In this paper we study the abstract quasi-linear evolution equation of second order
formula here
in a general Banach space Z. It is well-known that the abstract quasi-linear theory due
to Kato [10, 11] is widely applicable to quasi-linear partial differential equations of
second order and that his theory is based on the theory of semigroups of class (C0).
(For example, see the work of Hughes et al. [9] and Heard [8].) However, even in the
special case where A(t, w, v) = A is independent of (t, w, v), it is found in [2] and [14]
that there exist linear partial differential equations of second order for which Cauchy
problems are not solvable by the theory of semigroups of class (C0) but fit into the
mould of well-posed problems where the solution and its derivative depend
continuously on the initial data if the initial condition is measured in the graph norm
of a suitable power of A. (See also work by Krein and Khazan [13] and Fattorini [6,
Chapter 8].) This kind of Cauchy problem has recently been studied extensively, using
the theory of integrated semigroups or regularized semigroups. The theory of
integrated semigroups was studied intensively by Arendt [1] and that of regularized
semigroups was initiated by Da Prato [3] and renewed by Davies and Pang [4]. For
the theory of regularized semigroups we refer the reader to [5] and [16].
