Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions

TL;DR

This paper analyzes the long-time behavior of solutions to a class of dissipative linear hyperbolic systems in multiple dimensions, revealing decay rates and asymptotic profiles through Fourier analysis.

Contribution

It provides a detailed decomposition of solutions into asymptotic and exponentially decaying parts, with optimal decay rates established under symmetry conditions.

Findings

Solution decomposes into a parabolic profile and exponentially decaying part.

Asymptotic profile converges to a solution of a parabolic equation.

Decay rates are optimal and depend on symmetry properties.

Abstract

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf A:=(A^1,\dots,A^d)\in (\mathbb R^{n\times n})^d$ be a matrix-vector, where $A^j\in\mathbb R^{n\times n}$, and let $B\in \mathbb R^{n\times n}$ be not required to be symmetric but have one single eigenvalue zero, we consider the Cauchy problem for linear $n\times n$ systems having the form \begin{equation*} \partial_{t}u+\mathbf A\cdot \nabla_{\mathbf x} u+Bu=0,\qquad (\mathbf x,t)\in \mathbb R^d\times \mathbb R_+. \end{equation*} Under appropriate assumptions, we show that the solution $u$ is decomposed into $u=u^{(1)}+u^{(2)}$, where $u^{(1)}$ has the asymptotic profile which is the solution, denoted by $U$, of a parabolic equation and $u^{(1)}-U$ decays at the rate $t^{-\frac d2(\frac 1q-\frac 1p)-\frac 12}$ as $t\to +\infty$ in any $L^p$-norm, and $u^{(2)}$ decays exponentially in $L^2$-norm, provided $u(\cdot,0)\in L^q(\mathbb R^d)\cap L^2(\mathbb R^d)$ for $1\le q\le p\le \infty$. Moreover, $u^{(1)}-U$ decays at the optimal rate $t^{-\frac d2(\frac 1q-\frac 1p)-1}$ as $t\to +\infty$ if the system satisfies a symmetry property. The main proofs are based on asymptotic expansions of the solution $u$ in the frequency space and the Fourier analysis.