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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 398049, 10 pages
http://dx.doi.org/10.1155/2012/398049
Research Article

Boundedness of Global Solutions for a Heat Equation with Exponential Gradient Source

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

Received 21 September 2011; Accepted 4 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Zhengce Zhang and Yanyan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a one-dimensional semilinear parabolic equation with exponential gradient source and provide a complete classification of large time behavior of the classical solutions: either the space derivative of the solution blows up in finite time with the solution itself remaining bounded or the solution is global and converges in 𝐶 1 norm to the unique steady state. The main difficulty is to prove 𝐶 1 boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov's functional by carrying out the method of Zelenyak.