Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

Allaberen Ashyralyev,Asker Hanalyev

doi:10.1155/2014/519814

Abstract

The nonlocal boundary value problem for the parabolic differential equation v'(t) + A(t)v(t) = f(t) (0 ≤ t ≤ T), v(0) = v(λ) + φ, 0 < λ ≤ T in an arbitrary Banach space E with the dependent linear positive operator A(t) is investigated. The well-posedness of this problem is established in Banach spaces C 0 β,γ(E α−β) of all E α−β-valued continuous functions φ(t) on [0, T] satisfying a Hölder condition with a weight (t + τ)γ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

Highlights

A Cauchy ProblemIt is known that (see, e.g., [1,2,3,4,5] and the references given therein) several applied problems in fluid mechanics, physics, and mathematical biology were formulated into nonlocal mathematical models
It is known that several applied problems in fluid mechanics, physics, and mathematical biology were formulated into nonlocal mathematical models
Before going to discuss the well-posedness of nonlocal boundary value problem, we will give the definition of positive operators in a Banach space and introduce the fractional spaces generated by positive operators that will be needed in the sequel

Summary

A Cauchy Problem

It is known that (see, e.g., [1,2,3,4,5] and the references given therein) several applied problems in fluid mechanics, physics, and mathematical biology were formulated into nonlocal mathematical models. New exact estimates in Holder norms for the solution of three nonlocal boundary value problems for parabolic equations were obtained. New Schauder type exact estimates in Holder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established. If A(0)φ + f(λ) − f(0) ∈ Eα−γ, for the solution V(t) in C0β,γ(Eα−β) of the nonlocal boundary value problem (27) the coercive inequality. If A(0)φ + f(λ) − f(0) ∈ Eαβ−,γβ, for the solution V(t) in C0β,γ(Eα−β) of the nonlocal boundary value problem (27) the coercive inequality. Theorems 9 and 10 imply theorems on well-posedness of the nonlocal boundary value problem (27) in C0β,γ(E)(0 ≤ γ ≤ β, 0 < β < 1) which is established in the paper [41]

Applications

Conclusion