Approximation Analysis of a Parabolic-Parabolic Chemotaxis Model with Logarithmic Nonlinearity

We consider the Keller-Segel system with logical source \begin{align*} \begin{cases} u_t = \nabla \cdot (\phi(u)\nabla u) - \nabla \cdot (\psi(u)\nabla v)+f(u), & x \in \Omega, \; t > 0, v_t = \Delta v - v + u, & x \in \Omega, \; t > 0, \end{cases} \end{align*} in a smooth bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, the Neumann initial-boundary value problem admits a globally defined, uniformly bounded classic solution for all sufficiently regular non-negative initial data $u_0$ and $v_0$. In the first equation, assume that $\phi$ and $\psi$ are dominated by a logarithmic function and a polynomial respectively. The logical source $f$ representing the natural growth and decay of cells satisfies $f \in W^{1,\infty}_{\mathrm{loc}}(\Omega)$ and $f(0) \geq 0$. Then we will see that the unique solution $u \in C^{2,1}((\overline{\Omega}) \times [0,T] )$ and $v \in W^{1,q}([0,T] ; C^{2,1}(\overline{\Omega}))$.