Nonlinear Volterra equation of parabolic type due to singular kernels
The author studies an integral equation of the form \[ u_ t (t,x) = \int^ t_ 0 k(t - s) \bigl( \sigma (u_ x) \bigr)_ x (s,x)ds + f(t,x) \] with a nonincreasing convex kernel \(k\) and \(\sigma\) increasing. Such equations occur in viscoelasticity. Sufficient conditions for the global existence of a strong solution are given. Sufficiently smooth solutions will be unique.
nonlinear Volterra equation of parabolic type
scientific article; zbMATH DE number 635869