Abstract
This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by Jleli et al. (Proc Am Math Soc 148:2579–2593, 2020). Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli et al. (2020) for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The manuscript has no associated data.
References
D. G. Aronson, J. Serrin, Local behavior of solutions of quasi-linear parabolic equations, Arch. Rational Mech. Anal. 25 (1967) pp. 81-123.
A. Attouchi, Gradient estimate and a Liouville theorem for a p-Laplacian evolution equation with a gradient nonlinearity, Differ. Integr. Equ. 29:1-2 (2016), 137–150.
C. Bandle, H. A. Levine, Q. Zhang, Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl. 251 (2000), 624–648.
M. Ben-Artzi, P. Souplet, F. B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl. 81 (2002), 343–378.
M. Borikhanov, B. T. Torebek, Nonexistence of global solutions for an inhomogeneous pseudo-parabolic equation, Appl. Math. Lett. 134 (2022), 108366.
M. Chipot, F. B. Weissler, Some blow up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal. 20:4 (1989), 886–907.
E. Di Benedetto, U. Gianazza, V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200:2 (2008), 181–209.
H. Fujita, On the blowing up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha },\) J. Fac. Sci., Univ. Tokyo, Sect. I. 13 (1966), 109–124.
V. A. Galaktionov, Conditions for global non-existence and localizations of solutions of the Cauchy problem for a class of non-linear parabolic equations, USSR Comput. Math. Math. Phys. 23:6 (1983), 36–44.
M. Jleli, B. Samet, P. Souplet, Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities, Proc. Am. Math. Soc. 148 (2020), 2579–2593.
M. Kardar, G. Parisi, Y. C. Zhang, Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986), 889–892.
J. Krug, H. Spohn, Universality classes for deterministic surface growth. Phys. Rev. A. 38 (1988), 4271–4283.
O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, 1968.
P. Laurençot, P. Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math. 89 (2003), 367–383.
T. Y. Lee, W. M. Ni, Global existence, large time behavior and life span on solution of a semilinear parabolic Cauchy problem, Trans. Am. Math. Soc. 333 (1992), 365–378.
S. Z. Lian, H. J. Yuan, C. L. Cao, W. J. Gao, X. J. Xu, On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source, J. Differ. Equ. 235 (2007), 544–585.
H. Lu, Z. Zhang, The Cauchy problem for a parabolic p-Laplacian equation with combined nonlinearities, J. Math. Anal. Appl. 514:2 (2022), 126329.
P. Quittner, P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, second ed., Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser/Springer, Cham, 2019.
H. F. Shang, F. Q. Li, On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source and measures as initial data, Nonlinear Anal. 72 (2010), 3396–3411.
S. Snoussi, S. Tayachi, F. B. Weissler, Asymptotically selfsimilar global solutions of a semilinear parabolic equation with a nonlinear gradient term, Proc. R. Soc. Edinb., Sect. A 129:6 (1999), 1291–1307.
S. Tayachi, H. Zaag, Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term, Trans. Am. Math. Soc. 317 (2019), 5899–5972.
X. Zeng, Blow-up results and global existence of positive solutions for the inhomogeneous evolution p-Laplacian equations, Nonlinear Anal. 66 (2007), 1290–1301.
Acknowledgements
The author would like to thank the reviewers for their valuable comments and remarks.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declare that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research has been/was/is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP14869090) and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). No new data was collected or generated during the course of research.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Torebek, B.T. Critical exponents for the p-Laplace heat equations with combined nonlinearities. J. Evol. Equ. 23, 71 (2023). https://doi.org/10.1007/s00028-023-00922-x
Accepted:
Published:
DOI: https://doi.org/10.1007/s00028-023-00922-x