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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

34E10 Perturbations, asymptotics of solutions to ordinary differential equations
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