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Well-posedness of higher order abstract Cauchy problems. (English) Zbl 0589.34004

Summary: The paper is concerned with differential equations of the type \[ (*)\quad u^{(n+1)}(t)-Au^{(n)}(t)-B_ 1u^{(n-1)}(t)-...-B_ nu(t)=0 \] in a Banach space E where A is a linear operator with dense domain D(A) and \(B_ 1,...,B_ n\) are closed linear operators with \(D(A)\subset D(B_ k)\) for \(1\leq k\leq n\). The main result is the equivalence of the following two statements: (a) A has nonempty resolvent set and for every initial value \((x_ 0,...,x_ n)\in (D(A))^{n+1}\) the equation (*) has a unique solution in \(C^{n+1}({\mathbb{R}}^+,E)\cap C^ n({\mathbb{R}}^ n,[D(A)])([D(A)]\) 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 \(B_ k\), 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, \(B_ k\). The results are then used to discuss strongly damped semilinear second order equations.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
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