Coercive Solvability of Nonlocal Boundary-Value Problems for Parabolic Equations

Abstract

In an arbitrary Banach space E, we consider the nonlocal problem

\( {\displaystyle \begin{array}{l}\upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\\ {}\upsilon (0)=\upsilon \left(\uplambda \right)+\mu \kern1em \left(0<\uplambda \le 1\right)\end{array}} \)

for an abstract parabolic equation with a linear unbounded strongly positive operator A(t) such that its domain D = D(A(t)) is independent of t and is everywhere dense in E. This operator generates an analytic semigroup exp{−sA(t)}(s ≥ 0).

We prove the coercive solvability of the problem in the Banach space \( {C}_0^{\alpha, \alpha}\left(\left[0,1\right],E\right)\left(0<\alpha <1\right) \) with weight (t + τ)^α. Earlier, this result was known only for constant operators. We consider applications in the class of parabolic functional differential equations with transformation of spatial variables and in the class of parabolic equations with nonlocal conditions on the boundary of the domain. Thus, this describes parabolic equations with nonlocal conditions both with respect to time and with respect to spatial variables.