Affiliations: Institute of Analysis and Numerics, Otto‐von‐Guericke‐University Magdeburg, PSF 4120, D‐39106 Magdeburg, Germany E‐mail: [email protected] | Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, I–00161 Rome, Italy E‐mail: [email protected]
Note: [] Current address: Department of Mathematics, UCLA, Los Angeles, CA 90095‐1555, USA.
Abstract: We study the large time behavior of the solutions to the Cauchy problem for the system with relaxation source ut+vx=0, vt+a2ux=f(u)−v, with $(x,t)\in{\mathbb{R}} \times{\mathbb{R}} ^{+}$, for the initial data (u,v)=(u0,v0) at t=0, with $u_{0},v_{0}\in L^{1}({\mathbb{R}})\cap L^{\infty}({\mathbb{R}})$, f(u)=αu2/2 and |v0|≤au0. Under the sub‐characteristic condition we show that, as t→∞, the component u tends towards a fundamental solution of the convection‐diffusion equation ut+f(u)x=a2uxx in the Lp norm, at a rate faster than t−(p−1)/2p.
