Note on propagation speed of travelling waves for a weakly coupled parabolic system. (Q5934268)

Consider the system of two coupled reaction-diffusion equations NEWLINE\[NEWLINEw_t= Dw_{xx}+ h(w)\tag{\(*\)}NEWLINE\]NEWLINE with \(D= \text{diag}(1,d)\), \(d> 0\), \(x\in\mathbb R\).NEWLINENEWLINELet \(w= u(\xi)= u(x-ct)\) be a travelling wave solution of \((*)\) satisfying NEWLINE\[NEWLINE\lim_{\xi\to\pm\infty}\, u(\xi)= u_0,\quad h(u_0)= 0.NEWLINE\]NEWLINE The author proves that if \((*)\) has a nonconstant travelling wave and if \(u_0\) is an exponentially stable equilibrium of \(\dot w= h(w)\), where \(h\) satisfies two additional assumptions, then it holds \(c= 0\).