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Stable blow-up patterns. (English) Zbl 0770.35010

Summary: For the semilinear heat equation \(u_ t=\Delta u+e^ u\) in a convex domain \(\Omega\subset\mathbb{R}^ n\), given any \(b\in\Omega\) we show the existence of solutions which blow up in finite time exactly at \(b\) and whose final profile has the form \(u(T,x)\approx-2\ln| x- b|+\ln|\ln| x-b| |+\ln 8\), \(T\) being the blow-up time. Using a suitable set of rescaled coordinates, this asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35K55 Nonlinear parabolic equations
35K05 Heat equation
35B45 A priori estimates in context of PDEs
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