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.     

拟线性双曲型方程组Cauchy问题行波解的稳定性

当拟线性双曲系统线性退化时,其Cauchy问题最左族和最右族行波解是稳定的.而其中间族行波解未必稳定.我们在弱线性退化条件下,证明了拟线性双曲系统Cauchy问题适当小的W~(1,1)∩L~∞范数适当小的行波解是稳定的,并将此稳定性应用于可对角化的拟线性双曲系统和Chaplygin气体动力学方程组.     

Local-in-time well-posedness of a regularized mathematical model for silicon MESFET

We prove the local-in-time well-posedness of the initial boundary value problem for a system of quasilinear equations. This system is used for finding numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). The initial boundary value problem has the following peculiarities: the quasilinear system is not a Cauchy-Kovalevskaya-type system; the boundary is a non-smooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to an integro-differential system. The local-in-time existence of a weakened generalized solution of this system is then proved by the fixed-point argument.     

Existence and multiplicity results for quasilinear hemivariational inequalities at resonance

In this paper we consider quasilinear hemivariational inequality at resonance. We prove existence results for strongly resonant quasilinear problem, resonant problem under a Tang‐type condition as well as two multiplicity results. The method of the proofs is based on the nonsmooth critical point theory for locally Lipschitz functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local-in-time well-posedness of a regularized mathematical model for silicon MESFET

We prove the local-in-time well-posedness of the initial boundary value problem for a system of quasilinear equations. This system is used for finding numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). The initial boundary value problem has the following peculiarities: the quasilinear system is not a Cauchy-Kovalevskaya-type system; the boundary is a non-smooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to an integro-differential system. The local-in-time existence of a weakened generalized solution of this system is then proved by the fixed-point argument.     

The first eigenvalue for a quasilinear Schrödinger operator and its application

In this paper, we study the first eigenvalue for a quasilinear Schrödinger operator, which is greater than the first eigenvalue of the usual laplacian operator. As an application we treat a quasilinear resonance problem involving a subcritical growth perturbation.     

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.     

Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains,with applications to quasilinear Riccati type equations

A weighted norm inequality of Muckenhoupt–Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leads to a resolution of an existence problem for quasilinear Riccati type equations with a gradient source term of arbitrary power law growth.     

A quasilinear method in the theory of small eigenfunctions for nonlinear periodic boundary-value problems

Small eigenfunctions of a nonlinear periodic boundary-value problem are studied for the case of double degeneration of the eigenvalue of the linearized problem; the quasilinear representation is used.     

Sobolev Regularity for t > 0 in Quasilinear Parabolic Equations

We establish a regularity property for the solutions to the quasilinear parabolicinitial-boundary value problem (1.4) below, showing that for t > 0 they belong to the same space to which the solutions of the second order hyperbolic problem (1.5), which is a singular perturbation of (1.4), belong. This result provides another illustration of the asymptotically parabolic nature ofproblem (1.5), and would be needed to establish the diffusion phenomenon for quasilinear dissipative wave equations in Sobolev spaces.     

Global Exisienoe of Classical Solutions to the Typical Free Boundary Problem for General Quasilinear Hyperbolic Systems and Its Applications

In this paper the authors prove the existence and uniqueness of global classical solutions to the typical free boundary problem for general quasilinear hyperbolic systems. As an application, a unique global discontinuous solution only containing n shocks on t \leq 0 is obtained for a class of generalized Riemann problem for the quasilinear hyperbolic system of n conservation laws.     

Forward–backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs

We obtain an existence and uniqueness theorem for fully coupled forward–backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al. (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial–boundary value problem for non-local quasilinear parabolic second-order PDEs.     

On a mixed finite element formulation of a second‐order quasilinear problem in the plane

We analyze a mixed finite element discretization of a second‐order quasilinear problem based on the Raviart‐Thomas space. We prove that the discrete problem is solvable and provide a local uniqueness result for the solution. We also obtain optimal order L²‐error estimates for both the scalar variable and the associated flux. The main feature of our method is that it is free from the boundness conditions required in previous works on the coefficients of the quasilinear operator. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 90–103, 2004.     

On the solvability of the dirichlet problem for elliptic nondivergent equations of the second order in a domain with conical point

We study the problem of solvability of the Dirichlet problem for second-order linear and quasilinear uniformly elliptic equations in a bounded domain whose boundary contains a conical point. We prove new theorems on the unique solvability of a linear problem under minimal smoothness conditions for the coefficients, right-hand sides, and the boundary of the domain. We find classes of solvability of the problem for quasilinear equations under natural conditions.     

On the quasilinear stationary Ventzel boundary-value problem

A priori estimates for gradients of solutions of a boundary-value problem for a quasilinear nondivergent elliptic equation with the quasilinear Ventzel boundary condition are established. By these estimates, existence theorems in the Hölder and Sobolev spaces are proved. Bibliography:11 titles.     

Deterministic homogenization of quasilinear damped hyperbolic equations

Deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale highly oscillatory hyperbolic problems converges to the solution to a homogenized quasilinear hyperbolic problem.     

Hölder estimates for solutions to the degenerate boundary-value Venttsel' problem for parabolic and elliptic equations of nondivergence type

The authors continue to study the Venttsel' problem, i.e., the boundary-value problem for a parabolic or elliptic equation with the boundary condition in the form of a parabolic or elliptic equation with respect to tangent variables. A priori estimates for the Hölder norms of solutions are established in the case of quasilinear equations of nondivergence form with a quasilinear degenerate boundary Venttsel' condition. Bibliography: 16 titles.

     

Formation of Singularities of Solutions for Cauchy Problem of Quasilinear Hyperbolic Systems with Dissipative Terms

ln this paper, for a class of 2 × 2 quasilinear hyperbolic systems, we get existence theorems of the global smooth solutions of its Cauchy problem, under a certain hypotheses. In addition, Tor two concrete quasilinear hyperbolic systems, we study the formation of the singularities of the C¹-solution to its Cauchy problem.     

A product formula for semigroups of Lipschitz operators associated with abstract quasilinear evolution equations

The notion of semigroups of Lipschitz operators associated with abstract quasilinear evolution equations is introduced and a product formula for such semigroups is established. The product formula obtained in the paper is applied to the solvability of the Cauchy problem for a first order quasilinear system through a finite difference scheme of the Lax‐Friedrichs type.     

Interaction of weak discontinuities and the hodograph method as applied to electric field fractionation of a two-component mixture

The hodograph method is used to construct a solution describing the interaction of weak discontinuities (rarefaction waves) for the problem of mass transfer by an electric field (zonal electrophoresis). Mathematically, the problem is reduced to the study of a system of two first-order quasilinear hyperbolic partial differential equations with data on characteristics (Goursat problem). The solution is constructed analytically in the form of implicit relations. An efficient numerical algorithm is described that reduces the system of quasilinear partial differential equations to ordinary differential equations. For the zonal electrophoresis equations, the Riemann problem with initial discontinuities specified at two different spatial points is completely solved.