Variable domain second order evolution equations

Abstract

Well-posedness criteria are given abstract evolution equations of the form

for with Cauchy data

Here A(t), P(t), N(t) are linear operators on a Hilbert space with N(t) bounded. These are the kinds of equations 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:

Here α,β,γ,δ,∊ are smooth real-valued functions on with β positive; a,b, are smooth real-valued functions on and f ₁,f ₂ are smooth complex-valued functions on [0,1] with .

^†Supported by National Science Foundation Grant GP-28652.

^†Supported by National Science Foundation Grant GP-28652.

Notes

^†Supported by National Science Foundation Grant GP-28652.