非线性反应-对流-扩散方程的奇异点集和解的渐近性研究

This project is concerned with nonlinear reaction-convection-diffusion equations, which are derived from the reaction-convection-diffusion progress existing widely in physics, differential geometry, ecology, population dynamics and so on. They have both profound practical background and important theoretical value. We will investigate the singular sets and asymptotic properties of solutions for such kind of equations, including removable singularities of solutions, classification of singular solutions, especially complete classification, periodic solutions and stationary solutions, and asymptotic behavior of solutions. We expect to establish the critical exponents theories, which can be used to discribe the asymptotic behavior of solutions, and give the effection of reaction, convection and diffusion on them. Existence of the multiple nonlinearities, degeneracies and singularities makes the models more actual, but, at the same time, leads to essential difficulties. So our research not only needs the classical mathematical tools, but also needs to make innovations in the research ideas and methods, and it will enrich and perfect the theory of partial differential equations to some extent.

本项目拟研究非线性反应-对流-扩散方程, 这类方程来源于物理学、微分几何、生态学以及种群动力学等领域中广泛存在的反应-对流-扩散现象, 既有深刻实际背景也有重要理论价值. 本项目将研究此类方程的奇异点集和解的渐近性, 包括解的奇异性可去问题, 奇异解分类特别是完全分类, 周期解和稳态解以及解的渐近行为. 本项目将建立利用方程或初边值条件中指数或系数刻画解性态变化的临界指标理论, 明确给出反应项、对流项与扩散项对解性态的影响. 多重非线性与退化、奇异的存在, 这使得模型更接近于实际, 同时也为研究带来本质的困难. 因此本项目的研究既需要经典的数学工具, 也需要在研究思路与方法上有所创新, 将在一定程度上丰富和完善已有的偏微分方程理论.

本项目研究了具有鲜明背景的非线性偏微分方程的奇异点集和解的渐近行为。具体地，我们针对含非线性Kato类位势的退化抛物方程建立了孤立奇点可去条件，得出从孤立奇异性的角度，非线性Kato类位势是一类好位势；关于含径向对称非线性Kato类位势的加权p-Laplace椭圆方程（包括经典情形和非经典情形），对正解在孤立奇点处的渐近行为建立了完全分类和存在性；证明了几种（固态、液态）血管化肿瘤生长模型偏微分方程自由边界问题（径向对称和非径向对称）稳态解的存在性，包括含死核结构、含抑制因子等情形，揭示了血管生成、抑制因子对肿瘤大小和侵袭能力的影响；讨论了一类含小源的加权(p,q)-Laplace椭圆方程组Dirichlet边值问题无穷多解的存在性；给出了两类含非线性源的非线性快扩散方程组解的熄灭条件。本项目研究在结果与方法上均有不同程度的创新，丰富了已有的偏微分方程理论。