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.