Stability of travelling-wave solutions for reaction-diffusion-convection systems (Q5929986)

The paper deals with the asymptotic behaviour of the solutions of the system NEWLINE\[NEWLINEu_t= Au_{xx}+ f(u,u_x),\quad x\in\mathbb{R},\quad t>0,\quad u(x,t)\in \mathbb{R}^N,\quad u(x,0)= \varphi(x),NEWLINE\]NEWLINE where \(A\) is a positive definite diagonal matrix and \(f\) is a ``bistable'' nonlinearity such that the comparison principle holds. Under certain assumptions on \(\varphi\), it is shown that there exists \(x\in \mathbb{R}\) such that NEWLINE\[NEWLINE\|u(\cdot,t)- w(\cdot+ x-ct)\|_{\text{BUC}^1}\to 0\quad\text{as }t\to\infty.NEWLINE\]