An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

A convex-concave problem with a nonlinear boundary condition

In this paper we study the existence of nontrivial solutions of the problem

     

The thermostat problem with a nonlocal nonlinear boundary condition

The thermostat controller for an air-conditioning system isusually placed in a position at some distance from the unitand this can lead to large swings in temperature. This paperaddresses this question by studying a paradigm—a one-dimensionalheat conduction equation with and without heat loss, and wherethe flux of heat extracted or input by the unit is consideredto be a function of the temperature at the other end. The essential results are that the system can be unstable andthat this is exacerbated both by a more powerful air-conditioningunit and by more efficient insulation.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

The evolution dam problem for a compressible fluid with nonlinear Darcy's law and Dirichlet boundary condition

In this paper, we consider the evolution dam problem (P) related to a compressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet boundary conditions on some part of the boundary. We establish existence of a solution for this problem. We choose a convenient regularized problem (P_?) for which we prove the existence and uniqueness of solution using the comparison Lemma 2.1 and the Schauder fixed‐point theorem. Then, we pass to the limit, when ? goes to 0, to get a solution for our problem. Moreover, we will see another approach for the incompressible case where we pass to the limit in (P), when α goes to 0, to get a solution.     

非线性边值条件非线性方程的奇摄动问题

研究了三阶非线性边值条件的非线性方程奇摄动问题,先用合成展开法对形式近似解进行构造,再利用相关微分不等式理论给出所得解的存在性及一致有效性的证明.     

Finite element approximation of a problem with a nonlinear Newton boundary condition

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed. Received April 15, 1996 / Revised version received November 22, 1996     

Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition

The main purpose of this paper is to present the existence results of solutions and positive solutions of nonlinear high-order fractional boundary value problems with integral boundary condition. By using the Banach fixed point theorem and the Krasnosel’skii fixed point theorem, we obtain the existence and uniqueness of real solution. By the Guo–Krasnosel’skii fixed point theorem on the cone, we obtain a desired result for guaranteeing the existence of positive solution. Several interesting examples relevant to the main results are also considered.     

An optimization problem for nonlinear Steklov eigenvalues with a boundary potential

In this paper, we analyze an optimization problem for the first (nonlinear) Steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed L¹$L^1$-norm.     

Convergence to equilibrium for some nonlinear evolution equations with dynamical boundary condition

In this short note we present some convergence results of solutions to certain equilibrium for some nonlinear evolution equations subject to dynamical boundary condition by using the Łojasiewicz-Simon approach. The work was done in collaboration with S. Zheng and M. Grasselli. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An initial boundary-value problem with a noncoercive boundary condition in domains with edges

We consider an initial boundary-value problem for a second order parabolic equation in a domain with edges. We assume that on a part of the boundary the unknown function satisfies a boundary condition of the type (where is the external normal, φ is a given function). In the case of more than one space variable the existence results of the general theory of parabolic initial boundary-value problems do not apply to the problems with boundary conditions of this type. Unique solvability of the problem is established in weighted Sobolev spaces where the weight multipliers are certain powers of the distance to the edge. Bibliography: 17 titles. Dedicated to V. A. Solonikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 206–223.     

An inverse problem for a linear diffusion equation with non-linear boundary condition

In this paper we consider the determination of an unknown radiation term in the non-linear boundary condition of a linear diffusion equation from an overspecified condition. It is shown that the solutions of this inverse problem is unique and stable.     

The Sturm-Liouville problem with a nonlocal boundary condition

In this paper, we consider the Sturm-Liouville problem with one classical and another nonlocal boundary condition. We investigate general properties of the characteristic function and spectrum for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues, analyze the dependence of the spectrum on parameters of the boundary condition, and describe the qualitative behavior of all eigenvalues subject to of the nonlocal boundary condition. Dedicated to N. S. Bakhvalov (1934–2005) Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 410–428, July–September, 2007.     

Heat conduction with nonlinear boundary condition

The paper is concerned with determination of the solution of the one-dimensional heat equation in a semi-infinite region subject to a nonlinear boundary condition at the surface. The exact solution is obtained and is expressed in infinite series. It is shown that the series is absolutely and uniformly convergent. Two special cases of Newton's cooling and of Stefan-Boltzmann's radiation at the boundary are discussed.

---

Zusammenfassung Diese Arbeit befaßt sich mit der Lösung der Gleichung der eindimensionalen Wärmeleitung in einem Halbraum mit nicht-linearen Randbedingungen an der Oberfläche. Es wird eine genaue Lösung gefunden, die die Form einer unendlichen Reihe besitzt. Es wird gezeigt, daß die Reihe absolut und gleichmäßig konvergiert. Als zwei Spezialfälle werden Newton-Kühlung und Stefan-Boltzmann-Strahlung behandelt.

     

Error analysis of finite element approximation of a stefan problem with nonlinear free boundary condition

By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations inL2, H¹ and H² normed spaces.     

On an optimal boundary control problem with a dynamic boundary condition

We are the first solve the optimization problem for a boundary displacement control of string vibrations at one endpoint in the case of a nonstationary boundary condition containing a directional derivative at the other endpoint.