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Finite extinction time for nonlinear parabolic equations with nonlinear mixed boundary data. (English) Zbl 0910.35069
The finite extinction phenomenon for nonlinear diffusion equations has drawn the interests of many authors. Using comparison arguments the authors study necessary and sufficient conditions on $\varphi$, $f$ and $g$ for a (weak) solution of $$u_t= \Delta \varphi (u)\pm f(x,t,u) \quad \text{in } \Omega \times (0,T),$$ with ${\partial \over \partial n} \varphi (u)= \mp g(x,t,u)$ on $\partial\Omega \times(0,T)$ and $u(x,0)= u_0(x)\ge 0$, either to have a finite extinction time or to stay positive for all $t\in (0,T)$. Special attention is given to the balance between the source $(+)$ and the absorption $(-)$ term.

35K60Nonlinear initial value problems for linear parabolic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI
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