# zbMATH — the first resource for mathematics

Asymptotic dead cores for reaction-diffusion equations. (English) Zbl 0706.34052
The behaviour of nonnegative solutions of the Dirichlet problem $$(*)_{\kappa}$$ $$u''=\phi^ 2g_{\kappa}(u)$$, $$u(-1)=u(1)=1$$ is considered as the parameter $$\kappa\to 0$$, where $$g_{\kappa}(u)\to u^{1-q}$$ for $$0<q<2$$ as $$\kappa\to 0$$, uniformly on compact subsets of (0,1]. In general, $$(*)_ 0$$ possesses one fewer positive, classical solution than does $$(*)_{\kappa}$$. However, $$(*)_ 0$$ has a nonnegative solution $$u_ 0$$ with a dead core (where $$u\equiv 0)$$ on a suitable subinterval of (-1,1); we show that $$(*)_{\kappa}$$ possesses a solution $$u_{\kappa}$$ such that $$u_{\kappa}\to u_ 0$$ uniformly on [-1,1] as $$\kappa\to 0$$.
Reviewer: L.E.Bobisud

##### MSC:
 3.4e+11 Perturbations, asymptotics of solutions to ordinary differential equations
##### Keywords:
reaction-diffusion equations; Dirichlet problem; dead core
Full Text: