On the 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 the \(p(x)\) -Laplacian with one spatial variable in the presence of gradient terms that donot satisfy the Bernstein–Nagumo condition. A class of gradient nonlinearities is defined for whichthe existence of a viscosity solution that is Lipschitz continuous in \(x\) and Hölder continuous in \(t\) is proven.