Banach空间中一类周期非线性受控系统最优控制的存在性

研究一类强非线性发展方程的周期解及相应的最优控制问题的存在性，首先，证明了Banach空间中一类包含非线性单调算子和非线性非单调扰动的强非线性发展方程周期解的存在性；其次，给出了保证相应的Lagrange最优控制的充分条件；最后，举例说明理论结果在拟线笥抛物方程周期问题及相应的最优控制问题中的应用。     

On a multiscale analysis of an inverse problem of nonlinear transfer law identification in periodic microstructure

This paper is devoted to the multiscale analysis of a homogenization inverse problem of the heat exchange law identification, which is governed by parabolic equations with nonlinear transmission conditions in a periodic heterogeneous medium. The aim of this work is to transform this inverse problem with nonlinear transmission conditions into a new one governed by a less complex nonlinear parabolic equation, while preserving the same form and physical properties of the heat exchange law that it will be identified, based on periodic homogenization theory. For this, we reformulate first the encountered homogenization inverse problem to an optimal control one. Then, we study the well-posedness of the state problem using the Leray–Schauder topological degrees and we also check the existence of the solution for the obtained optimal control problem. Finally, using the periodic homogenization theory and priori estimates, with justified choise of test functions, we reduce our inverse problem to a less complex one in a homogeneous medium.     

Approximate periodic solutions of a nonlinear parabolic system and an identification problem

This work continues our study in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005] on the identification problem for the coefficients for the lower order terms in a parabolic system, through its approximate periodic solutions. Different from the work in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005], our system now is nonlinear and the coefficients to be detected are from the first order term. From the application point of view, we now try to determine the diffusion coefficients for the system by the observation over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper.     

一类非线性抛物方程Cauchy问题解的存在性条件

研究了带源项的非线性抛物方程Cauchy问题解存在的必要条件以及解所应具有的性质,通过把文献中的线性算子推广到形式较一般的带源项的非线性抛物算子,利用其中处理线性问题的方法来处理非线性问题．     

Periodic solutions of a class of degenerate parabolic system with delays

This paper is concerned with a class of periodic degenerate parabolic system with time delays in a bounded domain under mixed boundary condition. Under locally Lipschitz condition on reaction functions, we apply Schauder fixed point theorem to obtain the existence of periodic solutions of the periodic problem. With quasi-monotonicity in addition, we also show that the periodic problem has a maximal and a minimal periodic solutions. Applications of the obtained results are also given to some nonlinear diffusion models arising from ecology.     

Periodic solutions to history-dependent differential hemivariational inequalities with applications

In this article, we investigate a hybrid model combined by a parabolic differential equation and a parabolic hemivariational inequality (so-called differential hemivariational inequality of parabolic–parabolic type) in general infinite dimensional spaces which includes the history-dependent operator. The solvability of initial value problems as well as the periodic problems of the hemivariational inequality and the differential hemivariational inequality have been proved. In application, we study a contact problem with normal compliance driven by a history-dependent dynamical system.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Estimates of the solutions of two-point boundary-value problems for systems of first-order integrodifferential equations and the method of lines

One proves theorems on the estimates of the solutions of the systems of first-order integrodifferential equations with the boundary conditions On the basis of these theorems, one suggests a method for estimating the norms of integrodifferential equations by the method of the lines for the solutions of the periodic boundary-value problems for second-order integrodifferential equations of parabolic type. On the basis of the established theorem, on the solvability and on the estimate of the solution of the nonlinear equation $$Tx + F\left( x \right) = 0$$ in a Banach space X, where T is a linear unbounded operator, one investigates the convergence of the method of lines for solving the periodic boundary-value problem for a second-order nonlinear integrodifferential equation of parabolic type.     

Existence of solutions to various rock types odel model of two-phase flow in porous media

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.     

Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation

In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a fourth-order nonlinear parabolic equation.     

Bifurcation of periodic solutions of periodic evolution equations in a cone

We consider a class of nonlinear periodic evolution equations, leaving invariant a cone in a real Banach space, and satisfying appropriate monotonicity requirements. A necessary and sufficient condition is found for the existence of a (unique, globally attractive) nontrivial periodic solution within the cone. Such a condition is expressed in terms of an associated linear equation. To establish this result, an abstract version of the familiar comparison techniques for parabolic equations is worked out. Applications and examples are also discussed.     

Positive solutions of a nonlinear impulsive equation with periodic boundary conditions

In this paper, we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions. Sign properties of an associated Green’s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse. Upper and lower bounds for positive solutions are also given. The results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.     

To the theory of entropy sub-solutions of degenerate nonlinear parabolic equations

We prove existence of the largest entropy sub-solution and the smallest entropy super-solution to the Cauchy problem for a nonlinear degenerate parabolic equation with only continuous flux and diffusion functions. Applying this result, we establish the uniqueness of entropy solution with periodic initial data. The more general comparison principle is also proved in the case when at least one of the initial functions is periodic.     

An inverse coefficient problem with nonlinear parabolic equation

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.     

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

This paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs. Copyright © 2014 John Wiley & Sons, Ltd.     

On Hölder solutions of the integro-differential Zakai equation

The existence and uniqueness of solutions of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. The equation considered arises in a nonlinear filtering problem with a jump signal process and continuous observation.     

Existence of positive radial solutions for a nonlinear elliptic problem in annulus domains

In this paper, we study a nonlinear elliptic problem in an annulus domain. For which, the nonexistence of nontrivial solutions has been obtained by some authors in star‐shaped domain. Although we note that it may admit non‐trivial solutions if the domain is non‐star shaped. With the use of a variational method, we establish the existence of positive radial solutions in an annulus domain. On the basis of this, we also extend this result to the periodic problem of the corresponding degenerate parabolic problem. Copyright © 2012 John Wiley & Sons, Ltd.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

We prove the global existence and blow-up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.     

Nonlinear parabolic boundary value problems with upper and lower solutions

The existence of at least one periodic solution of a very general second order nonlinear parabolic boundary value problem is proved under the assumption that a lower solution ϕ and an upper solution ψ with ϕ≦ψ are known.