## Critical exponent and critical blow-up for quasilinear parabolic equations.(English)Zbl 0880.35057

This paper deals with nonlinear parabolic equations of the following type $u_t-\Delta u^m= u^p\quad\text{in }D\times (0,T),$
$u(x,t)= 0\quad\text{on }\partial D\times (0,T),\quad u(x,0)= u_0(x)\geq 0.$ Here $$D$$ is either $$\mathbb{R}^N$$ or an exterior domain. It is shown that for certain ranges of $$p$$ and $$m$$ no nontrivial solution exists for all times $$t>0$$. If a solution ceases to exist, it blows up. The authors use a variant of the method of Fourier coefficients which is common in this type of studies. They obtain interesting estimates for the solutions.
Reviewer: C.Bandle (Basel)

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations

### Keywords:

unbounded domains; method of Fourier coefficients
Full Text:

### References:

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