【作者】 杨志坚；

【导师】 陈国旺；

【作者基本信息】 郑州大学 ， 基础数学， 2000， 博士

【摘要】 本文致力于研究两类非线性数学物理模型方程初边值问题的适定性与解的Blowup问题。 在本文的第一部分，我们讨论出自粘弹性力学的具强阻尼、非线性应变、非线性阻尼和非线性力源项的多维拟线性发展方程初边值问题的整体解的存在性、解的渐近性、稳定性及解的Blowup问题。 我们利用位势井方法，单调性方法研究了在小初始能量情况下，上述初边值问题的整体弱解的存在性和衰减性，用Galerkin方法和紧致性方法研究了在大初值情况下上述初这值问题的整体弱解的存在性，分别利用补偿能量方法和能量方法研究了上述初边值问题的整体解的不存在性问题。我们证明，上述方程中诸非线性项的增长阶（假定它们均具有不超过多项式的增长阶）指数、初始能量的状态和相应的定解问题的整体解存在与解在有限时刻发生Blowup现象之间存在着密切的联系，得到了增长指数、初始能量的状态与整体解的存在与不存在之间的类似于门槛的结果。 我们利用H_o~k-Galerkin方法，进一步研究了具强阻尼、非线性应变和非线性外力项的多维非线性发展方程的初边值问题。证明了在小初始能量情况下，上述初边值问题的整体古典解的存在性、稳定性和衰减性，得到了一系列新的结果。 在本文的第二部分，我们讨论出自浅水波理论及等离子物理的“坏的”Boussinesq型方程初边值问题的局部解的存在性及解的Blowup问题，利用一种新的数学思想、即将抽象初值问题的解视为抽象空间中由初始点出发的“流”。通过建立一系列等距同构的Hilbert空间，利用这些空间的拓扑不变性、利用Galenkin逼近和解的逐次延拓，证明了在相当宽松的条件下，上述问题存在局部广义解。 我们分别利用Jensen不等式、常微分方程的比较原理、能量方法和Fourier变换方法，证明了在一定条件下，“坏的”Boussinesq型方程的上述初边值问题的局部解必在有限时刻发生Blowup现象。更多还原

【Abstract】 AbstractThe paper studies the existence, non-existence, asymptotic property and stability of global solutions of the initial boundary value problems to two classes of nonlinear model equations in mathematical physics.The paper is divided into two parts. In the first part, it discusses the existence of global weak solutions of the initial boundary value problems to a class of multidimensional quasi linear evolution equations with strong damping, nonlinear strain, nonlinear damping and source terms, which come from Viscoelasticity Mechanics. Making use of the potential well method as well as the monotonicity method, the paper arrives at the global existence and asymptotic property of weak solutions to the problems under the assumptions that the initial energy is properly small. Under the circumstances of large initial data, taking advantage of the Galerkin method a well as the compactness method, the paper gets the global weak solutions of the problems. And using the compensating energy method and the energy method respectively, the paper studies the non-existence of global weak solutions to the problems. These results implies that there exist close relations or thresholds among the growth indexes of the nonlinear terms, assuming that their growth indexes are no more than that of polynomials respectively, the value of initial energy, and the existence and non-existence of global weak solutions of the above-mentioned problems. Furthermore, the paper studies the initial boundary value problems to a class of multidimensional nonlinear evolution equations with strong damping, nonlinear strain and nonlinear external force terms. Under the assumptions that initial data is properly small, by virtue of the H -Galerkin method, the paper obtains theglobal existence, asymptotic property and stability of the classical solutions to the problemsand gains a series of new results.In the second part, the paper studies the local existence and the blowup of solutions of the initial boundary value problems to the ad?Boussinesq type equations, which arise from the theory of shallow waves and the Plasma Physics. Based on a new mathematical idea, i.e., viewing the solutions of the problems as streams starting from their initial points, by establishing a series of isometricly isomorphic Hilbert spaces, by the topological invariance of these spaces, and by exploiting the Galerkin approximation and the continuation of solutions step by step, the paper proved that under rather mild conditions for the nonlinear terms and the initial data, the initial boundary value problems of the ad?Boussinesq type equation have local generalized solutions. And by using the Jensen inequality, the comparison principle of ordinaiy differential equations, the energy method and the Fourier transform method respectively, the paper proved that under certain conditions the above-mentioned solutions will blow up in finite time.更多还原