Sixth Mississippi State Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 15 (2007), pp. 51-65. 

Title: Existence and non-existence results for a nonlinear heat equation

Author: Canan Celik (Univ. Sogutozu, Ankara, Turkey)
 
Abstract: 
 In this study, we consider the
 nonlinear heat equation
 $$\displaylines{
 u_{t}(x,t) = \Delta u(x,t) + u(x,t)^p \quad
 \hbox{in }  \Omega \times (0,T),\cr
 Bu(x,t) = 0    \quad \hbox{on }  \partial\Omega \times (0,T),\cr
 u(x,0) = u_0(x) \quad \hbox{in }  \Omega,
}$$
 with Dirichlet and mixed boundary conditions, where
 $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain
 and $p = 1+ 2 /n$ is the critical exponent.
 For an initial condition $u_0 \in L^1$, we prove the non-existence
 of local solution in $L^1$ for
 the mixed boundary condition. Our proof is based on comparison
 principle for Dirichlet and mixed boundary value problems. We also
 establish the global existence in $L^{1+\epsilon}$ to the Dirichlet
 problem, for any fixed $\epsilon > 0$ with $\|u_0\|_{1+\epsilon}$
 sufficiently small.

Published February 28, 2007.
Math Subject Classifications: 35K55, 35K05, 35K57, 35B33.
Key Words: Nonlinear heat equation; mixed boundary condition;
           global existence; critical exponent.