Well-posedness of higher order abstract Cauchy problems

Abstract:

The paper is concerned with differential equations of the type \begin{equation}\tag {$\ast $} {u^{(n + 1)}}(t) - A{u^{(n)}}(t) - {B_1}{u^{(n - 1)}}(t) - \cdots - {B_n}u(t) = 0\end{equation} in a Banach space $E$ where $A$ is a linear operator with dense domain $D(A)$ and ${B_1}, \ldots ,{B_n}$ 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 $({x_0}, \ldots ,{x_n}) \in {(D(A))^{n + 1}}$ the equation $( \ast )$ has a unique solution in ${C^{n + 1}}({{\mathbf {R}}^ + },E) \cap {C^n}({{\mathbf {R}}^n},[D(A)])([D(A)]$ 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 $( \ast )$. 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.