Well-posedness and numerical study for solutions of a parabolic equation with variable-exponent nonlinearities (Q1656185)

Summary: We consider the following nonlinear parabolic equation: \(u_t - \operatorname{div}(| \nabla u |^{p(x) - 2} \nabla u) = f(x, t)\), where \(f : \Omega \times(0, T) \rightarrow \mathbb{R}\) and the exponent of nonlinearity \(p(\cdot)\) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.