A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.     

Qualitative difference between solutions for a model of the Boltzmann equation in the linear and nonlinear cases

We consider the Boltzmann equation in the framework of a nonlinear model for problems of the gas flow in a half-space (the Kramers problem). We prove the existence of a positive bounded solution and find the limit of this solution at infinity. We show that taking the nonlinear dependence of the collision integral on the distribution function into account leads to an asymptotically new solution of the initial equation. To illustrate the result, we present examples of functions describing the nonlinearity of the collision integral.     

Upper and Lower Solution Methods for Fully Nonlinear Boundary Value Problems

Sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions. The conditions assume the existence of a strong upper solution-lower solution pair, a concept that is defined in the paper. The differential equation has nonlinear dependence on all lower order derivatives of the unknown; in particular, appropriate Nagumo conditions are obtained and employed.     

Existence of radially symmetric solutions of the inhomogeneous <Emphasis Type="Italic">p</Emphasis>-Laplace equation

We consider the Dirichlet problem for the inhomogeneous p-Laplace equation with p nonlinear source. New sufficient conditions are established for the existence of weak bounded radially symmetric solutions as well as a priori estimates of solution and of the gradient of solution. We obtain an explicit formula that shows the dependence of the existence of these solutions on the dimension of the problem, the size of the domain, the exponent p, the nonlinear source, and the exterior mass forces.     

Some solvability problems for the Boltzmann equation in the framework of the Shakhov model

We consider the nonlinear Boltzmann equation in the framework of the Shakhov model for the classical problem of gas flow in a plane layer. The problem reduces to a system of nonlinear integral equations. The nonlinearity of the studied system can be partially simplified by passing to a new argument depending on the solution of the problem itself. We prove the existence theorem for a unique solution of the linear system and the existence theorem for a positive solution of the nonlinear Urysohn equation. We determine the temperature jumps on the lower and upper walls in the linear and nonlinear cases, and it turns out that the difference between them is rather small.     

On well-posedness of generalized neural field equations with impulsive control

We consider nonlinear nonlocal integral equation generalizing equations typically used in mathematical neuroscience. We investigate solutions tending to zero at any fixed moment with unbounded growth of the spatial variable (these solutions correspond to normal brain functioning). We consider an impulsive control problem, which models electrical stimulation used in the presence of various diseases of central nervous system. We define suitable complete metric space, where we obtain conditions for existence, uniqueness and extendability of solution to the problem as well as continuous dependence of this solution on the impulsive control.     

On an overdetermined problem of determining parameter in a degenerate parabolic equation

This work is concerned with the solvability of the ill-posed problem of determining the diffusion velocity in nonlinear diffusion process with an additional restriction. We first prove the continuous dependence of the solutions with respect to the nonlinearity of the equation and give an explicit error-estimate formula, and then, we show the existence of the solution to the ill-posed problem and estimate the range of the velocity.     

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.     

Global existence of solutions to a coupled coagulation system

We consider a model equations describing the coagulation process of a gas on a surface. The problem is modeled by two coupled equations. The first one is a nonlinear transport equation with bilinear coagulation operator while the second one is a nonlinear ordinary differential equation. The velocity and the boundary condition of the transport equation depend on the supersaturation function satisfying the nonlinear ode. We first prove global existence and uniqueness of solution to the nonlinear transport equation then, we consider the coupled problem and prove existence in the large of solutions to the full coagulation system.     

On the Stochastic Wave Equation with Nonlinear Damping

We discuss an initial boundary value problem for the stochastic wave equation with nonlinear damping. We establish the existence and uniqueness of a solution. Our method for the existence of pathwise solutions consists of regularization of the equation and data, the Galerkin approximation and an elementary measure-theoretic argument. We also prove the existence of an invariant measure when the equation has pure nonlinear damping.     

分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.     

On a nonlinear heat equation with functional dependence

We reduce the Cauchy problem for a heat equation with the nonlinear right-hand side which depends on some functionals to an equivalent integral equation. Considering mainly Banach spaces of continuous, bounded and exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We give a counterexample which shows that the Lipschitz condition is, in general, insufficient for the Cauchy problem with unbounded data and with functional dependence to guarantee an existence result     

The Neumann boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution to a nonlocal Cahn-Hilliard equation on a bounded domain. The equation generates a gradient flow for a free energy functional with nonlocal interaction. Also we apply a nonlinear Poincaré inequality to show the existence of an absorbing set in each constant mass affine space.     

Fourth-order problems with nonlinear boundary conditions

We develop a new method of lower and upper solutions for a fourth-order nonlinear boundary value problem where the differential equation has dependence on all lower-order derivatives. Our boundary conditions are nonlinear. We will assume the functions that define the nonlinear boundary conditions are either monotone or nonmonotone. As a result we obtain existence principles which improve recent results in the literature.     

Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument

We consider a problem for a quasilinear hyperbolic equation with a nonlocal condition that contains a retarded argument. By reducing this problem to a nonlinear integrofunctional equation, we prove the existence and uniqueness theorem for its solution. We pose an inverse problem of finding a solution-dependent coefficient of the equation on the basis of additional information on the solution; the information is given at a fixed point in space and is a function of time. We prove the uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation and analysis of an integro-functional equation for the difference of two solutions of the inverse problem.     

Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave

We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space L ₁[?r, r] for all finite r < +∞. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution.     

Mixed Problem for an Ultraparabolic Equation in Unbounded Domain

We investigate a mixed problem for a nonlinear ultraparabolic equation in a certain domain Q unbounded in the space variables. This equation degenerates on a part of the lateral surface on which boundary conditions are given. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation; these conditions do not depend on the behavior of the solution at infinity. The problem is investigated in generalized Lebesgue spaces.     

A Duality Approach to Nonlinear Diffusion Equations

We provide new existence results for a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity, under minimal growth and coercivity conditions. The results given in this paper prove that a generalized solution to the nonlinear equation is provided by a solution to an equivalent minimization problem for a convex functional involving the potential of the nonlinearity and its conjugate, in the case when the potential is time and space depending. If the potential is time depending only and it has a symmetry at infinity, the null minimizer in the minimization problem is found to coincide with a weak solution to the nonlinear equation.     

Elliptic-parabolic equation and the corresponding problem with free boundary I: Elliptic problem with parameter

We consider the elliptic problem with a parameter that occurs under the consideration of a broad class of nonlinear elliptic-parabolic problems and nonlinear evolutionary elliptic problems, in particular, elliptic-parabolic and elliptic problems with free boundary. The existence and coercive estimates of a smooth solution to this problem are proved, including the smooth dependence of the solutions on the parameter.