Study of nonlinear prabolic systems and related elliptic systems

(1)具有交叉扩散项的反应扩散系统的分析：讨论了具有交叉扩散项的反应扩散系统和Lotka-Volterra型反应扩散系统。这些系统出现在数学生物学中。在数学上，推导出时间整体解存在的充分条件并获得关于正定常解(生物学上，共存状态)的结构的信息是非常重要的。对于非定常问题，得到了一维和二维空间中的整体存在性结果。对于具有零Dirichlet边界条件的定常问题，我们研究了正定常解的唯一性、非唯一性及其存在的充分条件。证明了我们的系统允许正解的多重存在性。此外，数值模拟还显示了正定常解的复杂结构，例如从半平凡解到对称解的分支，以及从对称解到对称解的分支。(2)具有p-Laplace项和Logistic项的拟线性抛物型方程的分析：虽然p-Laplace项的非线性和退化带来了困难，但它也带来了显著的非线性现象。我们对高维定态解的结构以及一维定态解的结构有了满意的理解。特别是，我们还研究了静止解的轮廓，并证明了源于简并扩散的扁帽的有趣结果。此外，我们还可以展示关于非平稳解的时间和空间变化的有趣信息。

(1) Analysis of reaction diffusion systems with cross-diffusion terms : We have discussed reaction diffusion systems with cross-diffusion and reaction of Lotka-Volterra type. These systems appear in mathematical biology. Mathematically, it is very important to derive sufficient conditions for the existence of time-global solutions and get information on the structure of positive stationary solutions (biologically, coexistence states). As to the non-stationary problem a global existence result has been obtained in one and two space-dimensions. For the stationary problem with zero Dirichlet boundary condition, we have studied uniqueness and non-uniqueness of positive stationary solutions as well as sufficient conditions for their existence. It is proved that our system admits multiple existence of postive solutions. Moreover, numerical simulations exhibit complicate structure of positive stationary solutions such as bifurcation of symmetric solutions from semitrivial solutions and, additionally, bifurcation of asymmetric solutions from symmetric ones.(2) Analysis of quasilinear parabolic equations with p-Laplacian and logistic terms : Although the nonlinearity and degeneracy of p-Laplacian brings about the difficulty, it also gives remarkable nonlinear phenomenon. We have obtained satisfactory understanding on the structure of stationary solutions in higher space dimension as well as one dimension. In particular, we also have studied profiles of stationary solutions and proved interesting results on flat hats which stem from degenerate diffusion. Furthermore, we could show interesting information on the temporal and spatial change of non-stationary solutions.