We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation u(tt) A(t)u with general Wentzell boundary conditions A(t)u(j, t) + (-1)(j-1) beta(j) (t) partial derivative u/partial derivative x (j, t) + gamma(j) (t)u(j, t) = 0, for j = 0, 1. Here A(t)u := (a(x,t)u(x))(x), a(x,t) >= epsilon > 0 in [0, 1] x (0, + infinity) and, beta(j)(t) > 0, gamma(j) (t) >= 0, (gamma 0 (t), gamma 1 (t)) not equal (0, 0). Under suitable regularity conditions on a, beta(j), gamma(j) we prove the well-posedness in a suitable (energy) Hilbert space.
The non-autonomous wave equation with general Wentzell boundary conditions
ROMANELLI, Silvia
2005-01-01
Abstract
We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation u(tt) A(t)u with general Wentzell boundary conditions A(t)u(j, t) + (-1)(j-1) beta(j) (t) partial derivative u/partial derivative x (j, t) + gamma(j) (t)u(j, t) = 0, for j = 0, 1. Here A(t)u := (a(x,t)u(x))(x), a(x,t) >= epsilon > 0 in [0, 1] x (0, + infinity) and, beta(j)(t) > 0, gamma(j) (t) >= 0, (gamma 0 (t), gamma 1 (t)) not equal (0, 0). Under suitable regularity conditions on a, beta(j), gamma(j) we prove the well-posedness in a suitable (energy) Hilbert space.File in questo prodotto:
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