Memoire Online - Existence et comportement asymptotique des solutions d'une équation de viscoélasticité non linéaire de type hyperbolique

In this thesis we consider the following nonlinear viscoelastic hyperbolic problem

where Q is a bounded domain in R^N (N > 1), with smooth boundary F, a, b, w are positive constants, and n7, > 2, p > 2. The function g(t) is assumed to be a positive nonincreasing function defined on R^#177; and satisfies the following conditions:

In the physical point of view, this type of problems arise usually in viscoelasticity. This type of problems have been considered first by Dafermos [12], in 1970, where the general decay was discussed. A related problems to (1) have attracted a great deal of attention in the last two decades, and many results have been appeared on the existence and long time behavior of solutions. See in this directions [2, 3, 5 - 8, 18, 29, 33, 34, 38] and references therein.

In the absence of the strong damping Aut, that is for w = 0, and when the function g vanishes identically ( i.e. g = 0), then problem (1) reduced to the following initial boundary damped wave equation with nonlinear damping and nonlinear sources terms.

Some special cases of equation (2) arise in quantum field theory which describe the motion of charged mesons in an electromagnetic field.

Equation (2) together with initial and boundary conditions of Dirichlet type, has been extensively
studied and results concerning existence, blow up and asymptotic behavior of smooth, as well

as weak solutions have been established by several authors over the past three decades. Some interesting results have been summarized by Said-Houari in his master thesis [42].

For b = 0, that is in the absence of the source term, it is well known that the damping term a utj^m-2 ut assures global existence and decay of the solution energy for arbitrary initial data ( see for instance [17] and [21]).

For a = 0, the source term causes finite-time blow-up of solutions with a large initial data ( negative initial energy). That is to say, the norm of our solution u(t, x) in the energy space reaches +oc when the time t approaches certain value T^* called " the blow up time", ( see [1] and [20] for more details).

The interaction between the damping term a utj^m-2 ut and the source term b uj^p-2 u makes the problem more interesting. This situation was first considered by Levine [23,24] in the linear damping case (m = 2), where he showed that solutions with negative initial energy blow up in finite time T^*. The main ingredient used in [23] and [24] is the " concavity method" where the basic idea of this method is to construct a positive function L(t) of the solution and show that for some 'y > 0, the function L^-Y(t) is a positive concave function of t. In order to find such 'y, it suffices to verify that:

d²L^-Y(t)	= --yL^-"^-2(t) [LL^'' - (1 + y)L^'2(t)] 0, Vt ~ 0.
dt²

Georgiev and Todorova in their famous paper [14], extended Levine's result to the nonlinear damping case (m > 2). More precisely, in [14] and by combining the Galerkin approximation with the contraction mapping theorem, the authors showed that problem (2) in a bounded domain with initial and boundary conditions of Dirichlet type has a unique solution in the interval [0, T) provided that T is small enough. Also, they proved that the obtained solutions continue to exist globally in time if m ~ p and the initial data are small enough. Whereas for p > m the unique solution of problem (2) blows up in finite time provided that the initial data are large enough. ( i.e. the initial energy is sufficiently negative).

This later result has been pushed by Messaoudi in [35] to the situation where the initial energy E(0) < 0. For more general result in this direction, we refer the interested reader to the works of Vitillaro [47], Levine [25] and Serrin and Messaoudi and Said-Houari [32].

In the presence of the viscoelastic term (g =6 0) and for w = 0, our problem (1) becomes

For a = 0, problem (3) has been investigated by Berrimi and Messaoudi [3]. They established the local existence result by using the Galerkin method together with the contraction mapping theorem. Also, they showed that for a suitable initial data, then the local solution is global in time and in addition, they showed that the dissipation given by the viscoelastic integral term is strong enough to stabilize the oscillations of the solution with the same rate of decaying ( exponential or polynomial) of the kernel g. Also their result has been obtained under weaker conditions than those used by Cavalcanti et al [7], in which a similar problem has been addressed.

Messaoudi in [29], showed that under appropriate conditions between m, p and g a blow up and global existence result, of course his work generalizes the results by Georgiev and Todorova [14] and Messaoudi [29].

One of the main direction of the research in this field seems to find the minimal dissipation such that the solutions of problems similar to (3) decay uniformly to zero, as time goes to infinity. Consequently, several authors introduced different types of dissipative mechanisms to stabilize these problems. For example, a localized frictional linear damping of the form a(x)ut acting in sub-domain w c Q has been considered by Cavalcanti et al [7]. More precisely the authors in [6] looked into the following problem

utt -- Au +

g(t -- s)Au(s,x)ds + a(x)ut + lul^~ u = 0. (4)

for 7 > 0, g a positive function and a : Q -> IR⁺ a function, which may be null on a part of the domain Q.

By assuming a(x) > a0 > 0 on the sub-domain w c Q, the authors showed a decay result of an exponential rate, provided that the kernel g satisfies

This later result has been improved by Berrimi and Messaoudi [2], in which they showed that the viscoelastic dissipation alone is strong enough to stabilize the problem even with an exponential rate.

For a viscoelastic systems with oscillating kernels, we mention the work by Rivera et al [36]. In that work the authors proved that if the kernel satisfies g(0) > 0 and decays exponentially to zero, that is for p = 1 in (6), then the solution also decays exponentially to zero. On the other hand, if the kernel decays polynomially, i.e. (p > 1) in the inequality (6), then the solution also decays polynomially with the same rate of decay.

In the presence of the strong damping (w > 0) and in the absence of the viscoelastic term (g = 0), the problem (1) takes the following form

₈

<>>>>>>>

>>>>>>>:

utt ~ u ~ !~ut + a jutj^m-2 ut = b juj^p-2 u, x 2 ~, t > 0

u(0,x) = u0 (x), u2 (0,x) = u1 (x), x 2 ~ (7)

u(t,x) = 0, x 2 [', t > 0

Problem (7) represents the wave equation with a strong damping ~ut. When m = 2, this problem has been studied by Gazzola and Squassina [13]. In their work, the authors proved some results on well posedness and asymptotic behavior of solutions. They showed the global existence and polynomial decay property of solutions provided that the initial data is in the potential well.

The proof in [13] is based on a method used in [19]. Unfortunately their decay rate is not optimal, and their result has been improved by Gerbi and Said-Houari [16], by using an appropriate modification of the energy method and some differential and integral inequalities.

Introducing a strong damping term Iu makes the problem from that considered in [42] and [14], for this reason less results where known for the wave equation with strong damping and many problem remain unsolved. ( See [13] and the recent work by Gerbi and Said-Houari [15]).

In this thesis, we investigated problem (1), in which all the damping mechanism have been considered
in the same time ( i.e. w > 0, g =6 0, and m ~ 2), these assumptions make our problem different

form those studied in the literature, specially the blow up result / exponential growth of solutions (chapter4).

In this chapter we introduce some notation and prepare some material needed for our work. The main results of this chapter such as: the L^p-- spaces, the Sobolev spaces, differential and integral inequalities and other theorems of functional analysis, can found in the books [4] and [43]. Chapter2:

This chapter is devoted to the study of the local existence result, the main ingredient used in this chapter is the Galerkin approximations ( the compactness method) introduced in the book of Lions [26], together with the fix point method.

Indeed, we consider first for u 2 C ([0, T] , H¹ 0) given, the following problem

vtt - Lv - WLvt + _Zt g(t - s)Lv(s, x)ds + ajvtj^m~2 vt = bjuj^p~2 u, x 2 , t > 0 (8)

and we will show that problem (8) - (10) has a unique local solutions v by the Faedo-Galerkin method, which consists in constructing approximations of the solution, then we obtain a priori estimates necessary to guarantee the convergence of these approximations. We recall here that the presence of nonlinearity on the damping term a vtj^m~2 vt forces us to go until the second a priori estimate. We point out that the contraction semigroup method fails here, because of the presence of the nonlinear terms.

Once the local solution v exists, we will use the contraction mapping theorem to show the local existence of our problem (1). This will be done under the assumption that T is required to be small enough (see formula (2.77)).

First, we introduce a set W defined in (3.5) called " the potential well" or " stable set" and we show that if we restrict our initial data in this set, then our solution obtained in chapter 2 is global in time, that is to say, the norm

in the energy space L²(l) x H¹₀(l) of our solution is bounded by a constant independent of the time t.

Second, We show that, if our solution is global in time, ( i.e. by assuming that the initial data u0 2 W) and if our function g satisfies the condition (6) ( for p = 1), then our solution decays time asymptotically to zero. More precisely we prove that the decay rate is of the form (1 + t)21(2_m) if n-i > 2, whereas for n-i = 2, we obtain an exponential decay rate. (See Theorem 3.2.1). The main tool used in our proof is an inequality due to Nakao [37], in which this inequality has been introduced in order to study the stability of the wave equation, but it is still works in our problem. Chapter 4:

In this chapter we will prove that if the initial energy E(0) of our solution is negative ( this means that our initial data are large enough), then our local solutions in bounded and

as t tends to +oo. In fact it will be proved that the L^p--norm of the solution grows as an exponential function. An essential tool of the proof is an idea used by Gerbi and Said-Houari [15], which based on an auxiliary function ( which is a small perturbation of the total energy), in order to obtain a differential inequality leads to the exponential growth result provided that the following conditions

Existence et comportement asymptotique des solutions d'une équation de viscoélasticité non linéaire de type hyperbolique

Introduction