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 | References | Similar Articles | Additional Information

Abstract: We consider the solution operator

*group*of bounded operators on , and is a generally unbounded linear operator with being another reflexive Banach space (without loss of generality we take to be boundedly invertible). Let be given. We prove the following theorem: if is continuous , then in fact continuous , 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 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 , with nonhomogeneous term of class 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 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