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A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations


Authors: I. Lasiecka and R. Triggiani
Journal: Proc. Amer. Math. Soc. 104 (1988), 745-755
MSC: Primary 34G10; Secondary 35L10, 47A50, 47D05
DOI: https://doi.org/10.1090/S0002-9939-1988-0964851-1
MathSciNet review: 964851
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Abstract: We consider the solution operator

$\displaystyle (Lu)(t) = A\int_0^t {G(t - \tau )} {A^{ - 1}}Bu(\tau )d\tau $

corresponding to the abstract equation $ \dot x = Ax + Bu$ on a reflexive Banach space $ X$, where the linear operator $ A:X \supset \mathcal{D}(A) \to X$ is the infinitesimal generator of a (strongly continuous) group $ G(t)$ of bounded operators on $ X$, and $ B:U \supset \mathcal{D}(B) \to X$ is a generally unbounded linear operator with $ {A^{ - 1}}B \in \mathcal{L}(U,X),U$ being another reflexive Banach space (without loss of generality we take $ A$ to be boundedly invertible). Let $ 0 < T < \infty $ be given. We prove the following theorem: if $ L$ is continuous $ {L^p}(0,T;U) \to {L^p}(0,T;X), 1 < p < \infty$, then in fact $ L:$ continuous $ {L^p}(0,T;U) \to C[0,T];X$, a lifting regularity theorem in the time variable. Moreover, we show by a parabolic example with nonhomogeneous term in the Dirichlet boundary conditions that the theorem fails to be true, if $ G(t)$ is merely a s.c. semigroup even if holomorphic. Applications of the theorem include mixed hyperbolic problems, including second order scalar hyperbolic equations defined on an open bounded domain $ \Omega \subset {R^n},\partial \Omega = \Gamma $, with nonhomogeneous term of class $ {L^2}(0,T;{L^2}(\Gamma ))$ acting in the Dirichlet or in the Neumann boundary conditions. In the former case, the theorem recovers the authors' original procedure which yielded optimal regularity results for this dynamics [L-T.2]; in the latter, the theorem improves upon results of Lions-Magenes [L-M.1, vol. II]. Extension to $ T = \infty $ is also studied.

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DOI: https://doi.org/10.1090/S0002-9939-1988-0964851-1
Article copyright: © Copyright 1988 American Mathematical Society