Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 931-962

EXISTENCE OF VERY WEAK SOLUTIONS OF DOUBLY NONLINEAR PARABOLIC EQUATIONS WITH MEASURE DATA

Stefan Sturm

Paris-Lodron-Universität Salzburg, Fachbereich Mathematik
5020 Salzburg, Austria; Stefan.Sturm 'at' sbg.ac.at

Abstract. We deal with a Cauchy–Dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space-time cylinder for doubly nonlinear parabolic equations, whose prototype is

∂_tu – div(|u|^{m –
1}|Du|^{p – 2}Du) = μ

with a non-negative Radon measure μ on the right-hand side. Here, the doubly degenerate (p ≥ 2, m ≥ 1) and singular-degenerate (p ∈ (2n/(n + 2),2), m ≥ 1) cases are considered. The central objective is to establish the existence of a solution in the sense of distributions (see Theorem 1.4). The constructed solution is obtained by a limit of approximations, i.e. we use solutions of regularized Cauchy–Dirichlet problems and pass to the limit to receive a solution for the original Cauchy–Dirichlet problem.

2010 Mathematics Subject Classification: Primary 35K55, 35K20.

Key words: Doubly nonlinear parabolic equations, existence, measure data.

Reference to this article: S. Sturm: Ann. Acad. Sci. Fenn. Math. 42 (2017), 931-962.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4255