## 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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