Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

On the existence of optimal controls for quasilinear parabolic partial differential equations

A class of systems governed by quasilinear parabolic partial differential equations with first boundary conditions is considered. Existence of solutions for this class of systems and theira priori estimates are established. Further, a theorem on the existence of optimal controls for the corresponding control problem is obtained. Its proof is based on Filippov's implicit functions lemma. The control restraint setU is taken as a measurable multifunction.The authors wish to thank Professor L. Cesari for his most valuable comments and suggestions. In fact, a condition assumed in the original version of this paper was substantially relaxed by him. For details, see Remark 4.1.     

Boundary value problem for a class of degenerate quasilinear parabolic equations with singularity

This paper is concerned with a class of quasilinear parabolic equations with singularity and arbitrary degeneracy. The existence and uniqueness of generalized solutions to a kind of boundary value problem is established.     

Insensitizing controls for the parabolic equation with equivalued surface boundary conditions

This paper is devoted to the study of the existence of insensitizing controls for the parabolic equation with equivalued surface boundary conditions. The insensitizing problem consists in finding a control function such that some energy functional of the equation is locally insensitive to a perturbation of the initial data. As usual, this problem can be reduced to a partially null controllability problem for a cascade system of two parabolic equations with equivalued surface boundary conditions. Compared the problems with usual boundary conditions, in the present case we need to derive a new global Carleman estimate, for which, in particular one needs to construct a new weight function to match the equivalued surface boundary conditions.     

Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as of all weak (energy) solutions of a class of equations with the following model representative: with prescribed global energy function Here , , , Ω is a bounded smooth domain, . Particularly, in the case it is proved that the solution u remains uniformly bounded as in an arbitrary subdomain and the sharp upper estimate of when has been obtained depending on and . In the case for all , sharp sufficient conditions on degeneration of near that guarantee the above mentioned boundedness for an arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of the solution when has been obtained.     

Implicit-explicit multistep methods for quasilinear parabolic equations

Summary. Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation. Received March 10, 1997 / Revised version received March 2, 1998     

Viscosity solutions of a class of degenerate quasilinear parabolic equations with Dirichlet boundary condition

This paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous Dirichlet boundary condition. The equation under consideration arises from a number of practical model problems including reaction–diffusion processes in a porous medium. The degeneracy of the problem appears on the boundary and possibly in the interior of the domain. The goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them. An application of the above results is given to a porous-medium type of reaction–diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem.     

Central manifolds of quasilinear parabolic equations

We investigate central manifolds of quasilinear parabolic equations of arbitrary order in an unbounded domain. We suggest an algorithm for the construction of an approximate central manifold in the form of asymptotically convergent power series. We describe the application of the results obtained in the theory of stability. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 315–328, March, 1998.     

Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations

In this paper we study boundary layers of nonlinear characteristic parabolic equations as the viscosity goes to zero. We obtain and justify in small time a complete expansion of the solution with respect to the viscosity.     

Insensitizing controls for a forward stochastic heat equation

The existence of insensitizing controls for a forward stochastic heat equation is considered. To develop the duality, we obtain observability estimates for linear forward and backward coupled stochastic heat equations with general coefficients, by means of some global Carleman estimates. Furthermore, the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation. As far as we know, our paper is the first one to address the problem of insensitizing controls for stochastic partial differential equations.     

Regularity theory for a class of quasilinear equations

This paper addresses the analysis of the weak solution of in a bounded domain Ω subject to the boundary condition on , when the data f belongs to and . We prove existence and uniqueness of solution for this problem in the Nikolskii space . Moreover, we obtain energy estimates regarding the Nikolskii norm of ω in terms of the norm of f.     

Continuous dependence inequalities for a class of quasilinear parabolic problems

In this paper, we establish continuous dependence inequalities for the solutions u(x, t) of a class of nonlinear parabolic initial-boundary value problems and their gradients when the data are subject to variations. Dedicated to Joseph Hersch on the occasion of his 80th birthday (Received: February 24, 2005; revised: March 14, 2005)