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A Class of Abstract Quasi‐Linear Evolution Equations of Second Order
Author(s) -
Tanaka Naoki
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700008656
Subject(s) - mathematics , banach space , pure mathematics , order (exchange) , initial value problem , class (philosophy) , norm (philosophy) , cauchy distribution , linear differential equation , cauchy problem , semigroup , linear map , linear space , mathematical analysis , differential equation , discrete mathematics , computer science , finance , artificial intelligence , political science , law , economics
In this paper we study the abstract quasi‐linear evolution equation of second order{u ″ ( t ) = A ( t , u ( t ) , u ′ ( t ) ) u ( t ) for t ∈ [ 0 , T ]u ( 0 ) = Φ , u ′ ( 0 ) = ψ ( 1.1 )in a general Banach space Z . It is well‐known that the abstract quasi‐linear theory due to Kato [ 10 , 11 ] is widely applicable to quasi‐linear partial differential equations of second order and that his theory is based on the theory of semigroups of class ( C 0 ). (For example, see the work of Hughes et al. [ 9 ] and Heard [ 8 ].) However, even in the special case where A ( t, w, v ) = A is independent of ( t, w, v ), it is found in [ 2 ] and [ 14 ] that there exist linear partial differential equations of second order for which Cauchy problems are not solvable by the theory of semigroups of class ( C 0 ) but fit into the mould of well‐posed problems where the solution and its derivative depend continuously on the initial data if the initial condition is measured in the graph norm of a suitable power of A . (See also work by Krein and Khazan [ 13 ] and Fattorini [ 6 , Chapter 8].) This kind of Cauchy problem has recently been studied extensively, using the theory of integrated semigroups or regularized semigroups. The theory of integrated semigroups was studied intensively by Arendt [ 1 ] and that of regularized semigroups was initiated by Da Prato [ 3 ] and renewed by Davies and Pang [ 4 ]. For the theory of regularized semigroups we refer the reader to [ 5 ] and [ 16 ].