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Similarity stabilizes blow-up. (English) Zbl 1004.35062
Journées “Équations aux dérivées partielles”, Saint-Jean-de-Monts, France, 31 mai au 4 juin 1999. Exposés Nos. I–XIX. Nantes: Université de Nantes. Exp. No. XII, 7 p. (1999).
Summary: The blow-up of solutions to the quasilinear heat equation \(u_t=\Delta u^2+u^2\) is studied using a similarity transformation that turns the equation into a nonlocal equation whose steady solutions are stable. This allows energy methods to be used, instead of the comparison principles used previously. Among the questions discussed are the time and location of blow-up of perturbations of the steady blow-up profile.
For the entire collection see [Zbl 0990.00047].

MSC:
35K55 Nonlinear 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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