L¹ Decay estimates for dissipative wave equations

Abstract

SUMMARYLet u and v be, respectively, the solutions to the Cauchy problems for the dissipative wave equationand the heat equationWe show that, as t → +∞, the norms 9 k t D x u(· ; t) L 1 (R n ) and 9 k t D x v(· ; t) L 1 (R n ) decay to 0 with the same polynomial rate. This result, which is well known for decay rates in L p (R n ) with 26p6 + ∞, provides another illustration of the asymptotically parabolic nature of the hyperbolic equation (1).

Diffusion phenomenon for second order linear evolution equations

“…In 1997 Nishihara [8] described the so-called diffusion phenomenon for quasilinear damped wave equations on 1-dimensional Euclidian space R in a concrete context, and Han-Milani [2] extended Nishihara's results to the case of N -dimensional Euclidian space R N for any quasilinear damped wave equation (see also Milani-Han [7] for another type of diffusion phenomenon). Furthermore, in [6] Karch has discovered the asymptotic self-similarity as t → ∞ of solutions to the equation (1.1) with A = −∆ in R N (in fact, he deals with more general dissipative wave equations).…”

On the long time behaviour of solutions to dissipative wave equations in $$\mathbb{R}^{2} $$

“…The aim of this paper is to prove that parabolic techniques can be used in the study of the long time behaviour of the solution to (2). Many results on the asymptotic decay of dissipative wave equations have been proved recently, in order to explain the asymptotically parabolic nature of the problem (2), (see [3], [4], [12], [14], [13]): it has been proved indeed that the wave equation with damping (2) has a diffusive structure if the time t goes to infinity.…”

“…Chern showed in [3] that the solution to a class of hyperbolic conservation laws with relaxation approaches the diffusive wave as time goes to infinity, if the initial data are a sufficiently small perturbation of a costant equilibrium state. In 2001, Milani and Yang have investigated in [13] the asymptotic decay rate of the norm of the solution u to (2) and its derivatives in the space L 1 (R N ): they have proved that this rate is exactly the same as the rate of the solution to the heat equation, with different assumptions on the initial data.…”

“…In this paper we study the long time behaviour of the solution of (1) by using completely different techniques from those of preceding works (see [3], [4], [12], [13]). Since the solution to (2) has a diffusive structure as the time t goes to infinity, we use a variant of the procedure which has been applied previously to study the large time behaviour of the solution to the Cauchy problem for convection-diffusion equations in [2].…”

Lp Decay problem for the dissipative wave equation in even dimensions

“…On the other hand, using the representation formulas of the solution u(t) of (1)- (2) for N 63, we have obtained L 1 type estimates for t¿0 in References [3,4] (see also Reference [5] for N = 3), that is u(t) L 1 6 C( u 0 Our goal in this paper is to study the L p (16p¡2) decay problem of the solution u(t) of (1)- (2) in any even dimensions (cf. Reference [6] for odd dimensions, and Ponce [7] for the heat equation).…”