (g\circ \nabla \left (u\right ) \right ) \left (t\right ) \) are positive. Therefore,
where C is a positive constant which depends only on l. □
5 Asymptotic Behavior of Solutions
We have the following theorem on the asymptotic behavior of the solutions of the degenerate wave equation of Kirchhoff type.
Theorem 3
Assume that (40), \(\left (u_{0},u_{1}\right ) \in {H_{0}^{1}}\left ({\Omega } \right ) \cap H^{2}\left ({\Omega } \right ) \times {H_{0}^{1}}\left ({\Omega } \right ) \) with 0≠u0 ∈ Hγ and
are considered and furthermore an initial value \(\left (u_{0},u_{1}\right ) \) with u0≠ 0 is sufficiently small, that is (44) to be satisfy. If the following conditions are satisfied,
then the energy E(t) of the solution u(t) to (1)–(3) converges to zero exponentially.
Proof
By Theorem 1, there exists the unique local solution u(t) in time to (1)–(3) with the initial value (u0,u1). Set
where ε > 0 is a positive real number. Then,
by adding \(\varepsilon \xi \left (t\right ) E\left (t\right ) -\varepsilon \xi \left (t\right ) E\left (t\right ) \), we get
where ε > 0 is a positive real number. Then, by the Young and Poincaré inequalities, from (47) we have that
similarly
and for γ1 > 0, it is easy to show that,
Thus,
then
By choosing a sufficiently small ε > 0, we obtain that
Hence, from (44)
Thus, by choosing a sufficiently small δ > 0, we obtain that
On the other hand
Then, there exists a positive real number α > 0 such that
Thus, we have that
which completes the proof of the theorem. □
Remark 2
We illustrate the energy decay rate given by Theorem 3 through the following examples which are introduced by many authors (1) If
then (48) gives the exponential decay estimate
Similarly, if
then we obtain the polynomial decay estimate
(2) If
then (48) gives the exponential decay estimate
(3) If
then (48) gives the exponential decay estimate
Remark 3
This result generalizes and improves the results of [27, 47, 56]. In particular, in [27], it allows some Lipschitz function which satisfies
instead of the usual assumption in (7), \(M\in C\left (\left [ 0,+\infty \right ) ,\mathbb {R}_{+}\right ) \) be a nonnegative locally Lipschitz function and for positive constant m3 > 0,
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Abita, R. Existence and Asymptotic Behavior of Solutions for Degenerate Nonlinear Kirchhoff Strings with Variable-Exponent Nonlinearities. Acta Math Vietnam 46, 613–643 (2021). https://doi.org/10.1007/s40306-021-00420-7
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DOI: https://doi.org/10.1007/s40306-021-00420-7
Keywords
- Weakly damped wave equations
- Weak solutions
- Viscoelasticity
- Asymptotic behavior of solutions
- Lebesgue and Sobolev spaces with variable exponents