Asymptotic behavior of the solutions of certain parabolic equations

Abstract

Abstract Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) \documentclass{article}\pagestyle{empty}\begin{document}$$\frac{{\partial u}}{{\partial t}}=\frac{\partial}{{\partial x_j }}(\eta \frac{{\partial u}}{{ax_j}}-vju),$$\end{document} j≤m, where η and v j are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x , 0) = u o (x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § R m the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u 0 ( x ) ≥ 0 we derive bounds d 1 u 1 ≤ u(x, t) ≤ d 2 u 2 , Where d i , i = 1, 2, depend on bounds for η, v and G , and the u i are bounds on the initial condition u 0 .