Diffuse measures and nonlinear parabolic equations

Given a parabolic cylinder Q = (0, T ) × Ω, where is a bounded domain, we prove new properties of solutions of with Dirichlet boundary conditions, where  μ is a finite Radon measure in Q . We first prove a priori estimates on the p -parabolic capacity of level sets of u . We then show that diffuse m...

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Bibliographic Details
Published in:Journal of evolution equations 2011-12, Vol.11 (4), p.861-905
Main Authors: Petitta, Francesco, Ponce, Augusto C., Porretta, Alessio
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
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Summary:Given a parabolic cylinder Q = (0, T ) × Ω, where is a bounded domain, we prove new properties of solutions of with Dirichlet boundary conditions, where  μ is a finite Radon measure in Q . We first prove a priori estimates on the p -parabolic capacity of level sets of u . We then show that diffuse measures (i.e., measures which do not charge sets of zero parabolic p -capacity) can be strongly approximated by the measures  μ k = ( T k ( u )) t −Δ p ( T k ( u )), and we introduce a new notion of renormalized solution based on this property. We finally apply our new approach to prove the existence of solutions of for any function h such that h ( s ) s ≥ 0 and for any diffuse measure  μ ; when h is nondecreasing, we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-011-0115-1