On the equation of Barenblatt–Sobolev

Abstract

In this paper, we are interested in the following pseudoparabolic problem, known as the Barenblatt–Sobolev problem: f(∂ut) - Δu - ϵΔ∂ut = g with u(0, ⋅) = u0 where f is a non-monotone Lipschitz-continuous function, ϵ > 0 and . We show the existence of a critical value ϵ0 >0 such that: if ϵ > ϵ0, then the problem admits a unique solution; if ϵ = ϵ0, the solution is unique and it exists under an additional assumption on f; if ϵ < ϵ0, then the solution is not unique in general. Passing to the limit with ϵ to 0+, we prove the existence (and uniqueness) of the solution of the Barenblatt differential inclusion Δu + g ∈ f(∂ut) for a class of maximal monotone operators f. Next, we give an extension of the main result for a stochastic perturbation of the problem and we give some numerical illustrations of the Barenblatt and the Barenblatt–Sobolev equation.