SPATIAL DECAY ESTIMATES FOR A CLASS OF NONLINEAR DAMPED HYPERBOLIC EQUATIONS

This paper is concerned with investigating the spatial decay estimates for a class of nonlinear damped hyperbolic equations. In addition, we compare the solutions of two-dimensional wave equations with different damped coefficients and establish an explicit inequality which displays continuous dependence on this coefficient. 2000 Mathematics Subject Classification. 35B45, 35L70, 80A20, 30C80.

1. Introduction.Spatial decay estimates for several types of partial differential equations and systems have been the subject of extensive investigations in the literature for close to a century and a half.These studies were motivated by a desire to formulate Saint-Venant and Phragmén-Lindelöf type principles in elasticity and heat conduction.Roughly speaking, these estimates assert that the solution of the problem decays exponentially with distance from the boundary on which a mechanical or thermal "load" has been applied.In the case of elliptic problems, this work has been directed toward establishing a rational form of Saint-Venant's principle and has included studies in linear elasticity (see Toupin [18] and Knowles [9]), in nonlinear plane elasticity (see Roseman [16]) and in linear viscoelasticity (see Edelstein [4]).In a recent paper, Tahamtani [17] derived an explicit Saint-Venant type decay estimate for solutions of the Dirichlet problem for nonlinear biharmonic equations defined in a semi-infinite cylinder in R n with homogeneous Dirichlet data on the lateral surface of the cylinder.
A spatial decay estimate for transient heat conduction was first given by Edelstein [3].The result has been consistently improved by the studies completed by Knowles [10], Horgan et al. [7], and Chiriţǎ [2].
Very little attention has been devoted to the study of hyperbolic differential equations.Horgan and Knowles [6] and Horgan [5] pointed out the paucity of Saint-Venant type results for hyperbolic system of the kind describing elastic wave propagation.The only previous work known to us on questions like this for the hyperbolic differential equations is that of Quintanilla [15].He considered the transient solutions of the damped wave equation and established a spatial decay estimate of the kind described by Knowles [10] for the heat conduction equation.The results we present here generalize the work in [15] to nonlinear damped hyperbolic equations and obtain stronger results involving an exponential decay of energy functional.
In this paper, we show that if the solution is bounded in an energy norm, then it must decay exponentially in energy norm as the distance from the near end tends to infinity.Finally, we compare the solutions of two damped wave equations with different damped coefficients and establish an explicit inequality which displays continuous dependence on this coefficient.

Preliminaries.
In this paper, we derive a spatial decay estimate for a functional defined on the solutions of the equation where α and ν are two positive numbers, ∆ is the Laplace operator, and f is a nonlinear function satisfying the inequalities Our attention is focused on the initial-boundary problem for (2.1) in the space-time region Ω × (0, ∞), where is the semi-infinite prismatic cylinder and σ x 1 denotes the open, bounded, and simply connected cross section of Ω.In addition, u(x 1 ,x ,t) is required to satisfy the initial and boundary conditions ) where the function g(x ,t) is a prescribed function and vanishes on the boundary ∂σ x 1 .For convenience, we introduce the notation We describe the quantity where C 1 0 (D) is the set of functions that are continuously differentiable with compact support in D. In [13] examples are given, where for an analogous λ p , a lower estimate can be found by means of the first eigenvalues of some elliptic boundary-value problem on D. We note that for p = 2, (2.8) is the Poincaré-Friedrich's inequality see [11].Young's inequality is used often in this article.It states that for x, y > 0 and arbitrary ε > 0.

A decay theorem.
In this section, we state a spatial decay estimate for the solution of the problem defined by (2.1), (2.4), (2.5), and (2.6).We recall that the following equalities: are satisfied for all solutions of the nonlinear equation (2.1).Let δ = ν/(1+α); we may consider To obtain our estimates, it is suitable to recall that (see [15, Lemma 2.1, page 80]) where ) Applying Hölder's and Young's inequalities we can estimate the second and fourth terms of (3.4) as follows: ) Using the quantity in (2.8), we find from (3.6) Dropping the first term on the right-hand side of (3.4), then from (3.4), (3.7), and (3.8) we obtain where C Ωτ is a positive constant depending only on Ω τ .
If we now set w = u − v, then w satisfies Using the methods of [12], we can treat the case in which v ≠ u on the end x 1 = 0. Calculations similar to those used in Section 3 lead to the equalities Similar to the definition of F in (3.2), we may define where δ is a positive constant to be specified later.By similar calculation techniques of the previous section, from (4.4) we deduce where Taking δ > 0 so small that ν > (ε/2) δλ −2 (Ω τ ), we obtain where We now choose(M 1 (τ) − β)M −1 0 (τ) = ω −1/2 (τ).But (4.14) may then be rewritten as the solution of the same problem with ν replaced by ν.Then for arbitrary τ ≥ 0, t ≥ 0 the closeness of u and v in energy measure satisfies the following inequality: We have thus established the following theorem.Theorem 4.1.Let u be the solution of the problem (4.1),(2.4),(2.5),and (2.6) and v