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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
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