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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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