A Note on the Nonlocal Boundary Value Problem for Hyperbolic-parabolic Differential Equations

The nonlocal boundary value problem        d 2 u(t) dt 2 + Au(t) = f (t)(0 ≤ t ≤ 1), du(t) dt + Au(t) = g(t)(−1 ≤ t ≤ 0), u(−1) = αu (µ) + βu (λ) + ϕ, |α|, |β| ≤ 1, 0 < µ, λ ≤ 1 for differential equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for the solutions of the mixed type boundary value problems for hyperbolic-parabolic equations are obtained. 1. Introduction. It is known that some problems in fluid mechanics (model of the motion of an ideal fluid filling, exhibiting both viscous and non-viscous phases) and other areas of physics and mathematical biology(taxis-diffusion-reaction model) lead to partial differential equations of the hyperbolic-parabolic type. Methods of solutions of the nonlocal boundary value problems for hyperbolic-parabolic differential equations have been studied extensively by many researches (see, e. and the references given therein). In the present paper we consider the nonlocal boundary value problem