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.