Asymptotic behavior of the solution of a nonlinear integro-differential diffusion equation

We study the asymptotic behavior as t → ∞ of the solution of the initial-boundary value problem for the nonlinear integro-differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{{\partial U}} {{\partial t}} = \frac{\partial } {{\partial x}}\left[ {a\left( {\mathop \smallint \limits_0^t \left( {\frac{{\partial U}} {{\partial x}}} \right)^2 d\tau } \right)\frac{{\partial U}} {{\partial x}}} \right], $$\end{document} where a(S) = (1 + S)p, 0 < p ≤ 1. We consider problems with homogeneous boundary conditions as well as with a nonhomogeneous boundary condition on part of the boundary. The orders of convergence are established.