Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

Abstract

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained.In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.

Large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation

“…Remark 1.3. The similar results as (1.17) and (1.18) with l = 0 are obtained by the first author, for the generalized KdV-Burgers equation [3] and the damped wave equation with a nonlinear convection term [5]. We emphasize that our results (1.17) and (1.18) are more general and include the estimates for derivatives of the solutions.…”

“…Next, we introduce the L p -decay estimate for the heat kernel G(x, t) defined by (1.21), and the estimate for the convolution G(t) * φ (for the proof, see Lemma 7.1 in [24] and Lemma 2.4 in [5]).…”

“…Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. To obtain the second asymptotic profiles, we develop the method used in [4,5] to study the Cauchy problem of the KdV-Burgers equation [4] and of the damped wave equation with a nonlinear convection term [5]. Now, let us state our first main result which generalizes the results given in [6] and treat the case of the initial data decays more slowly: Theorem 1.1.…”

“…In view of the second asymptotic profiles, we are able to investigate the optimality of the asymptotic rate to the nonlinear diffusion wave χ(x, t) (see Corollary 1.4 below). Actually, an upper bound and a lower bound in L ∞ -norm of the functions Z(x, t) and V (x, t) are obtained in [4,5]. More precisely, there are strictly positive constants ν 0 and ν 1 such that…”

Localisation of perturbations of a constant state in a traffic flow model

“…Then Theorem 1.1 can not be applied to the case ρ i = ε(χ(x − A) + χ(x + A)), u i = 0 for ε > 0 small, uniformly in A (it would require A ≪ 1 ε ). The stability has been shown for a larger space of perturbation than Theorem 1.1, but for simplier hyperbolic systems, see [7] and [15].…”