Integrated Study for Nonlinear Evolution Equations and Nonlinear Elliptic Equations

(i)本文将L^∞-能量方法应用于含时间导数非线性项的非线性抛物型方程，证明了局部解的存在性。现有的方法由于缺乏规律性，使得唯一性的验证比较困难。然而，这种方法通过确保解的高度正则性而使其成为可能。此外，该方法对于具有趋化性的非线性抛物方程组和具有滞后效应的系统也是非常有效的，因为它可以在比以往更弱的条件下确保解的存在唯一性。（ii）在L^2中构造了无穷维全局吸引子，它吸引p-Laplacian控制的拟线性抛物型方程初边值问题的所有轨道。对于半线性抛物方程，永远不会观察到无穷维全局吸引子，因此这个非常新的观察似乎非常重要。 ...更多信息 重要.另一方面，对于一类特殊的含Laplacian和p-Laplacian的拟线性抛物方程，证明了存在一个具有有限分维的指数吸引子，它能指数吸引从某类特殊初值开始的所有轨道.由此得出全局吸引子的有限维性。这些结果表明，与半线性方程相比，拟线性抛物方程应该存在某种结构，它控制着整体吸引子的有限维和无限维，这是一个非常有意义的研究对象。iii）证明了对于由含时次微分算子控制的抽象发展方程的Cauchy问题和周期问题，若逼近次微分算子序列收敛于原算子序列，则相应的逼近解收敛于原方程的解。对于周期性问题，对长期以来悬而未决的问题给予肯定的回答是非常有意义的。少

(i) L^∞-energy Method, developed in this research, is applied to the nonlinear parabolic equations with nonlinear terms involving the time derivative to show the existence of the unique local solution. The verification for the uniqueness was difficult for the existing methods because of the lack of regularity. However this method makes it possible by assuring the high regularity of solutions. Furthermore this method turns out to be very effective also for nonlinear parabolic systems for chemotaxis and systems with the hysteresis effect by the fact that it can assure the existence an uniqueness of solution under much weaker conditions than ever(ii) The infinite dimensional global attractor is constructed in L^2, which attracts all orbits for the initial boundary value problem for the quasi-linear parabolic equation governed by the p-Laplacian. The infinite dimensional global attractor is never observed for the semilinear parabolic equations, so this very new observation seems to be very … More important. On the other hand, the existence of the exponential attractor with finite fractal dimension , which attracts all orbits starting from some special class of initial data exponentially, is shown for some special quasilinear parabolic equations involving Laplacian and p-Laplacian., whence follows the finite dimensionality of the global attractor. These observations suggest that in contrast with semilinear equations, there should exist some structure in quasilinear parabolic equations which controls the finite-dimensionality and infinite-dimensionality of global attractors, which gives a very interesting future object ton study.iii) It is shown that for Cauchy problem and periodic problem for the abstract evolution equation governed by time-dependent subdifferential operators, if the sequence of approximating subdifferential operators converges to the original one, then the corresponding approximating solutions converge to the solution of the original equation. As for the periodic problem, it is very meaningful to give an affirmative answer to the open problem left long. Less