Wave Equations with Lipschitz Nonlinearity of Combined Power and Logarithmic Type: Existence of Global Solutions

In this paper, we investigate wave equations with a general class of Lipschitz nonlinearities combining power and logarithmic terms. Local existence and uniqueness of weak solutions are established via the Galerkin method. For nonlinearities of defocusing type, we prove global existence of the weak solutions for arbitrary initial energy. In particular, we provide examples of globally defined solutions with negative initial energy.