\(L_p\)-estimates of solutions to \(n\)-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation (Q2816490)

The authors study a parabolic-parabolic system of chemotaxis NEWLINE\[NEWLINEu_t=\Delta u-\chi\nabla\cdot(u\nabla v)+f(u),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\tau v_t=\Delta v-v+g(u),NEWLINE\]NEWLINE with a growth \(f(u)\) and a secretion \(g(u)\) terms, in bounded domains of \(\mathbb R^n\). Under assumptions that \(f(u)=u-\mu u^\alpha\), \(g(u)=u(1+u)^{\beta-1}\) with \(\alpha>1\), \(0<\beta\leq 2\) and \(2\beta<\alpha-1\), the global in time existence of solutions is proved in the \(L^p\) setting with \(p>n\). These results are extensions of the analysis of a related parabolic-elliptic (\(\tau=0\)) model by M. Winkler et al.