Regularity properties of solutions of linear integrodifferential equations with singular kernels
Let \({\mathbf H}\) be a Hilbert space with norm \(|\cdot |\). Let \(A\) be a possibly unbounded linear operator in \({\mathbf H}\), and let \(a: [0,\infty)\to {\mathbf R}\) be a given locally integrable kernel function. In this note we study properties of solutions of the abstract linear integrodifferential equation \[ u'(t)+ a * Au(t)= 0,\qquad 0< t< T,\tag{1} \] in \({\mathbf H}\) . Here, the usual convolution notation is employed: \(f * g(t)= \int^ t_ 0 f(t- s) g(s) ds\), if one of the two functions is scalar-valued and the other one is vector-valued. We want to consider mild solutions of (1), i.e., continuous functions \(u(\cdot)\) for which \(1 * a * u(t)\in D(A)\) for all \(0\leq t\leq T\) and for which the integrated version \[ u(t)+ A(1 * a * u(t))= u(0) \] of (1) holds for all \(t\in [0, T]\). The goal of this note is to give conditions on the kernel function \(a\) under which such mild solutions satisfy an estimate of the form \[ | Au(t)|\leq \textstyle{{C\over t^ M}} | u(0)| \] for a suitable power \(M\). It will be proved that such an estimate is always true if the derivative \(a'\) is integrable and behaves like \(- t^{- 2\alpha}\) near zero, and that in this case \(M= 1/\alpha\) is a suitable exponent.
abstract linear integro- differential equation
scientific article; zbMATH DE number 784747
scientific article