  In this paper we study a rather wide class of quasilinear parabolic problems
with nonlinear boundary condition and nonstandard growth terms. It includes the
important case of equations with a $p(t,x)$-Laplacian. By means of the
localization method and De Giorgi's iteration technique we derive global a
priori bounds for weak solutions of such problems. Our results seem to be new
even in the constant exponent case.
