# American Institute of Mathematical Sciences

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

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.
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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