Error Estimates for Solutions of the Semilinear Parabolic Equation in Whole Space

Xiaomei Hu

doi:10.1155/2014/501280

Abstract

This paper is focused on the error estimates for solutions of the three-dimensional semilinear parabolic equation with initial datau0∈L2(ℝ3). Employing the energy methods and Fourier analysis technique, it is proved that the error between the solution of the semilinear parabolic equation and that of linear heat equation has the behavior asO((1+t)−3/8).

Highlights

In this study we consider the Cauchy problem of the following three-dimensional semilinear parabolic equation:∂tu − Δu + |u|p−2u = 0, (1) u (x, 0) = u0.Here p > 5. u(x, t) is the unknown function at the point (x, t) ∈ R3 × (0, ∞) and u0 is the initial data.As an important partial differential equation, the wellposedness and asymptotic behavior of solutions of semilinear parabolic equation has attracted more and more attention and many important results have been investigated
To state our main results, let us firstly recall the definition of the weak solutions of the semilinear parabolic equation (1)
It seems impossible to derive the asymptotic behavior of the difference between the semilinear parabolic equation (1) and the linear heat equation (16)

Summary

Introduction

In this study we consider the Cauchy problem of the following three-dimensional semilinear parabolic equation:. From the view on the mathematics point, the nonlinear damping |u|p−2u in (1) may increase the regularity of the weak solutions It will be the main obstacle on the asymptotic behavior of the solutions to the semilinear parabolic equation (1). Compared with the behavior of heat equation (2), it is an interesting problem to consider the influence of the linear damping |u|p−2u in the semilinear parabolic equation (1). Motivated by the asymptotic results on some nonlinear differential equations in [6,7,8,9], in this study we will investigate the asymptotic error estimates between the solutions of both the semilinear parabolic equation (1) and the linear parabolic equation (2).

Preliminaries and Main Results

Error Estimates