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Existence and stabilization results for a singular parabolic equation involving the fractional laplacian
J. Giacomoni, , K. Sreenadh
Published in
2019
Volume: 12
   
Issue: 2
Pages: 311 - 337
Abstract
In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity (Pt s) {ut + (−∆)su = u−q + f(x, u), u > 0 in (0, T) × Ω, u = 0 in (0, T) × (Rn \ Ω), u(0, x) = u0(x) in Rn, where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n > 2s, s ∈ (0, 1), q > 0, q(2s − 1) < (2s + 1), u0 ∈ L∞(Ω) ∩ X0(Ω) and T > 0. We suppose that the map (x, y) ∈ Ω × R+ → f(x, y) is a bounded from below Carathéodary function, locally Lipschitz with respect to the second variable and uniformly for x ∈ Ω and it satisfies lim supf(x, y) < λs 1(Ω), y→+∞ y (0.1) where λs 1(Ω) is the first eigenvalue of (−∆)s in Ω with homogeneous Dirichlet boundary condition in Rn\Ω. We prove the existence and uniqueness of a weak solution to (Pt s) on assuming u0 satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results. We also show additional regularity on the solution of (Pt s) when we regularize our initial function u0. © American Institute of Mathematical Sciences. All rights reserved.}, author_keywords={Fractional Laplacian; Non-local operator; Parabolic equation; Singular nonlinearity; Sub-super solution}, funding_details={Indo-French Centre for Applied MathematicsIndo-French Centre for Applied Mathematics, IFCAM}, funding_details={College of Natural Resources and Sciences, Humboldt State UniversityCollege of Natural Resources and Sciences, Humboldt State University, CNRS, 3494}, funding_text_1={The authors were funded by IFCAM (Indo-French Centre for Applied Mathematics) UMI CNRS 3494 under the project “Singular phenomena in reaction diffusion equations and in conservation laws”}, funding_text_2={Acknowledgments. 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JournalDiscrete and Continuous Dynamical Systems - Series S