Pseudo-Parabolic Regularization of Forward-Backward Parabolic Equations with Bounded Nonlinearities

We study the initial-boundary value problem { ut = φ u xx + ε ψ u txx in Ω × 0 T φ u + ε ψ u t = 0 in ∂ Ω × 0 T u = u 0 ≥ 0 in Ω × 0 , with Radon measure-valued initial data, by assuming that the regularizing term ψ is bounded and increasing (the cases of power-type or logarithmic ψ were examined in...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2018-12, Vol.235 (4), p.536-555
Main Author: Tesei, A.
Format: Article
Language:English
Subjects:
Boundary value problems
Infinity
Mathematics
Mathematics and Statistics
Radon
Regularization
Singularity (mathematics)
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Summary:We study the initial-boundary value problem { ut = φ u xx + ε ψ u txx in Ω × 0 T φ u + ε ψ u t = 0 in ∂ Ω × 0 T u = u 0 ≥ 0 in Ω × 0 , with Radon measure-valued initial data, by assuming that the regularizing term ψ is bounded and increasing (the cases of power-type or logarithmic ψ were examined in [ 2 , 3 ] for spaces on any dimension). The function φ is nonmonotone and bounded, and either (i) decreases and vanishes at infinity, or (ii) increases at infinity. The existence of solutions in a space of positive Radon measures is proved in both cases. Moreover, a general result on the spontaneous appearance of singularities in he case (i) is presented. The case of a cubic-like φ is also discussed to point out the influence of the behavior at infinity of φ on the regularity of solutions.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-018-4084-6