A new result for global existence and boundedness of solutions to a parabolic--parabolic Keller--Segel system with logistic source

Abstract

We consider the following fully parabolic Keller--Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(u\nabla v)+ au-μu^2,\quad x\in Ω, t>0, \disp{v_t=Δv- v +u},\quad x\in Ω, t>0, \end{array}\right.\eqno(KS) $$ over a bounded domain $\Omega \subset {\mathbb{R}}^N(N\ge 1)$, with smooth boundary $\partial \Omega$, the parameters $a\in \mathbb{R},\mu >0,\chi >0$. It is proved that if $\mu >0$, then $(KS)$ admits a global weak solution, while if $\mu >\frac{(N-2{)}_{+}}{N}\chi C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$, then $(KS)$ possesses a global classical solution which is bounded, where $C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$ is a positive constant which is corresponding to the maximal Sobolev regularity. Apart from this, we also show that if $a=0$ and $\mu >\frac{(N-2{)}_{+}}{N}\chi C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$, then both $u(\cdot ,t)$ and $v(\cdot ,t)$ decay to zero with respect to the norm in $L^{\infty }(\Omega )$ as $t\to \infty$.

References in corpus (1)

• New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models