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

Pages: 574 - 582, Issue Special, September 2009

 Abstract        Full Text (167.9K)              

Jason R. Morris - Department of Mathematics, The College at Brockport, State University of New York, Brockport, NY 14420, United States (email)

Abstract: 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:  semilinear, parabolic, global existence, topological degree, Fredholm operator
Mathematics Subject Classification:  Primary: 35B40, 35B45, 35K15, 35K55; Secondary: 47A53, 47H11

Received: July 2008;      Revised: April 2009;      Published: September 2009.