Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation

Xie Li; Zhaoyin Xiang

doi:10.3934/cpaa.2014.13.1465

Article Contents

2014, Volume 13, Issue 4: 1465-1480. Doi: 10.3934/cpaa.2014.13.1465

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Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation

Xie Li^1, and
Zhaoyin Xiang^1,

1.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731

Received: April 30, 2013

Revised: November 30, 2013

Published: June 2015

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Abstract

In this paper, we study the existence of local/global solutions to the Cauchy problem \begin{eqnarray} \rho(x)u_t=\Delta u+q(x)u^p, (x,t)\in R^N \times (0,T),\\ u(x,0)=u_{0}(x)\ge 0, x \in R^N \end{eqnarray} with $p > 0$ and $N\ge 3$. We describe the sharp decay conditions on $\rho, q$ and the data $u_0$ at infinity that guarantee the local/global existence of nonnegative solutions.

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Mathematics Subject Classification: Primary: 35K55, 35Q92, 35Q35; Secondary: 92C17.

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