Existence of solution for a class of heat equation involving the p(x) Laplacian with triple regime

Abstract

In this paper, we study the local and global existence of solution and the blow-up phenomena for a class of heat equation involving the p(x)-Laplacian with triple regime.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Diening, L., Nägele, P., Råz̆ička, M.: Monotone operator theory for unsteady problems in variable exponent spaces. Complex Var. Elliptic Equ. 57, 1209–1231 (2012)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Kobayashi, J., Otani, M.: The principle of symmetric criticality for non-differentiable mappings. J. Funct. Anal. 214, 428–449 (2004)

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Alves, C.O., Rădulescu, V.: The Lane–Emden equation with variable double-phase and multiple regime. Proc. Am. Math. Soc. 148, 2937–2952 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Alves, C.O., Simsen, J., Simsen, M.: Parabolic problems in \(\mathbb{R}^N\) with spatially variable exponents. Asymptot. Anal. 93, 51–64 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Lourêdo, A.T., Miranda, M.M., Clark, M.R.: Variable exponent perturbation of a parabolic equation with \(p(.)\)-Laplacian. Electron. J. Qual. Theory Differ. Equ. 60, 1–14 (2019)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics. Springer-Verlag, Berlin (1983)

    Book  Google Scholar 

  8. 8.

    Pan, N., Zhang, B., Cao, J.: Weak solutions for parabolic equations with \(p(x)-\)growth. Electron. J. Differ. Equ. 209, 1–15 (2016)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Rădulescu, V.D., Repovš, D.D.: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press. Taylor & Francis Group, Boca Raton FL (2015)

    Book  Google Scholar 

  10. 10.

    Rajagopal, K.R., Ruzicka, M.: Mathematical modelling of electrorheological fluids. Contin. Mech. Thermodyn. 13, 59–78 (2001)

    Article  Google Scholar 

  11. 11.

    Showalter, E. R.: Monotone operators in Banach spaces and nonlinear partial differential equations (Mathematical surveys and monographs vol. 49). American Mathematical Society, Providence

  12. 12.

    Simon, J.: Compact sets in the space \(L^{p}(0, T, B)\). Ann. Math. Pura. Appl. 4(146), 65–96 (1987)

    MATH  Google Scholar 

  13. 13.

    Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics. Springer-Verlag, Berlin (2000)

    Book  Google Scholar 

  14. 14.

    Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Alves, C.O., Ferreira, M.C.: Multi-bump solutions for a class of quasilinear problems involving variable exponents. Ann. Math. Pura Appl. 194, 1563–1593 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Antontsev, S., Shmarev, S.: Evolution PDEs with Nonstandard Growth Conditions, Atlantis Studies in Differential Equations, vol. 4. Atlantics Press, Paris (2015)

    Book  Google Scholar 

  17. 17.

    Autouri, G., Pucci, P.: Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces. Complex Var. Elliptic Equ. 56, 715–753 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Brézis, H.: Opérateurs Maximaux Monotones et semi-groupes des contractions dans les espaces de Hilbert. North-Holland/American Elsevier, Amsterdam/London/New York (1971)

    MATH  Google Scholar 

  19. 19.

    Galaktionov, V.A., Levine, H.A.: A general approach to critical Fujita exponents in nonlinear parabolic problems. Nonlinear Anal. 34, 1005–1027 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Galaktionov, V.A., Vázquez, J.L.: Regional blow up in a semilinear heat equation with convergence to a Hamilton–Jacobi equation. SIAM J. Math. Anal. 24, 1254–1276 (1993)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Gazzola, F., Weth, T.: Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differ. Integral Equ. 18(9), 961–990 (2005)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Akagia, G., Matsuura, K.: Well-posedness and large-time behaviors of solutions for a parabolic equation involving \(p(x)\)-Laplacian. Contin. Dyn. Syst. 1, 22–31 (2011)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Alves, C.O., Boudjeriou, T.: Existence of solution for a class of nonvariational Kirchhoff type problem via dynamical methods. Nonlinear Anal. 197, 1–17 (2020)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Alves, C.O., Ferreira, M.C.: Existence of solutions for a class of \(p(x)\)- Laplacian equations involving a concave–convex nonlinearity with critical growth in \(\mathbb{R}^{N}\). Topol. Methods Nonlinear Anal. 45, 399–422 (2015)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Xu, R.: Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Q. Appl. Math. 68(3), 459–468 (2010)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Zheng, S.: Nonlinear Evolution Equations. Monographs and Surveys in Pure and Applied Mathematics, vol. 133. Chapman & Hall/CRC, Boca Raton, FL (2004)

    Google Scholar 

  27. 27.

    Antontsev, S., Shmarev, S.: Anisotropic parabolic equations with variable nonlinearity. Publ. Math. 53, 355–399 (2009)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Diening, L., Hästo, P., Harjulehto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents, Springer Lecture Notes, vol. 2017. Springer-Verlag, Berlin (2011)

    Book  Google Scholar 

  29. 29.

    Galaktionov, V.A., Vázquez, J.L.: The problem of blow-up in nonlinear parabolic equations. Discrete Contin. Dyn. Syst. 8, 399–433 (2002)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Guo, W., Gao, J.: Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the \(p(x, t)\)-Laplace operator and a non-local term. Discrete. Contin. Dyn. Syst. 36, 715–730 (2016)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Halsey, T.C.: Electrorheological fluids. Science 258, 761–766 (1992)

    Article  Google Scholar 

  32. 32.

    Alves, C.O., Shmarev, S., Simsen, J., Simsen, M.: The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: existence and asymptotic behavior. J. Math. Anal. Appl. 443, 265–294 (2016)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Fan, X.L., Shen, J.S., Zhao, D.: Sobolev embedding theorems for spaces \(W^{k, p(x)}(\Omega )\). J. Math. Anal. Appl. 262, 749–760 (2001)

    MathSciNet  Article  Google Scholar 

  34. 34.

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

    MathSciNet  Article  Google Scholar 

  35. 35.

    Mihǎilescu, M., Rădulescu, V.: Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz–Sobolev space setting. J. Math. Anal. Appl. 330, 416–432 (2007)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Mihǎilescu, M., Rădulescu, V.: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc. Lond. Ser. A 462, 2625–2641 (2006)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tahir Boudjeriou.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alves, C.O., Boudjeriou, T. Existence of solution for a class of heat equation involving the p(x) Laplacian with triple regime. Z. Angew. Math. Phys. 72, 2 (2021). https://doi.org/10.1007/s00033-020-01430-5

Download citation

Mathematics Subject Classification

  • 35K59
  • 65M60
  • 35B44

Keywords

  • Quasilinear parabolic equations
  • Galerkin Methods
  • Blow-up