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A note on the nonlocal boundary value problem for elliptic-parabolic equations. (English) Zbl 1124.47056
The paper considers the abstract nonlocal boundary value problem for elliptic-parabolic equations: \[ \begin{cases} {du(t)\over dt}+ Au(t)= f(t),& 0\leq t\leq 1,\\ -{d^2u(t)\over dt^2}+ Au(t)= g(t), & -1\leq t\leq 0,\\ u(1)= u(-1)+ \mu,\end{cases}\tag{1} \] in a Hilbert space \(H\), with the self-adjoint positive definite operator \(A\).
By \(C([a, b],H)\) is denoted the Banach space of all continuous functions \(\varphi(t)\) defined on \([a, b]\) with values in \(H\), equipped with the norm \(\|\varphi\|_{C([a,b], H)}= \max_{a\leq t\leq b}\,\|\varphi(t)\|_H\).
By \(C^\alpha([a,b], H)\), \(0<\alpha< 1\), is denoted the Banach space obtained by completion of the set of all smooth \(H\)-valued functions \(\phi(t)\) on \([a, b]\) in the norm \[ \|\varphi\|_{C^\alpha([a, b],H)}= \|\varphi\|_{C([a, b],H)^+}+ \sup_{a< t< t+\tau< b}\,{\|\varphi(t+\tau)- \varphi(t)\|_H\over \tau^\alpha}. \] In the main theorem, under some conditions, the well-posedness of the boundary value problem (1) in a Hölder space \(C^\alpha([-1,1], H)\) is established and coercive stability estimates for the solutions are obtained.
Later, some applications of this theorem to the mixed boundary value problems for elliptic-parabolic equations are given.

47N20 Applications of operator theory to differential and integral equations
47D06 One-parameter semigroups and linear evolution equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces