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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:

du(t) dt+Au(t)=f(t),0t1,-d 2 u(t) dt 2 +Au(t)=g(t),-1t0,u(1)=u(-1)+μ,(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 ϕ(t) defined on [a,b] with values in H, equipped with the norm ϕ C([a,b],H) =max atb ϕ(t) H .

By C α ([a,b],H), 0<α<1, is denoted the Banach space obtained by completion of the set of all smooth H-valued functions φ(t) on [a,b] in the norm

ϕ C α ([a,b],H) =ϕ C([a,b],H) + +sup a<t<t+τ<b ϕ(t+τ)-ϕ(t) H τ α ·

In the main theorem, under some conditions, the well-posedness of the boundary value problem (1) in a Hölder space C α ([-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:
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