Abstract
In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity
where \(\Omega \subset {\mathbb {R}}^N \, ( N\ge 1)\) is a bounded domain with Lipschitz boundary and \(2\le p< \infty \). The local existence will be done using the Galerkin approximations. By combining the potential well theory with the Nehari manifold, we establish the existence of global solutions. Then by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.
Similar content being viewed by others
References
Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications, volume 20 of Lecture Notes of the Unione Mathematica Italiana. Springer, Cham. Unione Mathematica Italiana, Bologna (2016)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 519–527 (2012)
Caffarelli, L.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symposia 7, 37–52 (2012)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)
Applebaum, D.: Lévy processes from probability to finance and quantum groups. Not. Am. Math. Soc. 51, 1336–1347 (2004)
Alves, C.O., Boudjeriou, T.: Existence of solution for a class of nonvariational Kirchhoff type problem via dynamical methots. Nonlinear Anal. 197, 1–17 (2020)
Ball, J.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. Oxf. Ser. 28, 473–486 (1977)
Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273–303 (1975)
Tan, Z.: Global solution and blow-up of semilinear heat equation with critical Sobolev exponent. Commun. Partial Differ. Equ. 26, 717–741 (2001)
Liu, Y., Zhao, J.: On the potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. 64(12), 2665–2687 (2006)
Zloshchastiev, K.G.: Logarithmic nonlinearity in the theories of quantum gravity: origin of time and observational consequences. Grav. Cosmol. 16, 288–297 (2017)
Le, C.N., Le, X.T.: Global solution and blow-up for a class of \(p\)-Laplacian evolution equation with logarithmic nonlinearity. Acta. Appl. Math. 151, 149–169 (2017)
Le, C.N., Le, X.T.: Global solution and blow-up for a class of pseudo \(p\)-Laplacian evolution equations with logarithmic nonlinearity. Comput. Math. Appl. 151, 149–169 (2017)
Chen, H., Luo, P., Liu, G.: Global solution and below-up of a semilinear heat equation with logarithmic nonlinearity. J. Math. Anal. Appl. 422, 84–98 (2015)
Del Pino, M., Dolbeault, J.: Asymptotic behaviour of nonlinear diffusion equation. C. R. Acad. Sci. Paris Sei. I Math. 334, 365–370 (2002)
Alves, C.O., de Morais Filho, D.C.: Existence of concentration of positive solutions for a Schrödinger logarithmic equation. Z. Angew. Math. Phys. 69, 144 (2018)
Squassina, M., Szulkin, A.: Multiple solution to logarithmic Schrödinger equations with periodic potential. Calc. Var. Partial Differ. Equ. 54, 585–597 (2015)
d’Avenia, P., Squassina, M., Zenari, M.: Fractional logarithmic Schrödinger equations. Math. Methods Appl. Sci. 38, 5207–5216 (2015)
Chen, W., Deng, S.: The Nehari manifold for a fractional \(p\)-Laplacian system involving concave-convex nonlinearities. Nonlinear Anal. Real World Appl. 27, 80–92 (2016)
Le, X.T.: The Nehari manifold for fractional \(p-\) Laplacian equation with logarithmic nonlinearity on whole space. Comput. Math. Appl. 78, 3931–3940 (2019)
Le, X.T.: The Nehari manifold for a class of Shrödinger equation involving fractional \(p\)-Laplacian and sing-changing logarithmic nonlinearity. J. Math. Phys. 60, 111505 (2019)
Gal, C.G., Warma, M.: Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. DCDS 36, 1279–1319 (2016)
Gal, C.G., Warma, M.: On some degenerate non-local parabolic equation associated with the fractional \(p\)-Laplacian. Dyn. Partial Differ. Equ 14, 47–77 (2017)
Giacomoni, J., Tiwari, S.: Existence and global behaviorr of solutions to fractional \(p-\) Laplacian parabolic problems. EJDE 44, 1–20 (2018)
Perra, K., Squassina, M., Yang, Y.: Critical fractional p-Laplacian problems with possibly vanishing potentials. J. Math. Anal. Appl. 433, 818–831 (2016)
Jiang, R., Zhou, J.: Blow-up and global existence of solutions to a parabolic equation associated with fractional \(p\)-Laplacian. Commun. Pure. Appl Anal. 18, 1205–1226 (2019)
Fu, Y., Pucci, P.: On solutions of space-fractional diffusion equations by means of potential wells. Electron. J. Qual. Theory Differ. Equ. 70, 1–17 (2016)
Ding, H., Zhou, J.: Local existence, global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem. Nonlinearity 33(1046), 1046–1063 (2020)
Pan, N., Zhang, B., Coa, J.: Degenerate Kirchhoff-type diffusion problems involving the fractional \(p\)-Laplacian. Nonlinear Anal. Real World Appl. 37, 56–70 (2017)
Mingqi, X., Rǎdulescu, D.V., Zhang, B.L.: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity 31, 3228–3250 (2018)
Mazón, J.M., Rossi, J.D., Toledo, J.: Fractional p-Laplacian evolution equation. J. Math. Pures Appl. 9, 810–844 (2016)
Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Math. 30, 148–172 (1968)
Ishii, H.: Asymptotic stability and blowing up of solutions of some nonlinear equations. J. Differ. Equ. 26, 291–319 (1977)
Puhst, D.: On the evolutionary fractional \(p\)-Laplacian. Appl. Math. Res. Express 2, 253–273 (2015)
Emmrich, E., Puhst, D.: Measure-valued and weak solution to the nonlinear peridynamic model in nonlocal elastodynamics. Nonlinearity 28, 285–307 (2015)
Evans, L.C.: Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)
Tartar, L.: An introduction to Sobolev spaces and interpolations spaces. In: Lect. Notes Unione Mat. Ital, vol. 3. Springer, Berlin (2007)
Drabek, P., Pohozaev, S.I.: Postive solutions for the p-Laplacian : application of the fibering method. Proc. R. Soc. Edinb. A 127, 703–726 (1997)
Lions, J.L.: Quelques Méthodes de Résolution des Problémes aux limites non linéaires. Dounod, Paris (1969)
Zheng, S.: Nonlinear Evolution Equations, Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall, Boca Raton (2004)
Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var. 4, 419–444 (1999)
Acknowledgements
The author warmly thanks the anonymous referee for his/her useful and nice comments on the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Boudjeriou, T. Global Existence and Blow-Up for the Fractional p-Laplacian with Logarithmic Nonlinearity. Mediterr. J. Math. 17, 162 (2020). https://doi.org/10.1007/s00009-020-01584-6
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-01584-6