【摘要】 本文研究一类具有奇异势和记忆项的四阶抛物方程在有界域上的初边值问题.当初值在稳定集中,初始能量在正有界范围内,根据Faedo-Galerkin方法结合Hardy-Sobolev不等式得到了问题解的整体存在性并建立了能量泛函的衰减估计;当初始能量为负时,利用凸方法证明了问题的解在有限时刻爆破并估计了爆破时间上界,该上界依赖于初始能量;当初值位于不稳定集,初始能量有上界时,通过构造辅助泛函获得了一个与初始能量无关的爆破时间上界.更多还原

【Abstract】 This paper is concerned with the initial boundary value problem of a class of fourth order parabolic equations with singular potential and memory terms which are in a bounded domain. Firstly,the global existence of solutions are derived and the decay rate of the energy functional is estimated by the Faedo-Galerkin method and Hardy Sobolev inequality when the inital value is in the stable set and the initial energy is positive and bounded. In addition, for the negative initial energy, the convex method is used to prove that the solutions blow up in finite time and give the upper bound of the blow-up time,which depends on the initial energy; Finally, when the inital value is in the unstable set and the initial energy has an upper bound. By constructing an auxiliary functional, the upper bound of the blow-up time is obtained, which is independent of the initial energy.更多还原