Sign-changing solutions of a quasilinear heat equation with a source term

Guirong Liu; Yuanwei Qi

doi:10.3934/dcdsb.2013.18.1389

Article Contents

2013, Volume 18, Issue 5: 1389-1414. Doi: 10.3934/dcdsb.2013.18.1389

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Sign-changing solutions of a quasilinear heat equation with a source term

Guirong Liu^1, and
Yuanwei Qi^2,

1.
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006
2.
Department of Mathematics, University of Central Florida, Orlando, FL 32816

Received: June 2012

Revised: August 2012

Published: July 2013

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Abstract

The Cauchy problem of a heat equation with a source term $$ \psi_t=\Delta \left(|\psi|^{m-1}\psi\right)+|\psi|^{\gamma-1}\psi\ \ \mbox{in}\ \ (0, \infty)\times R^n $$ is considered, where $\gamma>m>1$. We are interested in global solutions with H$\ddot{o}$lder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time $ t > 0$. Through detailed analysis of the self-similarity problem, we prove the existence of two type of such solutions, one with compact support and the other decays to zero as $ | x| \rightarrow \infty$ with an algebraic rate determined uniquely by $ n, m$ and $\gamma$. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of $m = 1$.

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Mathematics Subject Classification: 35K05, 35B40.

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