The trace triple for nonnegative solutions of the porous medium equation
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It is shown that, for every non-negative weak solution u of the generalized porous medium Eq. (1) \(\Psi (u)\) takes a Radon measure as its lateral trace. When \(\Psi (u)\) is integrable, u has an initial trace consisting of a bottom measure and a corner measure. The initial trace (bottom and corner measures) and the lateral trace form the trace triple for the solution. The initial-Dirichlet problem of prescribing trace triple is solved for the porous medium equation in the slow diffusion case.
Mathematics Subject ClassificationPrimary 35Q35 Secondary 76A20 35B35 93D20 37B30
This work was partially supported by an Earmarked Grant for Research RGC CUHK 14301014, Hong Kong. The second author would like to thank the hospitality of the Mathematics Department of the Chinese University of Hong Kong during his visit there. The authors would like to thank one referee whose suggestions have improved the presentation of Sect. 4 of this paper.