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Sharp extensions for convoluted solutions of wave equations. (English) Zbl 1363.47131

Summary: In this paper, we give sharp extensions of convoluted solutions of wave equations in abstract Banach spaces. The main technique is to use the algebraic structure, for convolution products \(\ast\) and \(\ast_c\), of these solutions which are defined by a version of Duhamel’s formula. We define algebra homomorphisms for the convolution product \(\ast_c\) from a certain set of test-functions and apply our results to concrete examples of abstract wave equations.

MSC:

47N20 Applications of operator theory to differential and integral equations
35L05 Wave equation
47D62 Integrated semigroups
47D06 One-parameter semigroups and linear evolution equations
44A10 Laplace transform
44A35 Convolution as an integral transform
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