A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains

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2009, 2009(Special): 574-582. doi: 10.3934/proc.2009.2009.574

A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains

1.	Department of Mathematics, The College at Brockport, State University of New York, Brockport, NY 14420, United States

Received July 2008 Revised April 2009 Published September 2009

Abstract
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We present a Sobolev space approach for semilinear heat equations $u_t=\Delta u + F(u(t,x))$ for $t>0$ on a bounded domain $\Omega\subset\mathbf{R}^n$. By proving that there exists a solution in the anisotropic Sobolev space $W^{1,2}_p( \R_+\times\Omega)$, we can deduce more than just global existence in time. For example, both the solution and its time derivative are of class $L^p$, and the solution tends to zero in $L^\infty(\Omega)$ as $t\to\infty$. The main result shows that the existence of a solution in $W^{1,2}_p$ depends primarily on the existence of an appropriate a priori estimate on the $L^\infty$ norm of solutions as the initial data is deformed to zero.

Keywords: parabolic, Fredholm operator, semilinear, topological degree, global existence.

Mathematics Subject Classification: Primary: 35B40, 35B45, 35K15, 35K55; Secondary: 47A53, 47H11.

Citation: Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574

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