Nonlinear holomorphic semigroups

Thomas Lee Hayden and Frank Jones Massey

Vol. 57 (1975), No. 2, 423–439

DOI: 10.2140/pjm.1975.57.423

Abstract

Conditions are given on a nonlinear operator A in a Banach space X under which the semigroup, S(t), generated by −A has the property that S(t)x is analytic in t for |arg t| < 𝜃 for each fixed x ∈ cl(D(A)). Analyticity in t of solutions of u′ + Tu = Fu where −T generates a linear holomorphic semigroup in X and F maps D(T^α) analytically into X for some α < 1 is also established. These results are applied to establish analyticity in t of solutions to ∂u∕∂t + Lu + β(u) = 0 where β : R → R is real analytic, monotone increasing and β(0) = 0, and L is a second order elliptic operator.

Mathematical Subject Classification 2000

Primary: 47H15

Secondary: 35K15

Milestones

Received: 27 February 1974

Revised: 3 March 1975

Published: 1 April 1975