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

In this study, we consider the nonlinear heat equation
 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.

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Canan Celik
Departmane to Mathematics
TOBB Economics and Technology University
Sögütözü cad. No. 43. 06560 Sögütözü
Ankara, Turkey

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