The one dimensional parabolic \(p(x)\)-Laplace equation (Q506006)

The author studies the initial-boundary value problem for a parabolic equation involving the \(p(x)\)-Laplacian operator with \(p(x)\in(-1,0]\), i.e. in a possibly singular case of the variable exponent \(p\)-Laplacian. Under suitable assumptions on the term \(f\) in the one-space variable equation \(u_t=(|u_x|^{p(x)}u_x)_x+f(x,u,u_x)=0\) the existence and uniqueness of strong and weak solutions is proved using the parabolic regularization method.