Abstract Cauchy problems for quasilinear operators whose domains are not necessarily dense or constant. (October 2017) - Matsumoto, Toshitaka; Tanaka, Naoki
Abstract: The solvability of the abstract Cauchy problem for the quasilinear evolution equation u ′ ( t ) = A ( u ( t ) ) u ( t ) for t > 0 and u ( 0 ) = u 0 ∈ D is discussed. Here { A ( w ) ; w ∈ Y } is a family of closed linear operators in a real Banach space X such that Y ⊂ D ( A ( w ) ) ⊂ Y ¯ for w ∈ Y, Y is another Banach space which is continuously embedded in X, and D is a closed subset of Y . The existence and uniqueness of C 1 solutions to the Cauchy problem is proved without assuming that Y is dense in X or D ( A ( w ) ) is independent of w . The abstract result is applied to obtain an L 1 -valued C 1 -solution to a size-structured population model.