J. Nonl. Evol. Equ. Appl. 2019 (7), pp. 115-133, published on August 6, 2020:

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. Furthermore, we discuss the main properties of such solutions.
Keywords:A priori estimates, equi-diffuse measure, porous media equation, parabolic p−capacity, renormalized solution.
2010 Mathematics Subject Classification:   65J15, 28A12, 35B45, 35A35, 35Q35.

Download Full Text: JNEEA-vol.2019-no.7.pdf [272 kB]