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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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