A discontinuous Galerkin method with interior penalties is proposed for solving nonlinear Sobolev equations with evolution terms. In this sense, a symmetric semi-discrete and a family of symmetric fully-discrete time approximate schemes are formulated.

$Hp$-version error estimates are analyzed for these schemes. For the semi-discrete time scheme an a priori

${L}^{\infty}\left({H}^{1}\right)$ error estimate is derived and similarly,

${l}^{\infty}$ and

${l}^{2}\left({H}^{1}\right)$ error bounds are obtained for the fully-discrete time schemes. These results indicate that spatial rates in

${H}^{1}$ and time truncation errors in

${L}^{2}$ are optimal.