Abstract
The purpose of this paper is to investigate the time behavior of the solution of a weighted p-Laplacian evolution equation, given by
where \(n \in \mathbb {N}{\setminus } \{1\}\), \(p \in (1,\infty ){\setminus } \{2\}\), \(S\subseteq \mathbb {R}^{n}\) is an open, bounded and connected set of class \(C^{1}\), \(\eta \) is the unit outer normal on \(\partial S\), and \(\gamma :S\rightarrow (0,\infty )\) is a bounded function which can be extended to an \(A_{p}\)-Muckenhoupt weight on \(\mathbb {R}^{n}\). It will be proven that the solution of (0.1) converges in \(L^{1}(S)\) to the average of the initial value \(u_{0} \in L^{1}(S)\). Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andreu-Vaillo, F., Caselles, V., Mazón, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Birkhäuser, Basel (2010)
Andreu, F., Mazón, J.M., Rossi, J., Toledo, J.: Local and nonlocal weighted p-Laplacian evolution equations with Neumann boundary conditions. Publ. Math. 55, 27–66 (2011)
Andreu, F., Mazón, J.M., Segura de León, S., Toledo, J.: Quasi-linear elliptic and parabolic equations in \(L^{1}\) with nonlinear boundary conditions. Adv. Math. Sci. Appl. 7, 183–213 (1997)
Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.: An \(L^{1}\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. 22, 241–273 (1995)
Bénilan, P., Crandall, M.: Completely accretive operators. In: Clément, P., Mitidieri, E., de Pagter, B. (eds.) Semigroup Theory and Evolution Equation. Marcel Dekker Inc., New York, pp. 41–76 (1991)
Crandall, M. G., Pazy, A.: Nonlinear evolution equations in Banach spaces. Israel J. Math. 11(1), 57–94 (1972)
Birnir, B., Rowlett, J.: Mathematical models for erosion and the optimal transportation of sediment. Int. J. Nonlinear Sci. 14, 323–337 (2013)
Fang, Z., Li, G., Wang, M.: Extinction properties of solutions for a p-Laplacian evolution equation with nonlinear source and strong absorption. Math. Aeterna 3, 579–591 (2013)
Kilpeläinen, T.: Weighted Sobolev spaces and capacity. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 19, 95–113 (1994)
Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Academic Press Incorporation, New York (1986)
Acknowledgements
The present author is grateful to Prof. Dr. Wolfgang Arendt as well as to Prof. Dr. Evgeny Spodarev for their advices.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nerlich, A. Asymptotic results for solutions of a weighted \({\varvec{p}}\)-Laplacian evolution equation with Neumann boundary conditions. Nonlinear Differ. Equ. Appl. 24, 46 (2017). https://doi.org/10.1007/s00030-017-0468-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-017-0468-4