A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

GLOBAL EXISTENCE, UNIQUENESS, AND STABILITY FOR A NONLINEAR HYPERBOLIC-PARABOLIC PROBLEM IN PULSE COMBUSTION

A global existence theorem is established for an initial-boundary value problem,with time-dependent boundary data,arising in a lumped parameter model of pulse combustion; the model in question gives ri...     

源于DNA的非线性波动方程组的Cauchy问题(英文)

本文首先证明源于DNA的非线性波动方程组的周期边值问题局部古典解的存在性和唯一性.其次通过周期边值问题序列证明这个方程组的Cauchy问题存在唯一的局部古典解.     

First-order three-point boundary value problems at resonance

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer’s Fixed Point Theorem which is due to Miranda.     

THE INITIAL BOUNDARY VALUE PROBLEM FOR QUASI-LINEAR SCHRODINGER-POISSON EQUATIONS

In this article, the author studies the initial-(Dirichlet) boundary problem for a high-field version of the Schrodinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrodinger equations on the unit cube. A global existence and uniqueness is established for a solution to this problem.     

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.     

A dissipative model for hydrogen storage: existence and regularity results

We prove global existence of a solution to an initial and boundary‐value problem for a highly nonlinear PDE system. The problem arises from a thermo‐mechanical dissipative model describing hydrogen storage by use of metal hydrides. In order to treat the model from an analytical point of view, we formulate it as a phase transition phenomenon thanks to the introduction of a suitable phase variable. Continuum mechanics laws lead to an evolutionary problem involving three state variables: the temperature, the phase parameter and the pressure. The problem thus consists of three coupled partial differential equations combined with initial and boundary conditions. The existence and regularity of the solutions are here investigated by means of a time discretization—textita priori estimates—passage to the limit procedure joined with compactness and monotonicity arguments. Copyright © 2010 John Wiley & Sons, Ltd.     

Global Existence and Uniqueness of Solutions to Evolution p-Laplacian Systems with Nonlinear Sources

This paper presents the global existence and uniqueness of the initial and boundary value problem to a system of evolution p-Laplacian equations coupled with general nonlinear terms. The authors use skills of inequality estimation and themethod of regularization to construct a sequence of approximation solutions, hence obtain the global existence of solutions to a regularized system. Then the global existence of solutions to the system of evolution p-Laplacian equations is obtained with the application of a standard limiting process. The uniqueness of the solution is proven when the nonlinear terms are local Lipschitz continuous.     

Finite difference reaction-diffusion systems with coupled boundary conditions and time delays

This paper is concerned with finite difference solutions of a coupled system of reaction-diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction-diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction-diffusion equations are directly applicable to systems of parabolic-ordinary equations and to reaction-diffusion systems without time delays.     

A solvable hyperbolic free boundary problem modelling tumour regrowth

Recently, Tian and Friedman et al. developed a mathematical model on brain tumour recurrence after resection [J.P. Tian, A. Friedman, J. Wang and E.A. Chiocca, Modeling the effects of resection, radiation and chemotherapy in glioblastoma, J. Neuro-Oncol. 91(3) (2009), pp. 287–293]. The model is a free boundary problem with a hyperbolic system of nonlinear partial differential equations. In this article, we conduct a rigorous analysis on this hyperbolic system and prove the local and global existence and uniqueness of the solution. It is well known that most nonlinear free boundary problems are impossible to solve in terms of explicit analytical solutions. In contrast, the free boundary problem in this study is solvable, and the explicit solution is found using the backward characteristic curve method. This explicit solution is then validated by numerical simulation results. An interesting finding in this study is that the problem can be treated as a hyperbolic system defined on an infinite domain where the initial condition has a first-type discontinuity.     

A hyperbolic free boundary problem existence uniqueness and discretization

For a one phase free boundary problem for a linear hyperbolic system with constant coefficients in one space dimension with nonlinear boundary conditions we prove existence, uniqueness and continuous dependence of a Lipschitz continuous solution using the method of characteristics. A semidiscrete version of front tracking is shown to be linearly convergent.     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

Global existence for a nonlinear theory of bubbly liquids

Global existence of smooth solutions is proved for an effective theory of bubbly liquids for either the initial value problem or initial boundary value problem in one dimension. This shows that the theory does not describe shock waves or bubble collapse. Since the analysis is not for the steady boundary value problem, there is no discussion of resonance. The proof uses a semilinear form of the equations to get local existence. A priori bounds resulting from energy conservation and a nonlinear Gronwall-like inequality are then derived to prove global existence.     

Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls

Summary. An optimal control problem for impressed cathodic systems in electrochemistry is studied. The control in this problem is the current density on the anode. A matching objective functional is considered. We first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition. We then prove the existence of an optimal solution. Next, we derive a necessary condition of optimality and establish an optimality system of equations. Finally, we define a finite element algorithm and derive optimal error estimates. Received March 10, 1993 / Revised version received July 4, 1994     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

We prove global existence and uniqueness of solutions to a Cahn–Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown.     

一个肿瘤生长自由边界问题解的整体存在性和唯一性

该文研究一个反应扩散方程组的自由边界问题,它来源于描述抑制物作用下无坏死核肿瘤生长的数学模型.作者运用抛物型方程的L_p理论和压缩映照原理,证明了这个问题局部解的存在唯一性,然后用延拓方法得到了整体解的存在唯一性.     

Fredholm alternative and boundary value problems

This note presents a simple proof of A. Lasota's application of the nonlinear Fredholm alternative to the existence proofs of the boundary value problems involving ordinary differential equations. It then uses Lasota's result to get a stronger version of the theorem of Herzog and Lemmert on the Dirichlet boundary value problem for the second-order systems of ordinary differential equations.

     

Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg-de Vries-Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.