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Non-negative solutions of generalized porous medium equations. (English) Zbl 0644.35057
The authors study the non-negative solutions u of the evolution equation $$\partial u/\partial t= \Delta\phi(u)$$, $$x\in R^ n$$, $$0<t<T\leq +\infty$$, which arises in nonlinear filtration.
In the linear case $$(\partial u/\partial t=\Delta u)$$ D. Widder considered for each non-negative solution of corresponding non-negative Borel measure $$\mu$$ on $$R^ n$$ such that $$\lim_{t\searrow 0}\int_{R^ n}u(x,t)\theta (x)dx=\int_{R^ n}\theta dx$$ for all continuous function $$\theta(x)\in R^ n$$ with compact support. The measure $$\mu$$ is called the trace of u and determines the solution uniquely.
The aim of this work is to find the analogues of the results of the linear theory for a class of non-linearities $$\phi$$, wider than the class of pure powers (i.e. $$\phi(u)=u^ m$$, $$m>1)$$ which was studied before.
Reviewer: A.Carabineanu

##### MSC:
 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations 76S05 Flows in porous media; filtration; seepage
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