Creeping fronts in degenerate reaction-diffusion systems. (English) Zbl 1083.35047
The authors deal with the travelling waves for
\[
\begin{cases} \partial_tu=\partial^2_xu+f(u,v)\\ \partial_tv=g(u,v)\end{cases}
\]
and analyse how the missing diffusivity changes the qualitative picture. The authors are interested in front solutions that connect the states \(S_-=(0,1)\) with \(S_+=(1,0)\), that is
\[
\begin{aligned} & (u,v)(x)\to S_-\text{ for }x\to-\infty\\ & (u,v)(x)\to S_+\text{ for }x\to +\infty.\end{aligned}
\]
The authors study the stability of the blocked waves. It turns out that the stationary fronts are unstable and that a sublinear penetration may occur. Their global analysis shows that, with an arbitrary small perturbation of the front, an invasion of the \(V\)-component at a logarithmic rate is possible.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
35K57 | Reaction-diffusion equations |
35K65 | Degenerate parabolic equations |
35R35 | Free boundary problems for PDEs |
35K30 | Initial value problems for higher-order parabolic equations |
35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |