Parabolic problems in R n with spatially variable exponents

Abstract

Abstract

We study the asymptotic behavior of parabolic $\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}$ -Laplacian problems of the form $$\frac{\mathit{\partial }{\mathit{u}}_{\mathit{\lambda }}}{\mathit{\partial }\mathit{t}}-div\mathrm{(}{\mathit{D}}^{\mathit{\lambda }}|\nabla {\mathit{u}}_{\mathit{\lambda }}{|}^{\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}-2}\nabla {\mathit{u}}_{\mathit{\lambda }}\mathrm{)}+\mathit{a}{|{\mathit{u}}_{\mathit{\lambda }}|}^{\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}-2}{\mathit{u}}_{\mathit{\lambda }}=\mathit{B}\mathrm{(}{\mathit{u}}_{\mathit{\lambda }}\mathrm{)}$$ in ${\mathit{L}}^2\mathrm{(}{\mathbb{R}}^{\mathit{n}}\mathrm{)}$ , where $\mathit{n}⩾1$ , $\mathit{p}\in {\mathit{L}}^{\infty }\mathrm{(}{\mathbb{R}}^{\mathit{n}}\mathrm{)}$ such that $2\mathrm{<}{\mathit{p}}_{-}:=ess\hspace{0.17em}inf\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}⩽\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}⩽{\mathit{p}}_{+}:=ess\hspace{0.17em}sup\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}$ , ${\mathit{D}}^{\mathit{\lambda }}\in {\mathit{L}}^{\infty }\mathrm{(}{\mathbb{R}}^{\mathit{n}}\mathrm{)}$ , $\infty \mathrm{>}\mathit{M}⩾{\mathit{D}}^{\mathit{\lambda }}\mathrm{(}\mathit{x}\mathrm{)}⩾\mathit{\sigma }\mathrm{>}0$ a.e. in ${\mathbb{R}}^{\mathit{n}}$ , $\mathit{\lambda }\in [0\mathrm{,}\infty \mathrm{)}$ , $\mathit{B}:{\mathit{L}}^2\mathrm{(}{\mathbb{R}}^{\mathit{n}}\mathrm{)}\to {\mathit{L}}^2\mathrm{(}{\mathbb{R}}^{\mathit{n}}\mathrm{)}$ is a globally Lipschitz map and $\mathit{a}:{\mathbb{R}}^{\mathit{n}}\to \mathbb{R}$ is a non-negative continuous function such that there exists ${\mathit{R}}_1\mathrm{>}0$ with $\{\mathit{x}\in {\mathbb{R}}^{\mathit{n}};\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}=0\}\subset {\mathit{B}}_{{\mathit{R}}_1}\mathrm{(}0\mathrm{)}$ , $\underset{\mathit{x}\in {\mathbb{R}}^{\mathit{n}}\setminus {\mathit{B}}_{{\mathit{R}}_1}\mathrm{(}0\mathrm{)}}{inf}\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{>}0$ , and $${\int }_{{\mathbb{R}}^{\mathit{n}}\setminus {\mathit{B}}_{{\mathit{R}}_1}\mathrm{(}0\mathrm{)}}\frac{1}{\mathit{a}{\mathrm{(}\mathit{x}\mathrm{)}}^{2\mathrm{/}\mathrm{(}\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}-2\mathrm{)}}}\mathrm{d}\mathit{x}\mathrm{<}+\infty .$$ We also study the sensitivity of the problem according to the variation of the diffusion coefficients.