Convergence to traveling waves with decay rates for solutions of the initial boundary problem to a relaxation model
The authors consider the following particular relaxation model \[ \begin{cases} u_t+v_x=0,\\ v_t+au_x={\bigl(f(u)-v\bigr)\over\varepsilon},\;(x,t)\in \mathbb{R}_+ \times\mathbb{R}_+, \end{cases} \tag{1} \] by assuming the initial boundary conditions \[ \begin{cases} (u,v)|_{t=0}= (u_0,v_0)(x) \to(u_+,v_+), \quad\text{as }x \to\infty, \\ v|_{x=0}=g(t), \end{cases}\tag{2} \] where \((u_+,v_+)\), satisfying \(v_+=f(u_+)\), is one of the end constant states of the traveling wave solutions \((U,V)(x-st)\) of (1) corresponding to the Cauchy problem, with other end states given as \((u_-,v_-)\) and satisfying \(v_-=f'(u)\). The function \(f\in C^2\), is nonconvex and \(a\) is a positive constant. This problem has been studied by many authors relative to similar problems. Here Mei and Rubino prove, by analizing at first the case \(g(t)=v_-\) or \(v_+\) (with \(s\neq 0)\) and then the case \(g(t)=v_\mp\) \((s=0)\), that the solutions \((u,v)(x,t)\) of the problem (1), (2) tend to the traveling wave solutions of the corresponding Cauchy problem time-asymptotically.
scientific article; zbMATH DE number 1389120