An optimization problem for time lag distributed parabolic systems

We consider an optimal boundary control problem for a distributed parabolic system with boundary conditions involving multipletime-varying lags. Necessary and sufficient optimality conditionsfor the Neumann problem with the quadratic performance functionalsare derived. The results are illustrated with numerical solutionsof example tasks.     

Partial regularity and singular sets of solutions of higher order parabolic systems

In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems of the type

Second order optimality conditions for a class of control problems governed by non-linear integral equations with application to parabolic boundary control

In the paper necessary and sufficient second order optimality conditions for optimal control problems governed by weakly singular non linear Hammerstein integral equations are derived. They are applied to a semilinear parabolic boundary control problem for the one dimensional heat equation.     

Blow-up estimates for semilinear parabolic systems coupled in an equation and a boundary condition

This paper deals with the blow-up rate estimates of solutions for semilinear parabolic systems coupled in an equation and a boundary condition. The upper and lower bounds of blow-up rates have been obtained.     

Optimal control of distributed parabolic systems with multiple time delays given in the integral form

Various optimal control problems for linear parabolic systemswith multiple time delays given in the integral form are considered.Necessary and sufficient conditions of optimality are derivedfor the Neumann problem. The optimal control is obtained inthe feedback form. Making use of the results of Schwartz, therepresentation of the optimal feedback control is given. A simpleexample of application is also provided.     

Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non-smooth coefficients

We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C¹-boundary, the other stated for sets with finite perimeter.     

半线性抛物型方程组解的整体存在性与爆破速率估计

研究了具有非线性热源的半线性抛物型方程组的齐次neumann问题解的爆破性质.利用上下解方法得到了解整体存在的条件与爆破条件,并利用FriedmannMcleod方法建立了爆破速率估计.     

Schauder estimates for solutions to boundary value problems for second order elliptic systems in polyhedral domains

Boundary value problems for second order elliptic differential equations and systems in a polyhedral domain are considered. The authors prove Schauder estimates and obtain regularity assertions for the solutions.     

Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources

This paper deals with two classes of parabolic systems with localized nonlinear sources. The critical exponents as well as the estimates for blow-up rates and boundary layer profiles are determined.     

Upper-lower solutions for nonlinear parabolic systems and their applications

The method of upper-lower solutions for nonlinear parabolic systems without the assumption of quasi-monotonicity is obtained. An application is provided, by using the method developed in this paper, involving the existence of positive solutions to certain time-dependent interaction systems arising in biological and medical sciences. Furthermore, the existence of the ω-limit of given system is studied.     

A priori and universal estimates for global solutions of superlinear degenerate parabolic equations

We prove an a priori estimate and a universal bound for any global solution of the nonlinear degenerate reaction-diffusion equation u _t =Δu ^m +u ^p in a bounded domain with zero Dirichlet boundary conditions. Received: October 1, 2001?Published online: July 9, 2002     

On continuity of solutions for parabolic control systems and input-to-state stability

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.     

Global and blow‐up solutions for non‐linear degenerate parabolic systems

In this paper the degenerate parabolic system u_t=u(u_xx+av). vt=v(v_xx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α₁=α₂) of solution are analysed. For , the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.     

Existence of periodic solutions for Hamiltonian systems with super-linear and sign-changing nonlinearities

In this paper, we consider the existence of periodic solutions for the super quadratic second order Hamiltonian system, and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our result extends and improves some existed results in the literature.     

Hölder regularity for weak solutions to divergence form degenerate quasilinear parabolic systems

In this paper, we establish Hölder regularity of weak solutions to the diagonal divergence form degenerate quasilinear parabolic system related to Hörmander type vector fields by deriving a parabolic Poincaré inequality and the higher integrability of gradients of weak solutions.     

Decentralized stability for switched nonlinear large-scale systems with interval time-varying delays in interconnections

In this paper, the problem of decentralized stability of switched nonlinear large-scale systems with time-varying delays in interconnections is studied. The time delays are assumed to be any continuous functions belonging to a given interval. By constructing a set of new Lyapunov–Krasovskii functionals, which are mainly based on the information of the lower and upper delay bounds, a new delay-dependent sufficient condition for designing switching law of exponential stability is established in terms of linear matrix inequalities (LMIs). The developed method using new inequalities for lower bounding cross terms eliminate the need for overbounding and provide larger values of the admissible delay bound. Numerical examples are given to illustrate the effectiveness of the new theory.     

Existence and uniqueness of solutions of infinite systems of semilinear parabolic differential-functional equations in arbitrary domains in ordered banach spaces

We consider the Fourier first initial-boundary value problem for a weakly coupled infinite system of semilinear parabolic differential-functional equations of reaction-diffusion type in arbitrary (bounded or unbounded) domain. The right-hand sides of the system are functionals of unknown functions of the Volterra type. Differential-integral equations give examples of such equations. To prove the existence and uniqueness of the solutions, we apply the monotone iterative method. The underlying monotone iterative scheme can be used for the computation of numerical solution.     

On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives

In this paper, a class of nonlinear fractional order differential impulsive systems with Hadamard derivative is discussed. First, a reasonable concept on the solutions of fractional impulsive Cauchy problems with Hadamard derivative and the corresponding fractional integral equations are established. Second, two fundamental existence results are presented by using standard fixed point methods. Finally, two examples are given to illustrate our theoretical results.     

Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions

This paper concerns with blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space. We establish the rate estimates for blow-up solutions and prove that the blow-up set is under proper conditions on initial data. Furthermore, for N=1, more complete conclusions about such two topics are given.