Abstract
The aim of this paper was to study vanishing and blowing-up properties of the solutions to a homogeneous initial Dirichlet problem of a nonlinear diffusion equation involving the p(x)-Laplace operator and a nonlinear source. The authors point out that the results obtained are not trivial generalizations of similar problems in the case of constant exponent because the variable exponent p(x) brings some essential difficulties such as the failure of upper and lower solution method and scaling technique, the existence of a gap between the modular and the norm. To overcome these difficulties, the authors have to improve the regularity of solutions, to construct a new control functional and apply suitable embedding theorems to prove the blowing-up property of the solutions. In addition, the authors utilize an energy estimate method and a comparison principle for ODE to prove that the solution vanishes in finite time. At the same time, the critical extinction exponents and an extinction rate estimate to the solutions are also obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
DiBenedetto E.: Degenerate Parabolic Equations. Springer, New York (1993)
Philip J.R.: N-diffusion. Austral. J. Phys. 14, 1–13 (1961)
Atkinson C., Jones C.W.: Similarity solutions in some nonlinear diffusion problems and in boundary-layer flow of a pseudo-plastic fluid. Q. J. Mech. Appl. Math. 27, 193–211 (1974)
Ruzicka M.: Electrorheological Fluids: Modelling and Mathematical Theory. Lecture Notes in Math. 1748. Springer, Berlin (2000)
Diening L., Harjulehto P., Hästö P., Rûžička M.: Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)
Yin J.X., Jin C.H.: Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources. Math. Methods Appl. Sci. 30, 1147–1167 (2007)
Liu W.J., Wang M.X.: Blow-up of solutions for a p-Laplacian equation with positive initial energy. Acta Appl. Math. 103, 141–146 (2008)
Antontsev S.N., Shmarev S.I.: Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math. 234(9), 2633–2645 (2010)
Antontsev S.N., Shmarev S.I.: Anisotropic parabolic equations with variable nonlinearity. Pub. Math. 53, 355–399 (2009)
Wu Z.Q., Zhao J.N., Yin J.X., Li H.L.: Nonlinear Diffusion Equations. World Scientific, New Jersey (2001)
Lian S.Z., Gao W.J., Yuan H.J., Cao C.L.: Existence of solutions to initial Dirichlet problem of evolution p(x)-Laplace Equations. Ann. Inst. H. Poincare Anal. Non Lineaire 29(3), 377–399 (2012)
Fan X.L.: Remarks on eigenvalue problems involving the p(x)-Laplacian. J. Math. Anal. Appl. 352(1), 85–98 (2009)
Guo B., Gao W.J.: Blow-up of solutions to quasilinear hyperbolic equations with p(x,t)-Laplacian and positive initial energy. Comptes. Rendus. Mecan. 342(9), 513–519 (2014)
Guo B., Gao W.J.: Study of weak solutions for parabolic equations with nonstandard growth conditions. J. Math. Anal. Appl. 374(2), 374–384 (2011)
Simon J.: Compact sets in the space L p(0,T;B). Ann. Math. Pura. Appl. 4(146), 65–96 (1987)
Peterson A.C.: Comparison theorems and existence theorems for ordinary differential equations. J. Math. Anal. Appl. 55, 773–784 (1976)
Akagi G., Matsuura K.: Nonlinear diffusion equations driven by the p(·)-Laplacian. Nonlinear Differ. Equ. Appl. 20, 37–64 (2013)
Teman R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guo, B., Li, Y. & Gao, W. Singular phenomena of solutions for nonlinear diffusion equations involving p(x)-Laplace operator and nonlinear sources. Z. Angew. Math. Phys. 66, 989–1005 (2015). https://doi.org/10.1007/s00033-014-0463-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-014-0463-0