Article Contents
Article Contents

# 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.

 Citation:

•  [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg, 1976. [3] S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622.  doi: 10.1002/cpa.20195. [4] A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117. [5] G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axons, J. Theoret. Biol., 237 (2005), 316-322.  doi: 10.1016/j.jtbi.2005.04.018. [6] S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156.  doi: 10.1093/qjmam/4.2.129. [7] T. Hosono and T. Ogawa, Large time behavior and Lp-Lq estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.  doi: 10.1016/j.jde.2004.03.034. [8] M. Kac, A stochastic model related to the telegrapher's equation. Reprinting of an article published in 1956, Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972), Rocky Mountain J. Math., 4 (1974), 497–509. doi: 10.1216/RMJ-1974-4-3-497. [9] T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin, 1995. [10] S. Kawashima, Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, North-Holland Math. Stud., 98 (1984), 59-85.  doi: 10.1016/S0304-0208(08)71492-0. [11] P. Marcati and K. Nishihara, The Lp-Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469.  doi: 10.1016/S0022-0396(03)00026-3. [12] C. Mascia, Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes, J. Differential Equations, 260 (2016), 401-426.  doi: 10.1016/j.jde.2015.08.043. [13] C. Mascia and T. T. Nguyen, Lp-Lq decay estimates for dissipative linear hyperbolic systems in 1D, J. Differential Equations, 263 (2017), 6189-6230.  doi: 10.1016/j.jde.2017.07.011. [14] T. Narazaki, Lp-Lq estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.  doi: 10.2969/jmsj/1191418647. [15] K. Nishihara, Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.  doi: 10.1007/s00209-003-0516-0. [16] D. Serre, Matrices. Theory and Applications, Springer-Verlag, New York, 2002. [17] Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663. [18] Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal., 205 (2012), 239-266.  doi: 10.1007/s00205-012-0508-5. [19] T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.