  We carry out an analysis of the existence of solutions for a class of
nonlinear partial differential equations of parabolic type. The equation is
associated to a nonlocal initial condition, written in general form which
includes, as particular cases, the Cauchy multipoint problem, the weighted mean
value problem and the periodic problem. The dynamic is transformed into an
abstract setting and by combining an approximation technique with the
Leray-Schauder continuation principle, we prove global existence results. By
the compactness of the semigroup generated by the linear operator, we do not
assume any Lipschitzianity, nor compactness on the nonlinear term or on the
nonlocal initial condition. In addition, the exploited approximation technique
coupled to a Hartman-type inequality argument, allows to treat nonlinearities
with superlinear growth. Moreover, regarding the periodic case we are able to
prove the existence of at least one periodic solution on the half line.
