A perturbation theorem for evolution equations and some applications


The following well-known perturbation theorem is of fundamental importance in semigroup theory"A be -dissipative (i.e., A generates a (C0) contraction semigroup). If P1 is dissipative and A bounded with relative A bound less than one, and if P2 is bounded, then A + P1 + P2 generates a (C0) semigroup. This result is generalized to allow A, P1, P2 all to depend on a real parameter. Thus in many cases, establishing the well-posedness of the Cauchy problem for u’(t) = (A(t) + P1(t) + P2(t)u(t) (’ = d/dt) is reduced to proving the well-posedness of the Cauehy problem for u’(t) = A(t)u(t). Applications are given to temporally inhomogeneous scattering theory and to second order evolution equations of the form u’‘t) + B(t)u’(t) + C(t)u(t) = 0; here both B(t) and C(t) can be unbounded. Some concrete examples are given, including mixed problems for utt = αuxx + βux + γutx + δut + εu + φ with time dependent boundary conditions; here all the coefficients are smooth functions on {(t, x): -∞ < t < ∞, 0 ≤ x ≤ 1}, α is positive and γ is sufficiently small. © 1974, Board Trustees, University of Illinois.

Publication Title

Illinois Journal of Mathematics