Variable Domain Second Order Evolution Equations
Well-posedness criteria arc given for abstract evolution equations of the form [formula omitted] for t ∊ R with Cauchy data [formula omitted] Here A(t), P(t), N(t) are linear operators on a Hilbert space with N(t) bounded. These are the kinds of equations studied in [J. Functional Analysis 4 (1969), 50–70], but there it was assumed that each A(t) was self-adjoint and that Dom (A(t)) did not depend on t. These restrictions are relaxed in the present paper. As an application, the following simple but nontrivial mixed hyperbolic problem is solved: [formula omitted] Here α, β, γ, δ, ε are smooth real-valued functions on R x [0, 1] with β positive; a, b are smooth real-valued functions on R, and f1, f2 are smooth complex-valued functions on [0, 1] with f1'0) - a(0)f1 (0), f1' b(0)f1(1), f2'(0) a(0)f2(0) + a'(0)f1(0), f2' b(0) f2(1)+b'(0)f(1). © 1976, Taylor & Francis Group, LLC. All rights reserved.
Goldstein, J. (1976). Variable Domain Second Order Evolution Equations. Applicable Analysis, 5 (4), 283-291. https://doi.org/10.1080/00036817608839132