ABOUT SOLVABILITY OF ONE PROBLEM WITH NONLOCAL CONDITIONS FOR HYPERBOLIC EQUATION

In this article we consider a nonlocal problem with integral condition of the second kind for hyperbolic equation. The choice of a method for investigating problems with nonlocal conditions of the second kind depends on the type of nonintegral terms. In this article we consider the case when the nonintegral term is a trace of required function on the boundary of the domain. To investigate the solvability of the problem we use method of reduction for loaded equation with homogeneous boundary conditions. This method proved to be effective for defining a generalized solution, to obtain apriori estimates and to prove existence of unique generalized solution of the given problem.

ON SMOOTHNESS OF SOLUTION OF ONE NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION

In this paper we consider a nonlocal problem with integral boundary condition for hyperbolic equation. The conditions of the problem contain derivatives of the first order with respect to both x and t,, which can be interpreted as an elastic fixation of the right end rod in the presence of a certain damper, and since the conditions also contain integral of the desired solution, this condition is nonlocal. It is known that problems with nonlocal integral conditions are non-self-adjoint and, therefore, the study of solvability encounters difficulties that are not characteristic of self-adjoint problems. Additional difficulties arise also due to the fact that one of the conditions is dynamic. The attention of the article is focused on studying thesmoothness of the solution of the nonlocal problem. The concept of a generalized solution is introduced, and the existence of second-order derivatives and their belonging to the space L2 are proved. The proof is basedon apriori estimates obtained in this work.

p-Harmonic Functions in ℝN+ with Nonlinear Neumann Boundary Conditions and Measure Data

We study a concept of renormalized solution to the problem\begin{cases}-\Delta_{p}u=0&\mbox{in }{\mathbb{R}}^{N}_{+},\\ \lvert\nabla u\rvert^{p-2}u_{\nu}+g(u)=\mu&\mbox{on }\partial{\mathbb{R}}^{N}_% {+},\end{cases}where {1<p\leq N}, {N\geq 2}, {{\mathbb{R}}^{N}_{+}=\{(x^{\prime},x_{N}):x^{\prime}\in{\mathbb{R}}^{N-1},\,x% _{N}>0\}}, {u_{\nu}} is the normal derivative of u, μ is a bounded Radon measure, and {g:{\mathbb{R}}\rightarrow{\mathbb{R}}} is a continuous function. We prove stability results and, using the symmetry of the domain, apriori estimates on hyperplanes, and potential methods, we obtain several existence results. In particular, we show existence of solutions for problems with nonlinear terms of absorption type in both the subcritical and supercritical case. For the problem with source we study the power nonlinearity {g(u)=-u^{q}}, showing existence in the supercritical case, and nonexistence in the subcritical one. We also give a characterization of removable sets when {\mu\equiv 0} and {g(u)=-u^{q}} in the supercritical case.

EXISTENCE OF POSITIVE SOLUTION OF TWO-POINT BOUNDARY PROBLEM FOR ONE NONLINEAR ODE OF THE FOURTH ORDER

In the work sufficient conditions for existence at least one positive solution of two-point boundary problem for one class of strongly nonlinear differential equations of the fourth order are received. The problem is considered on a segment [0,1] (more general case of segment[0, a] is reduced to considered). On the ends of a segment the solution of y and its second derivative of y′′ areequal to zero. Right part of an equation f (x, y) isn’t negative at x\geq 0 andat all y. Performance of sufficient conditions is easily checked. Performance ofthese conditions is easily checked. In the proof of existence the theory of conesin banach space is used. Also apriori estimates of positive solution, which ispossible to use further at numerical construction of the solution are obtained.

NONLOCAL PROBLEM WITH INTEGRAL CONDITION FOR A FOURTH ORDER EQUATION

In this paper, we consider a nonlocal problem with integral condition with respect to spacial variable for a forth order partial differential equation. The conditions on the data for unique solvability of the problem in Sobolev space are determined. Proving of uniqueness of generalized solution is based on acquired apriori estimates. To prove the solvability we use a following scheme: sequence of approximate solutions using Galerkin procedure is built, apriory estimates that allow to extract from it a convergent subsequence are received, on the final stage it is shown that the limit of subsequence is the required generalized solution.

On one problem with dynamic nonlocal condition for ahyperbolic equation

In this article, boundary value problem for hyperbolic partial differential equation with nonlocal data in an integral of the second kind form is considered. The emergence of dynamic conditions may be due to the presence of a damping device. Existence and uniqueness of generalized solution is proved in a given cylindrical field. There is some limitation on the input data. The uniqueness of generalized solution is proved by apriori estimates. The existence is proved by Galerkin’s method and embedding theorems.

A NONLOCAL PROBLEM WITH INTEGRAL CONDITION FOR A FOURTH ORDER EQUATION

In this paper we consider initial-boundary problems with integral conditions for certain fourth order equation. Unique solvability of posed problems is proved. The proof is based on apriori estimates, regularization method, auxiliary problems method, embedding theorems.