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.


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
Full Text: DOI


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