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Semilinear evolution equations in Banach spaces

Journal of Functional Analysis
Publication Date
DOI: 10.1016/0022-1236(79)90040-5


Abstract We study the evolution equation u′( t) = Au( t) + J( u( t)), t ⩾ 0, where e tA is a C 0 semi-group on a Banach space E, and J is a “singular” non-linear mapping defined on a subset of E. In Sections 1 and 2 of the paper we suppress the map J and instead consider maps K t : E → E, t > 0, which heuristically are just e tA J. Under certain integrability conditions on the K t we prove existence and uniqueness of local solutions to the integral equation u( t) = e tA φ + ∝ 0 t K t − s ( u( s)) ds for all φ in E, and investigate the regularity of the solutions. Conditions which insure existence of global solutions are given. In Section 3 we recover the map J from the maps K t , and show that the generator of the semi-flow on E induced by the integral equation has dense domain. Finally, we apply these results to a large class of examples which includes polynomial perturbations to elliptic operators on a domain in R n .

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