On a nonlinear hyperbolic equation with anisotropy: Global existence and decay of solution

Abstract

We establish the existence of a global solution to an initial boundary value problem for the nonlinear anisotropic hyperbolic equation u ″ − ∑ i = 1 n ∂ ∂ x i ( | ∂ u ∂ x i | p i − 2 ∂ u ∂ x i ) − Δ u ′ + g ( x , u ) = f ( x , t ) . Depending on the range of the p i ’s, we derive an exponential and a polynomial decay for the global solution.