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Optimal estimates for blowup rate and behavior for nonlinear heat equations. (English) Zbl 0899.35044

The authors describe all positive and bounded solutions to the equation \[ {{\partial w}\over {\partial s}}=\Delta w -{1\over 2}y\cdot \nabla w -{w\over {p-1}}+ w^{p}, \] where \((y,s)\in {\mathbb R}^{N}\times {\mathbb R},\) \(p>1\) and \((N-2)p\leq N+2\). The results are then applied to the blowing-up solutions \(u(t)\) of the equation \[ {{\partial u}\over {\partial t}}=\Delta u+u^{p}. \] Uniform estimates at the blowup time are obtained for \(u(t)\) based on uniform space-time comparison with solutions of \(u'=u^{p}.\)

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
35K15 Initial value problems for second-order parabolic equations
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