The one dimensional parabolic p(x)-Laplace equation

The Dirichlet problem for the degenerate and singular parabolic p(x)-Laplace equation with one spatial variable is considered. We prove the existence of the unique weak solution such that the derivatives u _t and u _x of a solution u belong to ({L_{infty}}). Moreover for the singular case we prove the existence of the strong solution i.e. such that u _t , u _x and u _xx belong to ({L_{infty}}).