伪抛物型方程的若干定性问题

In this project, we study a class of pseudo-parabolic equations，which are characterized by the occurrence of mixed time and space derivative appearing in the highest-order term, and arise in the viscous diffusion processes that widely exist in the seepage in fissured porous medium, nonlinear dispersive long wave, viscous incompressible fluids etc. Moreover, they can be used as effective regularization for treating the non-wellposed models from image processing etc..Via studying several qualitative problems of pseudo-parabolic equations, this project aims to investigate the effect of the viscous term of pseudo-parabolic type, so as to reveal the difference and relation between the properties of solutions to that of the corresponding parabolic and hyperbolic models, and thus provide some more reasonable approximation processes. We mainly concern the dead-core, traveling wave solution, Pinning/Depinning in the propogation and the asymptotic behavior of solutions, which are lack of complete discussion in the research of pseudo-parabolic equations. According to the characteristics of pseudo-parabolic equations, especially to the degeneration and singularity occurring at the high-order term with mixed derivative, there will be essential difficulties in our research, hence we need to find new approaches, frameworks etc. To a certain extent, our results and methods will provide important reference for explaining some physical phenomena and enrich the theory of partial differential equations.

本项目拟研究伪抛物型方程,其特点是具有关于时-空混合导的高阶项.这类方程来源于裂隙多孔介质的渗流、非线性色散长波、粘弹性不可压缩流体等问题中广泛存在的阻性扩散过程,同时也可作为研究图像处理等问题中的非适定模型的有效的正则化逼近..本项目旨在通过研究这类方程的若干定性问题,考察伪抛物粘性对解的性态的影响,从而揭示其与对应抛物及双曲模型解的性态的区别与联系,为真实模型的理论分析和数值计算提供一种更合理的逼近过程.拟研究死核、行波解、传播的Pinning/Depinning以及解的渐近行为等伪抛物型方程领域尚无完整结果的问题.伪抛物型方程本身的特有结构，尤其是当退化性、奇异性出现在高阶混合导项时,会为研究带来本质性困难,需要寻找新的研究思路.本项目的研究结果和方法将对解释有关物理现象提供重要参考，并在一定程度上丰富和完善偏微分方程的理论.

本项目研究与物理学、生物学以及图像处理等领域问题有着密切关系的伪抛物型方程。这类方程不但具有鲜明的实际背景，并且由于其特殊的高阶项形式，使得其研究结果也颇具理论价值。本项目以大连理工大学为依托单位，在项目执行期间，项目组按照研究计划开展了研究工作，取得了一系列研究成果。在Journal of Differential Equations，Discrete and Continuous Dynamical Systems Series A等学术期刊上发表论文8篇，基本实现了预期目标。主要成果包括：小摄动、内部源及伪抛物型扩散机制对方程及方程组解的渐近行为的影响，退化伪抛物型方程解的长时间行为，伪抛物型Fisher-KPP方程的行波解，具弱周期源的伪抛物型方程的时间周期解问题。这些研究工作将在一定程度上丰富偏微分方程的理论并对解决实际问题提供重要参考。