Long‐time diffusive behavior of solutions to a hyperbolic relaxation system

We study the large time behavior of the solutions to the Cauchy problem for the system with relaxation source u_t+v_x=0, v_t+a²u_x=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 u_t+f(u)_x=a²u_xx in the L^p norm, at a rate faster than t^{−(p−1)/2p}.