# zbMATH — the first resource for mathematics

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

##### MSC:
 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
##### Keywords:
existence; porous medium equation; non-negative solutions
Full Text: