一类反应扩散方程的定性分析

QUALITATIVE ANALYSIS ON A CLASS OF R D EQUATIONS

有限维代数自治系统不存在除奇点或无穷大外的无界ω－ 或α－ 极限集．一类反应扩散方程存在非平凡极限集、孤立子和击波解．定义了无穷维条件奇异吸引子、无穷维动力系统的混沌现象并用来解释湍流现象．用无穷维动力系统存在非平凡极限集的必要条件。

Many nonlinear problems arising from various sciences deal with partial differential equations joint with certain initial and boundary conditions.In order to solve nonlinear equations or to know some properties of nonlinear equations,new theoretical treatment and new numerical schemes have been creating.Open problems are proposed to understand some strange phenomena such as chaos,turbulence and chemical vibrations.Qualitative analysis of partial differential equations and ordinary differential equations are main directions in theoretically discussing intrinsic phenomena of nonlinear sciences via the type Ⅰ limit set (in Poincare's sense) and the type Ⅱ limit set(in anthor's sense).The study on infinitely dimensional systems is a main stream in qualitative analysis of partial differential equations and is also a popular subject theses days.On e may encounter much difficulty in solving nonlinear problems by traditional methods.Many authors introduce some new techniques other than apriori bounds.The treatment in finitely dimensional dynamical systems can be applied to analyze special solutions such as traveling waves etc.Global attractors,strange attractors and conditional strange attractors are important concepts for the good explanation of strange phenomena in applications.Limit sets of dynamical systems play a critical role in leading to systematic study on nonlinear problems and chaos which is an essential phenomena in nonlinear sciences. In this paper,a class of reaction diffusion system is discussed while new ideas and notions are introduced by the author.Some interesting results are obtained as follows.Any finitely dimensional polynomial autonomous system has no unbounded omega or alpha limit sets other than critical points or the infinity.The existence of nontrivial limit sets,solitons and shooting waves of a class of reaction diffusion equations has been proved under certain conditions.The stability of equilibria(steady state solutions)is also considered.The conditional strange attractor or chaotic phenomena of infinitely dimensional dynamical systems have been defined to interpret the turbulence phenomena.Some typical examples via a necessary condition for the existence of nontrivial limit sets are discussed.The vector field homeomorphism is an important tool in proofs.Many interesting questions can be asked for theoretical purpose.