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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: $$\cases {du(t)\over dt}+ Au(t)= f(t),& 0\le t\le 1,\\ -{d^2u(t)\over dt^2}+ Au(t)= g(t), & -1\le t\le 0,\\ u(1)= u(-1)+ \mu,\endcases\tag1$$ 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 $\Vert\varphi\Vert_{C([a,b], H)}= \max_{a\le t\le b}\,\Vert\varphi(t)\Vert_H$. 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 $$\Vert\varphi\Vert_{C^\alpha([a, b],H)}= \Vert\varphi\Vert_{C([a, b],H)^+}+ \sup_{a< t< t+\tau< b}\,{\Vert\varphi(t+\tau)- \varphi(t)\Vert_H\over \tau^\alpha}.$$ 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.

47N20Applications of operator theory to differential and integral equations
47D06One-parameter semigroups and linear evolution equations
34D05Asymptotic stability of ODE
34G10Linear ODE in abstract spaces