We study the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions u m ∈ L 2 loc (0, T ; H 1 loc (Ω)) and u m+1 2 ∈ L 2 loc (0, T ; H 1 loc (Ω)) in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.
Abstract. On a cylindrical domain E T , we consider doubly nonlinear parabolic equations, whose prototype is ∂ t u − div(|u| m−1 |Du| p−2 Du) = μ, where μ is a non-negative Radon measure having finite total mass μ(E T ). The central objective is to establish pointwise estimates for weak solutions in terms of nonlinear parabolic potentials in the doubly degenerate case ( p ≥ 2, m > 1). Moreover, we will prove the sharpness of the estimates by giving an optimal Lorentz space criterion regarding the local uniform boundedness of weak solutions and by comparing them to the decay of the Barenblatt solution.
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 iswith 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 CauchyDirichlet problem.
We deal with the obstacle problem for the porous medium equation in the slow diffusion regime m > 1. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.