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Radon measure-valued solutions of quasilinear parabolic equations. (English) Zbl 1476.35124

Summary: We discuss some recent results concerning Radon measure-valued solutions of the Cauchy-Dirichlet problem for \(\partial_t u = \Delta\phi (u)\). The function \(\phi\) is continuous, nondecreasing, with a growth at most powerlike. Well-posedness and regularity results are described, which depend on whether the initial data charge sets of suitable capacity (determined both by the Laplacian and by the growth order of \(\phi)\), and on suitable compatibility conditions at \(\pm\infty\).

MSC:

35K59 Quasilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35R06 PDEs with measure
28A33 Spaces of measures, convergence of measures
28A50 Integration and disintegration of measures
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