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A blow-up result for nonlinear diffusion equations. (English) Zbl 0704.35071
This paper studies the nonexistence of global solutions to the initial boundary value problems $u_ t=\Delta (u^ m)+u^ P-g(u),\quad x\in D,\quad t>0,\quad u(x,t)=0,\quad x\in \partial D,\quad t>0,\quad u(x,0)=u_ 0(x)\geq 0,\quad x\in D$ where D is a smooth bounded domain in $${\mathbb{R}}^ N$$ and m,P are positive numbers, $$1<P$$, $$0<m\leq P$$. The main result of this paper is formulated as follows
H) Assume i) D is of class $$C^ 3$$; ii) $$u_ 0(x)$$ is non negative on D and $$u_ 0^ m\in H'_ 0(D)\cap L^{\infty}(D).$$
A) $$g\in C'[0,\infty)$$, $$g(0)=0$$, g(u)$$\geq 0$$ for $$u\geq 0$$ and $$G(u)=g(u^{1/m})$$ satisfies $$G'(u)\leq \theta G(u)$$, for $$0<\theta <P/m$$ and all $$u>0.$$
If D and $$u_ 0$$ satisfy (H), $$0<m<P$$, $$1<P$$, and g satisfies A then if $$u_ 0^ m\in B$$ (a certain unstable set) then $$u^ m\in B$$ for $$0\leq t<T_{\max}$$, where $$T_{\max}$$ is finite with computable upper bound.
Reviewer: B.D.Sleeman

##### MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs
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