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Global Existence and Blow-Up for the Fractional p-Laplacian with Logarithmic Nonlinearity

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Abstract

In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity

$$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+(-\Delta )^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} {\mathbb {R}}^{N}\backslash \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$

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.

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References

  1. 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)

  2. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 519–527 (2012)

    Article  Google Scholar 

  3. Caffarelli, L.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symposia 7, 37–52 (2012)

    Article  Google Scholar 

  4. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)

    Article  MathSciNet  Google Scholar 

  5. Applebaum, D.: Lévy processes from probability to finance and quantum groups. Not. Am. Math. Soc. 51, 1336–1347 (2004)

    MATH  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. Ball, J.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. Oxf. Ser. 28, 473–486 (1977)

    Article  MathSciNet  Google Scholar 

  8. Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273–303 (1975)

    Article  MathSciNet  Google Scholar 

  9. Tan, Z.: Global solution and blow-up of semilinear heat equation with critical Sobolev exponent. Commun. Partial Differ. Equ. 26, 717–741 (2001)

    Article  Google Scholar 

  10. Liu, Y., Zhao, J.: On the potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. 64(12), 2665–2687 (2006)

    Article  MathSciNet  Google Scholar 

  11. Zloshchastiev, K.G.: Logarithmic nonlinearity in the theories of quantum gravity: origin of time and observational consequences. Grav. Cosmol. 16, 288–297 (2017)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. 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)

    MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  Google Scholar 

  15. Del Pino, M., Dolbeault, J.: Asymptotic behaviour of nonlinear diffusion equation. C. R. Acad. Sci. Paris Sei. I Math. 334, 365–370 (2002)

    Article  Google Scholar 

  16. 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)

    Article  Google Scholar 

  17. Squassina, M., Szulkin, A.: Multiple solution to logarithmic Schrödinger equations with periodic potential. Calc. Var. Partial Differ. Equ. 54, 585–597 (2015)

    Article  Google Scholar 

  18. d’Avenia, P., Squassina, M., Zenari, M.: Fractional logarithmic Schrödinger equations. Math. Methods Appl. Sci. 38, 5207–5216 (2015)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Le, X.T.: The Nehari manifold for fractional \(p-\) Laplacian equation with logarithmic nonlinearity on whole space. Comput. Math. Appl. 78, 3931–3940 (2019)

    Article  MathSciNet  Google Scholar 

  21. 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)

    Article  MathSciNet  Google Scholar 

  22. Gal, C.G., Warma, M.: Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. DCDS 36, 1279–1319 (2016)

    Article  MathSciNet  Google Scholar 

  23. 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)

    Article  MathSciNet  Google Scholar 

  24. Giacomoni, J., Tiwari, S.: Existence and global behaviorr of solutions to fractional \(p-\) Laplacian parabolic problems. EJDE 44, 1–20 (2018)

    MATH  Google Scholar 

  25. Perra, K., Squassina, M., Yang, Y.: Critical fractional p-Laplacian problems with possibly vanishing potentials. J. Math. Anal. Appl. 433, 818–831 (2016)

    Article  MathSciNet  Google Scholar 

  26. 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)

    Article  MathSciNet  Google Scholar 

  27. 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)

    Article  MathSciNet  Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. 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)

    Article  MathSciNet  Google Scholar 

  30. Mingqi, X., Rǎdulescu, D.V., Zhang, B.L.: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity 31, 3228–3250 (2018)

    Article  MathSciNet  Google Scholar 

  31. Mazón, J.M., Rossi, J.D., Toledo, J.: Fractional p-Laplacian evolution equation. J. Math. Pures Appl. 9, 810–844 (2016)

    Article  MathSciNet  Google Scholar 

  32. Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Math. 30, 148–172 (1968)

    Article  MathSciNet  Google Scholar 

  33. Ishii, H.: Asymptotic stability and blowing up of solutions of some nonlinear equations. J. Differ. Equ. 26, 291–319 (1977)

    Article  MathSciNet  Google Scholar 

  34. Puhst, D.: On the evolutionary fractional \(p\)-Laplacian. Appl. Math. Res. Express 2, 253–273 (2015)

    Article  MathSciNet  Google Scholar 

  35. Emmrich, E., Puhst, D.: Measure-valued and weak solution to the nonlinear peridynamic model in nonlocal elastodynamics. Nonlinearity 28, 285–307 (2015)

    Article  MathSciNet  Google Scholar 

  36. Evans, L.C.: Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)

  37. Tartar, L.: An introduction to Sobolev spaces and interpolations spaces. In: Lect. Notes Unione Mat. Ital, vol. 3. Springer, Berlin (2007)

  38. 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)

    Article  Google Scholar 

  39. Lions, J.L.: Quelques Méthodes de Résolution des Problémes aux limites non linéaires. Dounod, Paris (1969)

    MATH  Google Scholar 

  40. Zheng, S.: Nonlinear Evolution Equations, Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall, Boca Raton (2004)

    Google Scholar 

  41. Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var. 4, 419–444 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author warmly thanks the anonymous referee for his/her useful and nice comments on the paper.

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  1. Laboratory of Applied Mathematics, Department of Mathematics, Faculty of Exact Sciences, University of Bejaia, Bejaïa, 6000, Algeria

    Tahir Boudjeriou

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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

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  • DOI: https://doi.org/10.1007/s00009-020-01584-6

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