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+1)^{m_2-1}\nabla v)+\nabla\cdot(u\nabla v)-v+u, \end{cases} \end{equation} in a smooth bounded domain $Ω\subseteq\mathbb{R}^n$, $n\in\mathbb{N}$ with nonnegative initial data and homogenuous Neumann boundary conditions. Moreover, we show boundedness of $u$ in $L^\infty((0,\infty);\:L^{m_1}(Ω))$ and $v$ in $L^\infty(Ω\times(0,\infty))$. The strategy will be to show global existence and boundedness of classical solutions $(u_\varepsilon,v_\varepsilon)$ to the approximating system \begin{equation} \begin{cases} u_{\varepsilon t}=\nabla\cdot((u_\varepsilon+1)^{m_1-1}\nabla u_\varepsilon) -\nabla\cdot(\frac{u_\varepsilon\nabla v_\varepsilon}{1+\varepsilon\vert\nabla v_\varepsilon\vert}), \\ v_{\varepsilon t}=\nabla\cdot((v_\varepsilon+1)^{m_2-1}\nabla v_\varepsilon)+\nabla\cdot(u_\varepsilon\nabla v_\varepsilon)-v_\varepsilon+u_\varepsilon. \end{cases} \end{equation} Passing to the limit $\varepsilon\searrow 0$, we obtain convergence of weak solutions of the second to weak solutions of the first system. One of the main challenges lie in the proof of weak convergence of $\frac{\nabla v_\varepsilon}{1+\varepsilon\vert\nabla v_\varepsilon\vert}$ to $\nabla v$ in $L^2(Ω\times(0,T))$ for $T\in(0,\infty)$.