一些退化椭圆和抛物方程的研究

Nonlinear degenerate equations, as an important class of partial differential equations, can model many problems arising in physics, chemistry, economics and image processing, etc. In the past few decades, many mathematicians have studied this kind of problems, and remarkable progress has been achieved on the existence and uniqueness of solutions, regularity and blow-up, etc. Until now, the study of such problems is still a very active research area.. In this project, we mainly study a class of nonlinear degenerate problems whose main operators goes to zero as the solution u goes to infinity(that is, the diffusion goes to zero when u goes to infinity). Such kind of equations could be seen as a reaction model which produces a saturation effect. The main difficulty comes from the fact that the main operator is degenerate and not coercive, which means that the classical Leray–Lions methods can’t be applied to prove the existence of solutions. We shall use the rearrangement theory, potential theory together with energy methods and compactness argument, to study the existence, regularity of solutions to such kinds of problems. Furthermore, based on the theory of variable exponents Sobolev space, we shall investigate the equations with variable exponent p(x), which coincide the case that p(x)=p of the above problems. The study of this project has important practical significance,which can enrich the theory of nonlinear degenerate equations and the partial differential equations with variable exponent.

非线性退化方程是一类重要的偏微分方程，可模拟许多来源于诸如物理、化学、经济学和图像处理等领域的实际问题。近年来，众多数学家研究了此类问题，并在解的存在与唯一性、正则性和爆破等方面取得了丰富的成果。目前，对此类问题的研究仍是一个十分活跃的领域。. 本项目拟研究几类非线性退化方程，其中当解u趋于无穷时，方程的主算子趋于零(即解u趋于无穷时，扩散现象消失)。这些方程可用来刻画产生饱和效应的反应模型，问题的难点主要在于主算子的退化非强制性，这意味着不能直接用经典的Leray–Lions理论来证明解的存在性。我们将利用重排理论、位势理论结合能量估计方法与紧性理论，来研究上述问题解的存在性与正则性，进一步利用变指Sololev空间理论研究与之对应的变指数p(x)方程。本项目的研究将进一步丰富非线性退化方程与变指数偏微分方程的理论，具有重要的现实意义。

非线性退化方程是一类重要的偏微分方程，可模拟许多来源于诸如物理、化学、经济学和图像处理等领域的实际问题。近年来，国内外众多数学家开展了此类问题的研究，并在解的存在与唯一性、正则性和爆破等方面取得了一系列结果，大大丰富了偏微分方程的理论。目前，对此类问题的研究仍是一个十分活跃的领域。. 本项目主要研究一类非线性退化方程，其中当解u趋于无穷时，方程的主算子趋于零(即解u趋于无穷时，扩散现象消失)。这些方程可用来刻画产生饱和效应的反应模型，问题的难点主要在于主算子的退化非强制性，这意味着不能直接用经典的Leray–Lions理论来证明解的存在性。我们先研究了一类含低阶项的非线性退化椭圆方程Dirichlet 边值问题，证明了问题解的存在性与正则性结果，揭示了低阶项对问题解的影响机制；其次，研究了一类含低阶项（不含低阶项）的非线性退化抛物方程Dirichlet 边值问题，证明了问题解的存在性与正则性；最后研究了具非标准增长条件的非线性退化方程，得到了解的一些正则性结果，推广并改进了以往文献的结果。本项目的研究结果不仅丰富了偏微分方程的理论，同时也具有重要的理论意义和应用价值。