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.     

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.     

On a problem for operator-differential second-order equations with nonlocal boundary condition

An operator-differential second-order equation with nonlocal boundary condition at zero is considered on the semiaxis. Here we give sufficient conditions on the operator coefficients for the regular solvability of the boundary-value problem. Moreover, we obtain conditions for the completeness andminimality of the derivative of the chain of eigen- and associated vectors generated by the boundary-value problem under study and establish the completeness and minimality of the decreasing elementary solutions of the operator-differential equation under consideration.     

Initial-boundary value problem with a nonlocal boundary condition for a multidimensional hyperbolic equation

We consider a mixed initial-boundary value problem for a multidimensional (with respect to the space variables) hyperbolic equation with a nonlocal boundary condition containing an integral of the desired solution. We prove the unique solvability of the problem in the space W ₂ ¹ .     

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.     

Inverse problem for an equation of parabolic-hyperbolic type with a nonlocal boundary condition

For an equation of the parabolic-hyperbolic type, we consider an inverse problem with a nonlocal condition relating solution derivatives that belong to different types of the equation in question. We justify a uniqueness criterion and prove the existence of a solution of the problem by the spectral analysis method. We prove the stability of the solution with respect to the nonlocal boundary condition.     

Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition

In this paper we consider a semilinear parabolic equation u_t=Δu−c(x,t)u^p for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣_{∂Ω×(0,∞)}=∫_Ωk(x,y,t)u^ldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.     

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

We investigate a multitype Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Assuming that the mean of the increment X of the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices is negative and the random variable Xe ^X has zero mean, we find the asymptotics of the survival probability at time n as n → ∞.     

A degenerate parabolic system with nonlocal boundary condition

In this article, we investigate the blow-up properties of the positive solutions to a degenerate parabolic system with nonlocal boundary condition. We give the criteria for finite time blow-up or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate for small weighted nonlocal boundary.     

An inverse problem for an equation of elliptic-hyperbolic type with a nonlocal boundary condition

We examine a mixed type equation with the Lavrent??ev-Bitsadze operator and an unknown right-hand side in a rectangle and study a nonlocal boundary value problem in which the values of the stream function on the lateral sides of the rectangle are equal. The solution is given as a sum of a biorthogonal series. The uniqueness criterion and stability of solutions with respect to the boundary data are established.     

Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition

An initial boundary-value problem for a quasilinear system of partial differential equations with a nonlocal boundary condition involving a delayed argument is considered. The existence of a unique solution to this problem is proved by reducing it to a system of nonlinear integral-functional equations. The inverse problem of finding a solution-dependent coefficient of the system from additional information on a solution component specified at a fixed point of space as a function of time is formulated. The uniqueness of the solution of the inverse problem is proved. The proof is based on the derivation and analysis of an integral-functional equation for the difference between two solutions of the inverse problem.     

A problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type

We study the boundary-value problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type. The unique solvability of the problem is proved.     

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.     

A convex-concave problem with a nonlinear boundary condition

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

     

A porous medium equation with nonlocal boundary condition and a localized source

This article studies the blow-up properties of solutions to a porous medium equation with nonlocal boundary condition and a general localized source. Conditions for the existence of global or blow-up solutions are obtained. Moreover, it is proved that the unique solution has global blow-up property whenever blow-up occurs. Blow-up rate estimates are also obtained for some special cases.