一类具源项的粘弹性波动方程Cauchy问题解的研究

Viscoelastic equation arises in mechanics, materials science, which describes the motion state of viscoelastic materials after the deformation. Because of nonlinearity of the equation and complexity of the described phenomena, the mathematical theory has been the focus of the research in nonlinear partial differential equations. There are a lot of works on the initial boundary value problem for the viscoelastic wave equation and many excellent results have been obtained. Recently, there are also many good researches on the Cauchy problem for the viscoelastic wave equation. Even so, for the case of the Cauchy problem for the viscoelastic wave equation with source term, fewer results are available up to now. This program mainly studies the Cauchy problem for the nonlinear viscoelastic wave equation, our main goal in this project is concerned on the solution of the global existence, global nonexistence, and the blow-up in the finite time.

粘弹性方程来源于力学、材料科学等，它用来描述粘弹性材料发生形变后的运动状态。由于方程的高度非线性和描述现象的复杂性，其数学理论研究是非线性偏微分方程的一个研究焦点。关于粘弹性波动方程初边值问题解的性质研究已经取得了有意义的进展。近年来，关于粘弹性波动方程Cauchy问题的研究也引起了国内外同行的关注并取得了一些很好的结果。虽然如此，对具有源项的粘弹性波动方程解的性质研究还很少。本项目主要讨论具有非线性阻尼项和源项的粘弹性波动方程Cauchy问题，主要研究解的整体存在性、整体不存在性及有限时刻爆破等性质。

本项目主要研究具有对数型源项的波动型方程的初边值问题解的相关性质。经过一年的努力，基本完成了项目的预期成果，取得成果概述如下：首先考虑对数型源项的Boussinesq方程的初边值问题，利用位势井理论和对数Sobolev不等式得到了解的整体成长性，并且当初值满足一定条件时，解呈指数型增长。其次，研究了一类具有对数型源项波动方程的初边值问题，利用Galerkin方法结合对数Sobolev不等式和对数Gronwall不等式，得到了解的整体存在性，并且利用位势井理论，给出了解在无穷远处爆破的充分条件，并且此方程还带有线性阻尼时，通过构造Lyapunov函数，得到了能量的衰减估计。另外，研究了一类具有时滞项波动方程解的适定性、整体吸引子和能量衰减估计等性质。