Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains

We investigate the Dirichlet problem for the parablic equation

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in a non-smooth domain . In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove -contraction estimation in general non-smooth domains.

     

Optimal control of n x n parabolic system involving time lag

In this paper, we study the optimal control problem for an nx n coupled system of second-order parabolic partial differentialequations with infinitely many variables and constant time lag.Making use of the Lions scheme (Lions, 1971, Optimal Controlof Systems Governed by Partial Differential Equations), necessaryand sufficient condition of optimality for the Neumann problemwith quadratic performance functional and constraint controlis derived. Finally, several mathematical examples for derivedoptimality conditions are presented.     

Global Existence and Blow-up of Solutions for Parabolic Systems Involving Cross-Diffusions and Nonlinear Boundary Conditions

Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.     

Geometry of conservation laws for a class of parabolic partial differential equations

I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is: Theorem. Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal ${\cal I}$ on the original 12-manifold M. I show that if a nontrivial conservation law exists, then has a deprolongation to an equivalent system on a 7-manifold N, and any conservation law for can be expressed as a closed 3-form on N that lies in . Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA (u _xx u _yy -u ² _xy )+Bu _xx +2Cu _xy +Du _yy +E = 0 where A, B, C, D, E are functions of x, y, t, u, u _x , u _y , u _t . I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.     

Semiclassical Parabolic Systems Related to M24

Semiclassical parabolic systems belong to diagram geometries whose rank two residues are generalized n-gons or a triple cover of the generalized 4-gon for Sp(4, 2), and the latter really occurs. So semiclassical parabolic systems are a generalization of Tits chamber systems of classsical type. We classify all such systems of rank four assuming in addition that there is some residue belonging to a subdiagram of type . Among others iteresting examples showing up are systems for Co₁ and M(24).     

Complicated dynamics of parabolic equations with simple gradient dependence

Let be a smooth bounded domain. Given positive integers , and , , ..., , consider the semilinear parabolic equation

where and are smooth functions. By refining and extending previous results of Polácik we show that arbitrary -jets of vector fields in can be realized in equations of the form (E). In particular, taking we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on .

     

On the cauchy problem for certain system of semilinear parabolic equations

In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations:

On the local in time well-posedness of an elliptic–parabolic ferroelectric phase-field model

We consider a state-of-the-art ferroelectric phase-field model arising from the engineering area in recent years, which is mathematically formulated as a coupled elliptic–parabolic differential system. We utilize the maximal parabolic regularity theory to show the local in time well-posedness of the ferroelectric problem in both 2D and 3D spaces, which is sharp in the sense that the local solution is unique and a blow-up criterion is present. The well-posedness result will firstly be proved under some general assumptions. Afterwards we give sufficient geometric and regularity conditions which will guarantee the fulfillment of the imposed assumptions.     

三类有效预处理子的关系及其优化

针对一类由时谐抛物方程约束的最优控制问题导出的分块$2\times2$复线性方程组,进一步研究了三类有效的块预处理子,推导了这三类预处理子间的关系,结论表明三个预处理矩阵的特征值由同一个矩阵确定.通过分析预处理矩阵的谱性质,获得了有效的参数选择策略,可以进一步改进和优化现有结果,同时获得了预处理矩阵的精确特征值分布,并证明了此结果是目前文献中最优结果.最后,给出实例,不仅验证了优化的预处理子和迭代方法的有效性,而且说明了理论结果是令人信服的.