# American Institute of Mathematical Sciences

April  2019, 39(4): 1651-1684. doi: 10.3934/dcds.2019073

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

 Gran Sasso Science Institute, Department of Mathematics, Viale Francesco Crispi 7, 67100 - L'Aquila, Italy

Received  August 2017 Published  January 2019

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in, for instance, the 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 and let
 $B\in \mathbb R^{n\times n}$
be not necessarily symmetric but have one single eigenvalue zero, we consider the Cauchy problem for
 $n\times n$
linear 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
 $u = u^{(1)}+u^{(2)}$
such that the asymptotic profile of
 $u^{(1)}$
denoted by
 $U$
is a solution to a parabolic equation,
 $u^{(1)}-U$
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 original system satisfies a symmetry property. The main proofs are based on the asymptotic expansions of the fundamental solution in the frequency space and the Fourier analysis.
Citation: Thinh Tien Nguyen. Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1651-1684. doi: 10.3934/dcds.2019073
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