Insensitizing controls for a class of quasilinear parabolic equations

This paper is addressed to showing the existence of insensitizing controls for a class of quasilinear parabolic equations with homogeneous Dirichlet boundary conditions. As usual, this insensitizing problem is reduced to a nonstandard null controllability problem of some nonlinear cascade system governed by a quasilinear parabolic equation and a linear parabolic equation. Nevertheless, in order to solve the later quasilinear controllability problem by the fixed point technique, we need to establish the null controllability of the linearized cascade parabolic system in the framework of classical solutions. The key point is to find the desired control function in a Hölder space for given data with certain regularities.     

On The Coupled Cahn-hilliard Equations

This paper is concerned with a system of nonlinear partial differential equations, in short, the coupled Cahn-Hilliard equations, which consists of a fourth order quasilinear parabolic equation and a second order quasilinear parabolic equation. This system was recently derived by Penrose and Fife and also by Alt and Pawlow to describe the non-isothermal phase separation of a two-component system. The global existence and uniqueness of classical solutions is proved. The results about the asymptotic behavior, as time goes to infinity, of solution and about the existence and multiplicity of solutions to the corresponding stationary problem, which is a nonlinear boundary value problem involving nonlocal term and constraints, are also obtained.     

Multiscale problems and homogenization for second-order Hamilton–Jacobi equations

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-scale homogenization of uniformly parabolic fully nonlinear PDEs.     

On a local in time solvability of the Neumann problem of quasilinear hyperbolic parabolic coupled systems

We prove a local in time existence theorem of classical solutions to some coupled system of quasilinear hyperbolic equations and quasilinear parabolic equations with Neumann boundary condition. This coupled system contains a non-linear thermoelastic equation as an important physical example.     

Propagation of Perturbations in Quasilinear Multidimensional Parabolic Equations with Convective Term

We establish estimates for the initial evolution of the supports of solutions of a broad class of quasilinear parabolic equations of arbitrary order that have the structure of the equation of strong nonlinear convective diffusion.     

Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator

We study an abstract nonlinear evolution equation governed by a time-dependent operator of subdifferential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.     

On nonlinear systems consisting of different types of differential equations

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity. Supported by the Hungarian NFSR under grant OTKA T 049819.     

Strong well‐posedness of a three phase problem with nonlinear transmission condition

We prove existence and uniqueness of strong solutions to a quasilinear parabolic‐elliptic system modelling an ionic exchanger. This chemical system consists of three phases connected with nonlinear boundary conditions. The most interesting difficulty of our problem manifests in the nonlinear transmission condition, as almost all quantities are non‐linearly involved in this boundary equation. Our approach is based on the contraction mapping principle, where maximal L_p‐regularity of the associated linear problem is used to obtain a fixed point equation of the starting problem. Copyright © 2012 John Wiley & Sons, Ltd.     

Construction of solutions to quasilinear partial differential equations

We propose a method for constructing solutions to a class of quasilinear parabolic partial differential equations (PDEs) basing on a new property of these equations. The method applies to quasilinear hyperbolic and elliptic equations as well. The results of this article broaden the class of exact solutions to the quasilinear equations, in particular, to the nonlinear heat equations, the equations of chemical kinetics and mathematical biology.     

Comparison principle and analogues of the phragmen—lindelöf theorem for solutions of parabolic inequalities

It is well known that a Monotonicity Condition and a Coerciveness Condition principally lie in the basis of most results of the Theory of PDE's. The necessity of these important assumptions for the validity of a comparison principle and analogues of the Phragmen-Lindelöf theorem for solutions of quasilinear parabolic inequalities is discussed in the paper. In the first part of the work we introduce a new concept of monotonicity for nonlinear differential operators-nonlinear monotonicity concept-and on its basis we obtain new phenomena for solutions, subsolutions and supersolutions of the well-known quasilinear differential equations. In the second part we omit the current coerciveness condition and change it by a weaker one. In spite of this we obtain a series of new qualitative properties of solutions for wide classes of quasilinear parabolic inequalities. Most of these properties are also new for solutions of the well-known equations, which we consider in the paper.     

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     

Optimal Regularity and Long-Time Behavior of Solutions for the Westervelt Equation

We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently small and regular, then there exists a unique global solution with optimal L _p -regularity. We show furthermore that the solution converges to zero at an exponential rate as time tends to infinity. Our techniques are based on maximal L _p -regularity for abstract quasilinear parabolic equations.     

General decay for a quasilinear system of viscoelastic equations with nonlinear damping

In this paper, we consider a system of coupled quasilinear viscoelastic equations with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.     

在纵向自由薄膜凝固期内温度分布的数值分析

考虑到薄膜表面的热通量主要是来自辐射，因而采用一个依赖时间的拟二维拟线性扩散方程的Ｓｔｅｆａｎ问题（混合初边值问题）作为该问题的数学模型。用一种具有Ｃｒａｎｋ－Ｎｉｃｈｏｌｓｏｎ格式的无条件稳定的有限差分析来求解抛物型偏微分方程的定解问题。用追赶法求解离散化的三对角方程组，然后用预估校正法求解拟线性扩散方程，从而避免了示解非线性差分方程组，琥珀亚硝酸酯在纵向自由薄膜凝固期内的温度分布数值计算结果和     

The Bismut-Elworthy formula for backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces

We consider a forward-backward system of stochastic evolution equations in a Hilbert space. Under nondegeneracy assumptions on the diffusion coefficient (that may be nonconstant) we prove an analogue of the well-known Bismut-Elworthy formula. Next, we consider a nonlinear version of the Kolmogorov equation, i.e. a deterministic quasilinear equation associated to the system according to Pardoux, E and Peng, S. (1992). "Backward stochastic differential equations and quasilinear parabolic partial differential equations". In: Rozowskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and Their Applications , Lecture Notes in Control Inf. Sci., Vol. 176, pp. 200-217. Springer: Berlin. The Bismut-Elworthy formula is applied to prove smoothing effect, i.e. to prove existence and uniqueness of a solution which is differentiable with respect to the space variable, even if the initial datum and (some) coefficients of the equation are not. The results are then applied to the Hamilton-Jacobi-Bellman equation of stochastic optimal control. This way we are able to characterize optimal controls by feedback laws for a class of infinite-dimensional control systems, including in particular the stochastic heat equation with state-dependent diffusion coefficient.     

Global solutions for quasilinear parabolic problems

Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions. Received December 21, 2000; accepted August 30, 2001.     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.     

Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form

Summary Conditions are given for the validity of long range quasi optimal error estimates of the Galerkin method for quasilinear parabolic equations. If the corresponding nonlinear elliptic operator satisfies a coercivity condition, the stationary problem may be approximated by the numerical solution of the parabolic problem when the time variablet is large enough, starting with an arbitrary initial function. Some practical applications are discussed.     

Global existence for fully parabolic boundary value problems

We present some results on the global existence of classical solutions for quasilinear parabolic equations with nonlinear dynamic boundary conditions in bounded domains with a smooth boundary.     

The Green's function method in stationary and nonstationary nonlinear heat-conduction problems

By constructing the Green's function we reduce quasilinear boundary-value problems for second-order parabolic and elliptic equations to nonlinear integral equations of second kind. The method is illustrated using the examples of the stationary and nonstationary heat-conduction problems in the case of radiant heat transfer.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 42–47.