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Nondegeneracy of blow up for semilinear heat equations. (English) Zbl 0703.35020
This paper is a rather comprehensive treatment of the blow up of solutions to the equation \[ u_ t-\Delta u-| u|^{p- 1}u=0\text{ in } D\times (0,T), \] with zero boundary data. Here D is a domain in \({\mathbb{R}}^ n\), u is scalar-valued, and \(p>1.\)
A first set of results is concerned more generally with the solutions of the parabolic differential inequality \(| v_ t-\Delta v| \leq k(1+| v|^ p),\) \(k>0\) in a cylinder \(Q_ r=B_ r(a)\times (t_ 1-r^ 2,t_ 1),\) \(0<r\leq 1\). For instance it is proved that if \(| v(x,t)| \leq \epsilon (t_ 1-t)^{-1/(p-1)}\) in \(Q_ r\) for some appropriate \(\epsilon =\epsilon (k,p,n)\), then \((a,t_ 1)\) is not a blow up point for v.
Another interesting conclusion regards a sufficient condition for excluding blow up of u at a given point in terms of the smallness of an energy-type functional. The behaviour of solutions near blow up points is characterized by the so-called blow up limit, showing that tending to blow up point (a,T) along parabolas \(x=a+y(T-t)^{1/2},\) the limit of \((T-t)^{1/(p-1)}u\) has to be \(\pm (p-1)^{-1/(p-1)}.\) In addition, the authors describe the structure of the blow up set and they present some extensions to more general equations.
Reviewer: A.Fasano

35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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