A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded
domains
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
Keywords: semilinear, parabolic, global existence, topological degree, Fredholm operator
Received: July 2008; Revised: April 2009; Published: September 2009. |