Higher-dimensional moving singularities in a superlinear parabolic equation

Higher-dimensional moving singularities in a superlinear parabolic equation J. Evol. Equ. © 2018 Springer International Publishing AG, Journal of Evolution part of Springer Nature Equations https://doi.org/10.1007/s00028-018-0452-4 Higher-dimensional moving singularities in a superlinear parabolic equation Khin Phyu Phyu Htoo, Jin Takahashi and Eiji Yanagida Abstract. This paper is concerned with the existence of singular solutions of a superlinear parabolic equa- tion. It is shown that under some growth conditions on the nonlinearity, there exists a solution whose singularity forms a one or higher-dimensional time-dependent set. Such solutions are constructed by mod- ifying singular solutions of the linear heat equation. 1. Introduction We study the existence of solutions with a time-dependent singular set for a nonlinear parabolic equation ∂ u − u = f (u) (1.1) n 1 1 in R , where f ∈ C (R) is a superlinear C -function of u satisfying f (0) = 0 (typically f (u) = u with p > 1). The singular set {M } is assumed to be a family t t ∈R of m-dimensional submanifolds in R with m ≥ 1 and n − m ≥ 3. Our aim is to construct a positive solution u of (1.1) that satisfies (1.1)on R \M in the classical sense and u(x http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Higher-dimensional moving singularities in a superlinear parabolic equation

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
D.O.I.
10.1007/s00028-018-0452-4
Publisher site
See Article on Publisher Site

Abstract

J. Evol. Equ. © 2018 Springer International Publishing AG, Journal of Evolution part of Springer Nature Equations https://doi.org/10.1007/s00028-018-0452-4 Higher-dimensional moving singularities in a superlinear parabolic equation Khin Phyu Phyu Htoo, Jin Takahashi and Eiji Yanagida Abstract. This paper is concerned with the existence of singular solutions of a superlinear parabolic equa- tion. It is shown that under some growth conditions on the nonlinearity, there exists a solution whose singularity forms a one or higher-dimensional time-dependent set. Such solutions are constructed by mod- ifying singular solutions of the linear heat equation. 1. Introduction We study the existence of solutions with a time-dependent singular set for a nonlinear parabolic equation ∂ u − u = f (u) (1.1) n 1 1 in R , where f ∈ C (R) is a superlinear C -function of u satisfying f (0) = 0 (typically f (u) = u with p > 1). The singular set {M } is assumed to be a family t t ∈R of m-dimensional submanifolds in R with m ≥ 1 and n − m ≥ 3. Our aim is to construct a positive solution u of (1.1) that satisfies (1.1)on R \M in the classical sense and u(x

Journal

Journal of Evolution EquationsSpringer Journals

Published: May 29, 2018

References

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