The authors deal with the approximate solution to the nonlinear Sobolev-type evolution equation \[ \frac{d}{dt} \left(u(t) + g(t,u(t)) \right) + Au(t) = f(t,u(t)), \quad t>0, \] for prescribed initial data in some separable Hilbert space, where \(A\) is a densely defined linear operator such that \(-A\) generates an analytic semigroup and \(g\), \(f\) are suitable continuous functions satisfying some Lipschitz conditions.
Whereas the case \(g \equiv 0\) now seems to be standard and the case with linear \(g\), which is related to degenerate parabolic problems, has been studied in detail by Showalter and Brill, the case with nonlinear \(g\) has been addressed to over the last few years.
Nonlinear evolution equations of the above type arise in many applications in physics and in the study of partial neutral functional-differential equations with an unbounded delay (as it may also appear in biological applications).
After reformulating the problem as an integral equation (Duhamel’s principle) and introducing some approximate integral equation, the Faedo-Galerkin method is applied. Existence, uniqueness, and convergence of the approximate solutions are shown. Finally, results are applied to an initial value problem of meta-parabolic type with fourth-order spatial derivative and a time derivative of \(w(x,t) - \Delta w(x,t)\).