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

We consider the initial-boundary value problem for a variant of the Keller-Segel chemotaxis system with doubly degenerate diffusion, i.e. we study $$\{\begin{array}{l}{u}_t=\mathrm{\nabla }\cdot (uv\mathrm{\nabla }u)-\mathrm{\nabla }\cdot (u^{2}v\mathrm{\nabla }v),\\ v_t=\mathrm{\Delta }v+u-v,\\ (uv\mathrm{\nabla }u-u^{2}v\mathrm{\nabla }v)\cdot \nu =\mathrm{\nabla }v\cdot \nu =0,\\ u(x,0)=u_0(x),v(x,0)=v_0(x),\end{array}$$ in a smoothly bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$. 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 $m_0=4\pi$. Moreover, in a radial setting the threshold number can be increased to $m_0=8\pi$.