Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

Abstract

We study the nonlinear Neumann boundary value problem for semilinear heat equation $$\begin{array}{c}\left\{\begin{array}{c}{\partial }_{\mathit{t}}\mathit{u}-\mathrm{\Delta }\mathit{u}=\mathit{\lambda }|\mathit{u}{|}^{\mathit{p}}\mathrm{,}\mathit{t}\mathrm{>}0\mathrm{,}\mathit{x}\in {\mathbb{R}}_{+}^{\mathit{n}}\mathrm{,}\hfill \\ \mathit{u}\mathrm{(}0\mathrm{,}\mathit{x}\mathrm{)}=\mathit{\epsilon }{\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\mathit{x}\in {\mathbb{R}}_{+}^{\mathit{n}}\mathrm{,}\hfill \\ -{\partial }_{\mathit{x}}\mathit{u}\mathrm{(}\mathit{t}\mathrm{,}{\mathit{x}}^{\prime }\mathrm{,}0\mathrm{)}=\mathit{\gamma }|\mathit{u}{|}^{\mathit{q}}\mathrm{(}\mathit{t}\mathrm{,}{\mathit{x}}^{\prime }\mathrm{,}0\mathrm{)}\mathrm{,}\mathit{t}\mathrm{>}0\mathrm{,}{\mathit{x}}^{\prime }\in {\mathbb{R}}^{\mathit{n}-1}\hfill \end{array}\right.\end{array}$$ where $\mathit{p}=1+\frac{2}{\mathit{n}}$ , $\mathit{q}=1+\frac{1}{\mathit{n}}$ and $\mathit{\epsilon }\mathrm{>}0$ is small enough. We investigate the life span of solutions for $\mathit{\lambda }\mathrm{,}\mathit{\gamma }\mathrm{>}0$ . Also we study the global in time existence and large time asymptotic behavior of solutions in the case of $\mathit{\lambda }\mathrm{,}\mathit{\gamma }\mathrm{<}0$ and ${\int }_{{\mathbb{R}}_{+}^{\mathit{n}}}{\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\mathit{d}\mathit{x}\mathrm{>}0$ .