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The profile near blowup time for solution of the heat equation with a nonlinear boundary condition. (English) Zbl 0823.35020

The authors study the blow-up profile near the blow-up time for the heat equation \(u_ t= \Delta u\) in \(\Omega\times (0,T)\) with the nonlinear boundary condition \(\partial u/\partial n= u^ p\) on \(\partial \Omega\times (0,T)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^ N\), \(p>1\) and the initial function \(u_ 0\) is nonnegative. Under certain assumptions on \(p\), \(u_ 0\) and \(\Omega\), they establish the exact rate of blow-up. The proof of this result is based on elliptic and parabolic estimates and on a non-existence result for elliptic equations. Moreover, the authors prove that blow-up does not occur in the interior of the domain and using the energy method they also study the asymptotic behavior of the solution near the blow-up point.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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