Global a priori bounds for weak solutions to quasilinear parabolic equations with nonstandard growth

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)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.