Well-posedness of higher order abstract cauchy problems1

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The paper is concerned with differential equations of the type (Formula Presented) in a Banach space E where A is a linear operator with dense domain D(A) and B1,…, Bn are closed linear operators with D(A) Ì D(Bk) for 1 £ k £ n. The main result is the equivalence of the following two statements: (a) A has nonempty resolvent set and for every initial value (x0,…, xn) Î (D(A))n+1 the equation (*) has a unique solution in Cn+1(R+, E) Ç Cn(Rn, [D(A)]) ([D(A)] denotes the Banach space D(A) endowed with the graph norm); (b) A is the generator of a strongly continuous semigroup. Under additional assumptions on the operators Bk, which are frequently fulfilled in applications, we obtain continuous dependence of the solutions on the initial data; i.e., well-posedness of (*). Using Laplace transform methods, we give explicit expressions for the solutions in terms of the operators A, Bk. The results are then used to discuss strongly damped semilinear second order equations. © 1986 American Mathematical Society.

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Transactions of the American Mathematical Society

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