Existence and non-existence results for a nonlinear heat equation

In this study, we consider the nonlinear heat equation $$displaylines{ u_{t}(x,t) = Delta u(x,t) + u(x,t)^p quad hbox{in } Omega imes (0,T),cr Bu(x,t) = 0 quad hbox{on } partialOmega imes (0,T),cr u(x,0) = u_0(x) quad hbox{in } Omega,}$$ 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