Subcritical-mass global solvability in a doubly degenerate Keller-Segel system with signal production

Abstract

We consider the initial-boundary value problem for a variant of the Keller-Segel chemotaxis system with doubly degenerate diffusion, i.e. we study \[ \left\{ \begin{array}{ll} u_t = \nabla \cdot (uv\nabla u) - \nabla \cdot (u^2v\nabla v),\\ v_t = Δv + u - v,\\ (uv\nabla u-u^2v\nabla v)\cdotν=\nabla v\cdotν=0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \end{array} \right. \] in a smoothly bounded domain $Ω\subset\mathbb{R}^2$. Crucially, we only assume the sufficiently regular initial data to be nonnegative, but allow those functions to be zero at non-trivial parts of the domain. We show that, despite possibly starting from a degenerate state, the system admits global solutions in a framework of generalized energy solutions, whenever the initial mass is below the threshold number $m_0=4π$. Moreover, in a radial setting the threshold number can be increased to $m_0=8π$.