On the Existence of Viscosity Solutionsfor Evolution$$p(x)$$-Laplace Equationwith One Spatial Variable

In this paper, we study the first boundary value problem for the $$ p(x) $$ -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.