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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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