Asymptotic behavior of the solutions of certain parabolic equations

Abstract

AbstractSome 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 vj 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) = uo(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § Rm 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 u0(x) ≥ 0 we derive boundsd1u1 ≤ u(x, t) ≤ d2u2,Where di, i = 1, 2, depend on bounds for η, v and G, and the ui are bounds on the initial condition u0.