A Note about Parabolic Systems and Analytic Semigroups

We investigate the question whether certain parabolic systems in the sense of Petrovskii fulfill the resolvent estimate required for the generation of an analytic semigroup and apply the result to a problem concerning the diffusion of gases.     

The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials

This paper is devoted to prove the controllability to trajectories of a system of n one-dimensional parabolic equations when the control is exerted on a part of the boundary by means of m controls. We give a general Kalman condition (necessary and sufficient) and also present a construction and sharp estimates of a biorthogonal family in L²(0,T;C) to {tje−Λkt}.     

New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems

The authors establish a general monotonicity formula for the following elliptic system
△ui＋fi（x,ui,…,um）=0 in Ω,
where Ω belong to belong to R^n is a regular domain, （fi（x, u1,... ,um）） = △↓F（x,→↑u）, F（x,→↑u ） is a given smooth function of x ∈ R^n and →↑u = （u1,…,um） ∈ R^m. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic system
δtui-△ui-fi（x,ui,…,um）=0 in（ti,t2）×R^n,
where t1 〈 t2 are two constants, （fi（x,→↑u ） is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonieity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.     

Bounds for Blow-Up Time in Nonlinear Parabolic Systems Under Various Boundary Conditions

We consider blow-up solutions to parabolic systems, coupled through their nonlinearities under various boundary conditions with nonlinearities depending on the gradient solution. To obtain a lower bound to blow up time t* for the vector solution, Sobolev-type inequalities are introduced to make use of a differential inequality technique. In addition for Dirichlet systems sufficient conditions are introduced to derive an upper bound for t* and to have a criterion for the global existence of the vector solution.     

具有奇异项的拟线性椭圆型方程组无穷多弱解的存在性

本文在一定条件下讨论了一类具有奇异项的,被两个pLaplacian算子控制的拟线性椭圆型方程组Dirichlet问题无穷多弱解的存在性.     

The Dependence of Solution Uniqueness Classes of Boundary Value Problems for General Parabolic Systems on the Geometry of an Unbounded Domain

General boundary value problems are considered for general parabolic (in the Douglas–Nirenberg–Solonnikov sense) systems. The dependence of solution uniqueness classes of these problems on the geometry of a nonbounded domain is established.     

Diffusion phenomena for partially dissipative hyperbolic systems

We consider a partially dissipative hyperbolic system with time-dependent coefficients and show that under natural assumptions its solutions behave like solutions to a parabolic problem modulo terms of faster decay. This generalises the well-known diffusion phenomenon for damped waves and gives some further insights into the structure of dissipative hyperbolic systems.     

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.     

一类非线性抛物型方程组的交替方向变网格有限元方法

本文对一类非线性抛物型方程组提出并分析了一类全离散交替方向变网格有限元格式，且在相当一般的情况下得到了最佳的L^2模误差估计。