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On the blow-up set for $$u_t=\Delta u^m+u^m$$, $$m>1$$. (English) Zbl 0916.35056
Summary: It is well known that every non-trivial solution of $$u_t= \Delta u^m+ u^m$$ in $$\mathbb{R}^N\times [0,\infty)$$, $$u(x,0)= u_0(x)\geq 0$$ on $$\mathbb{R}^N$$, with $$m>1$$, blows up in finite time. We study the blow-up set and the blow-up profile of a solution $$u(x,t)$$ to this equation with blow-up time $$T>0$$, under the assumption that $$u_0(x)$$ is compactly supported. We prove that, up to subsequences, $$(T- t)^{1/(m-1)} u(x,t)$$ converges as $$t\to T$$ to $$w(x)$$. Here $$w(x)$$ is a finite sum of translations with disjoint supports of the unique positive radially symmetric, compactly supported, solution of $$\Delta w^m+ w^m- w/(m-1)= 0$$. The centers of these supports do not go beyond the smallest ball containing the support of $$u_0$$ and, $$u(x,t)$$ remains uniformly bounded away from these supports. An estimate of the blow-up time is also provided.

##### MSC:
 35K65 Degenerate parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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