Global existence and gradient blowupof solutions for a semilinear parabolic equation with exponential source

Zhengce Zhang; Yan Li

doi:10.3934/dcdsb.2014.19.3019

Article Contents

2014, Volume 19, Issue 9: 3019-3029. Doi: 10.3934/dcdsb.2014.19.3019

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Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source

Zhengce Zhang^1, and
Yan Li^1,

1.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049

Received: April 30, 2013

Revised: August 31, 2013

Published: October 2014

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Abstract

Throughout this paper, we consider the equation \[u_t - \Delta u = e^{|\nabla u|}\] with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.

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Mathematics Subject Classification: Primary: 35K58; Secondary: 35A01, 35B40, 35B44.

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