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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.

##### MSC:
 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