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 (φ(u)\nabla u) - \nabla \cdot (ψ(u)\nabla v)+f(u), & x \in Ω, ; t > 0, v_t = Δv - v + u, & x \in Ω, ; t > 0, \end{cases} \end{align*} in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 2$, 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 $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\mathrm{\infty }}(\mathrm{\Omega })$ and $f(0)\ge 0$. Then we will see that the unique solution $u\in C^{2,1}((\bar{\mathrm{\Omega }})\times [0,T])$ and $v\in W^{1,q}([0,T];C^{2,1}(\bar{\mathrm{\Omega }}))$.