${\displaystyle L_{\infty }}$-estimates for solutions of nonlinear parabolic systems with gradient linear growth

Existence of weak solutions and an ${\displaystyle L_{\infty }}$-estimate are shown for nonlinear nondegenerate parabolic systems with linear growth conditions with respect to the gradient. The ${\displaystyle L_{\infty }}$-estimate is proved for equations with coefficients continuous with respect to x and t in the general main part, and for diagonal systems with coefficients satisfying the Carathéodory condition.