A homogeneous boundary condition is constructed for the parabolic equation (∂t + I − Δ)u = f in an arbitrary cylindrical domain Ω×R (Ω ⊂ Rn being a bounded domain, I and Δ being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator ∂t + I −Δ, but also f...

Abstract We know that interpolation spaces in terms of analytic semigroup have a significant role into the study strict Hölder regularity solutions classical abstract Cauchy problem (ACP). In this paper, we first construct solution operators fractional calculus and characterize these spaces. Then establish mild order ACP.

A classical theory is developed for the time development and scattering of a minimally coupled scalar eld on closed spacetimes that evolve from initial, to nal static states. The time development is obtained by reformulating the eld equation as an abstract Cauchy problem on a Hilbert space. Constraints are imposed on the metric that enable the use of semigroup theory, and the eld solution is ob...

This note is devoted to study an inverse Cauchy problem in a Hilbert space H for fractional abstract differential equations of the form; ), ( ) ( ) ( = ) ( t g t f t u A dt t u d    with the initial condition H u u = (0) 0 and the overdetermination condition: ), ( = ) ), ( ( t w v t u where (.,.) is the inner product in H , f is a real unknown function w is a given real function, 0 u , v ar...