Global existence of weak solutions in a Keller-Segel system with local anisotropy and nonlinear diffusion

We show global existence of weak solutions to the following system of partial differential equations with nonlinear diffusion in both equations and anisotropic signal production \begin{equation} \begin{cases} u_t=\nabla\cdot((u+1)^{m_1-1}\nabla u)-\nabla\cdot(u\nabla v), \\ v_t=\nabla\cdot((v+