Nonlinear Volterra integrodifferential equations
This is a continuation of the authors' earlier paper [J. Math. Anal. Appl. 139, No. 1, 78-109 (1989; Zbl 0681.45010)]. Consider the semilinear Volterra integro-differential equation \[ u'(t) + Au(t) = \int^ t_ 0 a(t,s)g \bigl( s,u(s) \bigr) ds + f \bigl( t,u(t) \bigr), \quad t > 0, \] with \(u(0) = u_ 0\), where \(-A\) generates an analytic semigroup in a Banach space \(X\), where \(g\) is a polynomial differential operator in divergence form having the same order as \(A\) and the kernel \(a\) is smooth on \(\mathbb{R}^ + \times \mathbb{R}^ +\). Under certain technical assumptions on \(u_ 0\), \(g,f\) the authors prove the existence of a unique local weak solution, i.e. of \(u\) such that \(u \in C ([t_ 0, t_ 0 + T]\); \(X_{{1 \over 2} + \gamma})\), \[ \begin{multlined} u(t) = T(t - t_ 0) u_ 0 + \int^ t_{t_ 0} A^{{1 \over 2}} T(t - s) \left( \int^ s_{t_ 0} a(s, \tau) q \bigl( \tau, u(\tau) \bigr) d \tau \right) ds + \int^ t_{t_ 0} T(t - s)f \bigl( s,u(s) \bigr) ds, \\ t_ 0 \leq t \leq t_ 0 + \delta, \end{multlined} \] where \(\gamma \in (0, {1 \over 2})\) and \(X_ \alpha = D(A^ \alpha)\) equipped with the graph norm. After a review of the theory of Bessel potential spaces the authors apply their result to the case \(A = - \Delta\).
semilinear Volterra integro-differential equation
scientific article; zbMATH DE number 710341