Asymptotic Behavior of Global Solutions to the Boussinesq Equation in Multidimensions

Yu-Zhu Wang,Qingnian Zhang

doi:10.1155/2013/154102

Abstract

The Cauchy problem for the Boussinesq equation in multidimensions is investigated. We prove the asymptotic behavior of the global solutions provided that the initial data are suitably small. Moreover, our global solutions can be approximated by the solutions to the corresponding linear equation as time tends to infinity when the dimension of space <svg style="vertical-align:-0.546pt;width:36.237499px;" id="M1" height="11.6" version="1.1" viewBox="0 0 36.237499 11.6" width="36.237499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
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Highlights

We investigate the Cauchy problem of the following damped Boussinesq equation in multidimensions: utt − aΔutt − 2bΔut − αΔ3u + βΔ2u − Δu = Δf (u) (1)with the initial value t = 0 : u = u0 (x), ut = u1 (x) . (2)Here u = u(x, t) is the unknown function of x = (x1, . . . , xn) ∈ Rn and t > 0, a, b, α, and β are positive constants
We prove the asymptotic behavior of the global solutions provided that the initial data are suitably small
We estimate the term L1, applying (30) with p = 2, j = 0, and l = 0 and (39), (40), we arrive at

Summary

Introduction

Wang [3] proved the global existence and asymptotic decay of solutions to the problem (1), (2). Their proof is based on the contraction mapping principle and makes use of the sharp decay estimates for the linearized problem. The main purpose of this paper is to establish the following optimal decay estimate of solutions to (1) and (2) by constructing the antiderivatives conditions: u1 (x) = ∂x1 V1 (x). The solution u of the problem (1), (2), which is constructed in Theorem 1, can be approximated by the linear solution uL as t → ∞.

Decay Property

Proof of Main Result