On Asymptotic Properties of Solutions of Diffusion Equations

In this work the authors study the conditions for the existence of diffusion equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\partial _t u\left( {x,t} \right) = 3DA\left( {x,\partial x} \right)u\left( {x,t} \right) + f\left( u \right),A\left( {x,\partial x} \right) \equiv \sum\limits_{i,j = 3D1}^n {\partial _{xj} \left( {a_{ij} \left( x \right)\partial _{xi} } \right)}$$ \end{document} in the cylinder Q = 3DΩ × \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}+, Ω ⊂ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}n, satisfying the homogeneous Dirichlet or Neumann conditions on the side boundary of the cylinder Q and decreasing with respect to t as a power for t → ∞.