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Well-posedness and spectral properties of heat and wave equations with non-local conditions

Abstract : We consider the one-dimensional heat and wave equations but – instead of boundary conditions – we impose on the solution certain non-local, integral constraints. An appropriate Hilbert setting leads to an integration-by-parts formula in Sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well-posedness, hence sharpening regularity results previously obtained by other authors. In doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions, like periodic ones. In the self-adjoint case, we even obtain eigenvalue asymptotics of so-called Weylʼs type.
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Delio Mugnolo, Serge Nicaise. Well-posedness and spectral properties of heat and wave equations with non-local conditions. Journal of Differential Equations, Elsevier, 2014, 256 (7), pp.2115-2151. ⟨10.1016/j.jde.2013.12.016⟩. ⟨hal-03163703⟩

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