ON THE EQUATION OF BARENBLATT–SOBOLEV
Résumé
In this paper, we are interested in the following pseudoparabolic problem, known as the Barenblatt–Sobolev problem: f(∂u t ) - Δu - ϵΔ∂u t = g with u(0, ⋅) = u 0 where f is a non-monotone Lipschitz-continuous function, ϵ > 0 and [Formula: see text]. We show the existence of a critical value ϵ 0 >0 such that: if ϵ > ϵ 0 , then the problem admits a unique solution; if ϵ = ϵ 0 , the solution is unique and it exists under an additional assumption on f; if ϵ < ϵ 0 , 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(∂u t ) 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.