Asymptotic Behavior of the Nonlinear Parabolic Equations

Abstract

This paper is concerned with the large time behavior for solutions of the nonlinear parabolic equations in whole spaces R^n. The spectral decomposition methods of Laplace operator are applied and it is proved that if the initial data u_0 ∈ L² ∩ L^r for 1 ≤ r ≤ 2, then the solutions decay in L² norm at t^{-\frac{n}{2}(\frac{1}{r}-\frac{1}{2})}. The decay rates are optimal in the sense that they coincide with the decay rates of the solutions to the heat equations with the same initial data.