Sixth Mississippi State Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 15 (2007), pp. 51-65.

Existence and non-existence results for a nonlinear heat equation

Canan Celik

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.

Show me the PDF file (275K), TEX file, and other files for this article.

Canan Celik
Departmane to Mathematics
TOBB Economics and Technology University
Sögütözü cad. No. 43. 06560 Sögütözü
Ankara, Turkey
email: canan.celik@etu.edu.tr

Return to the table of contents for this conference.
Return to the EJDE web page