Accepted for publication in JNEEA

Nonlinear parabolic equations with soft measure data

M. Abdellaoui, E. Azroul

University of Fez, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics, B.P. 1796, Atlas Fez, Morocco

Received on July 20, 2018, revised version on May 17, 2019
Accepted on June 30, 2019

Communicated by Alexander Pankov

Abstract.  In this paper we prove existence and uniqueness results for nonlinear parabolicproblems with Dirichlet boundary values whose model is
b(u)t − ∆pu = μ   in (0, T) × Ω,
b(u(0, x)) = b(u0)   in Ω,
u(t, x) = 0   on (0, T) × ∂Ω.
where pu = div(|∇u|p−2 ∇u) is the usual p−Laplace operator, b is a increasing C1−function and μ is a finite measure which does not charge sets of zero parabolic p−capacity, and we discuss their main properties.
Keywords: Porous media equation, parabolic p−capacity, renormalized solution, a priori estimates, equidiffuse measure.
2010 Mathematics Subject Classification:   65J15, 28A12, 35B45, 35A35, 35Q35.

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