Variable exponent spaces have found interesting applications in real world problems. In imaging models, the variable exponent can approach the critical value 1 and this poses unique challenges in proving existence of solutions for the corresponding PDEs. In this work, we develop some new functional framework to study time-dependent parabolic variable exponent flows. Specifically, we consider bounded vectorial partial variation ( B V P V ) space and its variable exponent counterpart. We then prove the existence of weak solutions to the critical vectorial p ( t , x ) -Laplacian flow in the variable exponent B V P V space. For time-independent critical vectorial p ( x ) -Laplacian flow we obtain a unique semigroup solution. Our results are in particular valid in the scalar case and solve a long standing open problem.