Integrated Study for Nonlinear Evolution Equations and Nonlinear Elliptic Equations

(1)将本研究发展的L^∞-能量法应用于具有非线性项的非线性抛物型方程，证明了其局部唯一解的存在性。现有的方法由于缺乏规律性，给唯一性的验证带来困难。然而，这种方法通过保证解的高正则性使之成为可能。此外，该方法对于具有趋化性的非线性抛物型方程组和具有滞后效应的非线性抛物型方程组也是非常有效的，因为它可以在比以往更弱的条件下保证解的存在性和唯一性。（ii）在L^2中构造了无限维全局吸引子，它吸引了由p-拉普拉斯算子控制的拟线性抛物型方程组的初始边值问题的所有轨道。无限维全局吸引子从来没有在半线性抛物方程中被观察到，所以这个新的观察似乎非常…更重要。另一方面，对于一些特殊的拟线性抛物方程，证明了具有有限分形维数的指数吸引子的存在性，它能以指数方式吸引从某一类特殊初始数据出发的所有轨道。，由此可知全局吸引子的有限维数。这些观察结果表明，与半线性方程相比，拟线性抛物方程中应该存在某种控制全局吸引子有限维和无限维的结构，这将为未来的研究提供一个非常有趣的对象。iii)证明了对于柯西问题和由时变子微分算子控制的抽象演化方程的周期问题，如果子微分算子的逼近序列收敛于原序列，则相应的逼近解收敛于原方程的解。对于周期问题，对遗留已久的开放性问题给出一个肯定的答案是很有意义的。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