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On solutions of a perturbed fast diffusion equation. (English) Zbl 0652.35064
The paper is concerned with the problem $v_ t-\Delta (v\quad m)=f(v)\quad in\quad D\times (0,T)$ with homogeneous Dirichlet boundary conditions and given initial distribution $$v_ 0(x).$$
Here, D is a bounded domain of R N with smooth boundary, $$0<m<1$$, F(0)$$\geq 0$$ and $$v_ 0\geq 0$$. The author first proves existence of a global solution assuming global Lipschitz continuity of f and smooth initial data. The main idea is to use a difference equation in t. He then proves among other things blow-up, local existence and finite extinction results for the above problem and closely related problems such as $$(\beta (u))_ t=\Delta u$$ in $$D\times (0,\infty)$$.
Reviewer: R.Sperb

MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35K20 Initial-boundary value problems for second-order parabolic equations
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References:
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