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Observations on blow up and dead cores for nonlinear parabolic equations. (English) Zbl 0661.35053
Consider the following problem \((P_ 1):\) \[ (1)\quad u_ t-\Delta u=u^ p-a| \nabla u|^ q\quad t>0,\quad x\in \Omega, \] \[ u=0,\quad t>0,\quad x\in \partial \Omega;\quad u(0,x)=\Phi_ 1(x),\quad x\in \Omega. \] It has been known for quite a while that solutions of a simpler problem (without the gradient term) can blow up in finite time. Recently Chipot and Weissler investigated whether the blow up effect persists when one adds a dissipative term to the equation. They discussed the case \(1<p\) and \(1<q\leq 2p/(p+1).\) In the present paper the case \(q=2\) is studied in some detail.
For positive a and \(1<p<2\) the gradient term is of no lower power than the source term \(u^ p\). Therefore the damping term does control the source term in the sense that there is no blow up. For any \(p>2\), however, one may still have blow up phenomena, so that the restriction on q of Chipot and Weissler appears to be technical.
Blow up phenomena for other functions f(u), instead of \(u^ p\), are studied as well.
An important step in the proofs is a transformation to so-called dead problems, such as \((P_ 2):\) \[ (2)\quad v_ t-\Delta v=-h(v)\quad t>0,\quad x\in \Omega, \] \[ v=1,\quad t>0,\quad x\in \partial \Omega;\quad v(0,x)=\Phi_ 2(x)\quad x\in \Omega. \] The function u blows up if and only if v tends to zero in finite time. Problem \((P_ 2)\) is analyzed by methods which are essentially due to Bandle and Stakgold.
Finally the case \(a<0\) in \((P_ 1)\) is treated. This causes an effect which enhances blow up in the sense that sometimes blow up will occur on all of \(\Omega\) and not in a single point only. iii) If \(p>2\), there is single point blow up.
The proof of this Theorem can be reduced to some known blow up results, once one finds the right transformation. If we set \(v(t,x)=e^{u(t,x)}- 1\), we obtain the transformed equation \[ v_ t-\Delta v=(1+v)(\log (1+v))^ p. \] Now the result follows immediately from the paper by A. A. Lacey [Proc. R. Soc. Edinb., Sect. A 104, 161-167 (1986; Zbl 0627.35047)]. Remark 7. After this paper was completed we were kindly informed by V. A. Galaktionov of the paper by himself and S. A. Posashkov [Exact solutions of parabolic equations with quadratic nonlinearities, preprint, Moscow, in Russian (1988)], which contains exact solutions of \(u_ t-u_{xx}=u^ 2+| u_ x|^ 2\) on \({\mathbb{R}}^+\times {\mathbb{R}}\).
Reviewer: B.Kawohl

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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