Get all the updates for this publication

Articles

Existence and stabilization results for a singular parabolic equation involving the fractional laplacianPublished in

2019

Volume: 12

Issue: 2

Pages: 311 - 337

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. 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”}, references={Abdellaoui, B., Medina, M., Peral, I., Primo, A., Optimal results for the fractional heat equation involving the hardy potential (2016) Nonlinear Anal, 140, pp. 166-207; Adimurthi, J.G., Santra, S., Positive solutions to a fractional equation with singular nonlinearity (2018) J. Differential Equations, 265, pp. 1191-1226; Alibaud, N., Imbert, C., Fractional semi-linear parabolic equations with unbounded data (2009) Transactions of The American Mathematical Society, 361, pp. 2527-2566; Amghibech, S., On the discrete version of picone's identity (2008) Discrete Applied Mathematics, 156, pp. 1-10; Avelin, B., Gianazza, U., Salsa, S., Boundary estimates for certain degenerate and singular parabolic equations (2016) Journal of The European Mathematical Society, 18, pp. 381-424; Badra, M., Bal, K., Giacomoni, J., A singular parabolic equation: Existence, stabilization (2012) J. Differential Equations, 252, pp. 5042-5075; Barbu, V., (2010) Nonlinear Differential Equations of Monotone Types in Banach Spaces, , 1st edition, Springer Monogr. Math., Springer, New York; Barrios, B., De Bonis, I., Medina, M., Peral, I., Semilinear problems for the fractional laplacian with a singular nonlinearity (2015) Open Math, 13, pp. 390-407; Bougherara, B., Giacomoni, J., Existence of mild solutions for a singular parabolic equation and stabilization (2015) Adv. Nonlinear Anal., 4, pp. 123-134; Cafarelli, L., Figalli, A., Regularity of solutions to the parabolic fractional obstacle problem (2013) Journal Für Die Reine und Angewandte Mathematik (Crelles Journal), 680, pp. 191-233; Dávila, J., Montenegro, M., Existence and asymptotic behavior for a singular parabolic equation (2005) Transactions of The American Mathematical Society, 357, pp. 1801-1828; Del Pezzo, L.M., Quaas, A.J., Non-resonant fredholm alternative and anti-maximum principle for the fractional p-laplacian (2017) Journal of Fixed Point Theory and Applications, 19, pp. 939-958; Fino, A., Karch, G., Decay of mass for nonlinear equation with fractional laplacian (2010) Monat-Shefte Für Mathematik, 160, pp. 375-384; Fragnelli, G., Mugnai, D., Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients (2017) Adv. Nonlinear Anal., 6, pp. 61-84; Frank, R.L., Seiringer, R., Non-linear ground state representations and sharp hardy inequalities (2008) Journal of Functional Analysis, 255, pp. 3407-3430; Giacomoni, J., Mukherjee, T., Sreenadh, K., Positive solutions of fractional elliptic equation with critical and singular nonlinearity (2016) Adv. Nonlinear Anal., 6, pp. 327-354; Kim, S., Lee, K.-A., Hölder estimates for singular non-local parabolic equations (2011) Journal of Functional Analysis, 261, pp. 3482-3518; Leonori, T., Peral, I., Primo, A., Soria, F., Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations (2015) Discrete Contin. Dyn. Syst., 35, pp. 6031-6068; Mukherjee, T., Sreenadh, K., Fractional elliptic equations with critical growth and singular nonlinearities (2016) Electronic Journal of Differential Equations, 54, pp. 1-23; Ros-Oton, X., Serra, J., The dirichlet problem for the fractional laplacian: Regularity up to the boundary (2014) Journal De Mathématiques Pures Et Appliquées, 101, pp. 275-302; Servadei, R., Valdinoci, E., The brezis-nirenberg result for the fractional laplacian (2015) Transactions of The American Mathematical Society, 367, pp. 67-102; Servadei, R., Valdinoci, E., Variational methods for non-local operators of elliptic type (2013) Discrete Contin. Dyn. Syst., 33, pp. 2105-2137; Silvestre, L., Regularity of the obstacle problem for a fractional power of the laplace operator (2007) Comm. Pure Appl. Math., 60, pp. 67-112; Simon, J., Compact sets in the space Lp(0, T; B) (1987) Ann. Mat. Pura Appl., 146, pp. 65-96; Vázquez, J.L., Nonlinear diffusion with fractional laplacian operators (2012) Nonlinear Partial Differential Equations, 7, pp. 271-298. , Holden, Helge and Karlsen, Kenneth H. eds., Springer; Vázquez, J.L., Recent progress in the theory of nonlinear diffusion with fractional laplacian operators (2014) Discrete and Continuous Dynamical Systems - Series S, 7, pp. 857-885}, correspondence_address1={Giacomoni, J.; Université de Pau et des Pays de l'Adour, avenue de l'université, France; email: jacques.giacomoni@univ-pau.fr}, publisher={American Institute of Mathematical Sciences}, issn={19371632}, language={English}, abbrev_source_title={Discrete Contin. Dyn. Syst. Ser. S}, document_type={Article}, source={Scopus},

Publisher Copy Version

Content may be subject to copyright,Check License

About the journal

Journal | Discrete and Continuous Dynamical Systems - Series S |
---|