Nonlocal Initial-boundary Value Problems Research Articles

B-Separable boundary value problems in Banach-valued function spaces

In this paper, the nonlocal boundary value problems for anisotropic partial differential-operator equations with dependent coefficients in Banach-valued Besov ( B) spaces are studied. The principal parts of the appropriate differential operators are non-self-adjoint. Several conditions for separability and Fredholmness are given. These results permit us to establish that the inverse of the corresponding differential operators belong to the Schatten q-class. The spectral properties of the appropriate differential operators are also investigated. In addition we study the maximal regularity of nonlocal initial boundary value problems for abstract parabolic equations, finite or infinite systems of parabolic equations and the separability of nonlocal boundary value problems for finite or infinite systems of quasi-elliptic equations in B spaces.

Maximal regular boundary value problems in Banach‐valuedfunction spaces and applications

The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non‐selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach‐valued Lp‐spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process

We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation u t=Δu+λf(u)/( ∫ Ω f(u) dx) 2, Ω⊂ R N, N=1,2 . Under specific conditions on f, there exists a λ ∗ such that for each 0<λ<λ ∗ there corresponds a unique steady-state solution and u= u( x, t; λ) is a global in time-bounded solution, which tends to the unique steady-state solution as t→∞ uniformly in x. Whereas for λ⩾λ ∗ there is no steady state and if λ>λ ∗ then u blows up globally. Here, we show that when (a) N=1,Ω=(−1,1) and f(s)>0, f′(s)<0, s⩾0 , or (b) N=2, Ω=B(0,1) and f( s)=e − s , the solution u ∗=u(x,t;λ ∗) is global in time and diverges in the sense ||u ∗(·,t)|| ∞→∞ , as t→∞. Moreover, it is proved that this divergence is global i.e. u ∗(x,t)→∞ as t→∞ for all x∈Ω. The asymptotic form of divergence is also discussed for some special cases.

Investigation of the nonlocal initial boundary value problems for some hyperbolic equations

In the present article we are interested in the analysis of nonlocal initial boundary value problems for some medium oscillation equations. More precisely, we investigate di¤erent types of nonlocal problems for one-dimensional oscillation equations and prove existence and uniqueness theorems. In some cases algorithms for direct construction of the solution are given. We also consider nonlocal problem for multidimensional hyperbolic equation and prove the uniqueness theorem for the formulated initial boundary value problem applying the theory of characteristics under rather general assumptions.

Monotone method for first order nonlocal hyperbolic initial-boundary value problems

In this paper, the monotone method is extended to nonlocal hyperbolic intial-boundary value problems of first order. A comparison principle is established, and linear convergence of monotone sequences of upper and lower solutions is shown.