The one dimensional parabolic <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)-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 \({L_{\infty}}\).