On the equation of Barenblatt–Sobolev

Adimurthi, . ; Seam, Ngonn ; Vallet, Guy (2011) On the equation of Barenblatt–Sobolev Communications in Contemporary Mathematics, 13 (05). pp. 843-862. ISSN 0219-1997

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Official URL: http://www.worldscientific.com/doi/abs/10.1142/S02...

Related URL: http://dx.doi.org/10.1142/S0219199711004476


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.

Item Type:Article
Source:Copyright of this article belongs to World Scientific Publishing Company.
Keywords:Pseudoparabolic; Existence and Uniqueness; Barenblatt's Equation, Sobolev's Equation
ID Code:112167
Deposited On:31 Jan 2018 04:31
Last Modified:31 Jan 2018 04:31

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