On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics

We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for $$u_t=u\mathrm{\Delta }u+u{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$$ in bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. We show that in this case the blow-up set coincides with $\bar{\mathrm{\Omega }}$, i.e. the finite-time blow-up is global. Key words: Degenerate diffusion, non-local nonlinearity, blow-up, evolutionary games, infinite dimensional replicator dynamics