Solutions of the porous medium equation in \({\mathbb{R}}^ N\) under optimal conditions on initial values. (English) Zbl 0552.35045

The authors prove an existence result for the porous medium equation \(u_ t=\Delta (| u|^{m-1}u)\) in \({\mathbb{R}}^ N\), \(m>1\) under the following condition on the initial data \(u_ 0\) \(\ell (u_ 0)=\sup_{R\geq 1}(\int_{B(x,R)}| u_ 0| dx)R^{-(N+2/(m- 1))}<+\infty\) and the interval of existence of the solution is (0,T) where \(T\geq c/\ell (u_ 0)^{m-1}\) where c is a constant depending only on m,N. This condition turns out to be optimal for the existence of a solution at least in the case of non-negative solutions.
Reviewer: M.Biroli


35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
76S05 Flows in porous media; filtration; seepage
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