On existence of viscosity solutions for evolution $p(x)$-Laplace equation with one spatial variable

In this paper, we study the first boundary value problem for $p(x)$-Laplacian with one spatial variable in the presence of gradient terms that do not satisfy the Bernstein—Nagumo condition. A class of gradient nonlinearities is defined, for which the existence of a viscosity solution that is Lipschitz continuous in $x$ and Hölder continuous in $t$ is proven.