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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
##### Keywords:
blow up; dissipative term; dead problems
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##### References:
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