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

• 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.

Mathematics Subject Classification: Primary: 35L45; Secondary: 35C20.

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