Wave propagation for linear integrodifferential equations in Banach space (Q796786)

Let \(A:D(A)\subset X\to X\) be the generator of a \(c_ 0\)-semigroup in the Banach space X, \(B(t):D(A)\subset X\to X\) closed and strongly continuously differentiable from \(R_+\) to X and \(x_ 0\in D(A)\). The authors give necessary and sufficient conditions that the integrodifferential equation \(x'(t)=Ax(t)+\int^{t}_{0}B(t-s)x(s)ds,\) \(t\geq 0\), \(x(0)=x_ 0\) be hyperbolic in the sense that solutions propagate with finite speed. Examples arising from electricity and heat conduction are given.