Global solvability and boundedness in the $N$-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

Abstract

We consider the following chemotaxis model %fully parabolic Keller-Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in \Omega, t>0, \disp{v_t-\Delta v=-uv },\quad x\in \Omega, t>0, %\disp{\tau w_t+\delta w=u },\quad %x\in \Omega, t>0, \disp{(\nabla D(u)-\chi u\cdot \nabla v)\cdot \nu=\frac{\partial v}{\partial\nu}=0},\quad x\in \partial\Omega, t>0, \disp{u(x,0)=u_0(x)},\quad v(x,0)=v_0(x),~~ x\in \Omega \end{array}\right. $$ on a bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partial\Omega, \chi$ and $\mu$ are positive constants. Besides appropriate smoothness assumptions, in this paper it is only required that $D(u)\geq C_{D}(u+1)^{m-1}$ for all $u\geq 0$ with some $C_{D} > 0$ and some $$ m>\left\{\begin{array}{ll} 1-\frac{\mu}{\chi[1+\lambda_{0}\|v_0\|_{L^\infty(\Omega)}2^{3}]}~~\mbox{if}~~ N\leq2, % >1+\frac{(N+2-2r)^+}{N+2}~~~~~~\mbox{if}~~ % \frac{N+2}{2}\geq r\geq\frac{N+2}{N}, 1~~~~~~\mbox{if}~~ N\geq3, \end{array}\right. $$ then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where $\lambda_{0}$ is a positive constant which is corresponding to the maximal sobolev regularity. The results of this paper extends the results of Jin (J. Diff. Eqns., 263(9)(2017), 5759-5772), who proved the possibility of boundness of weak solutions, in the case $m>1$ and $N=3$.