Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production

Abstract

<p style='text-indent:20px;'>This paper deals with a quasilinear parabolic-elliptic chemo-repulsion system with nonlinear signal production</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} & u_t = \nabla\cdot(\phi(u)\nabla u)+\chi\nabla\cdot(u(u+1)^{\alpha-1}\nabla v)+f(u), & (x,t)\in \Omega\times (0,\infty), \\ & 0 = \Delta v-v+u^{\beta}, & (x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset \mathbb{R}^{n}(n\geq1), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ \chi,\beta>0,\alpha\in\mathbb{R}, $\end{document}</tex-math></inline-formula> the nonlinear diffusion <inline-formula><tex-math id="M3">\begin{document}$ \phi\in C^{2}([0,\infty)) $\end{document}</tex-math></inline-formula> satisfies <inline-formula><tex-math id="M4">\begin{document}$ \phi(u)\geq(u+1)^{m} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ m\in\mathbb{R}, $\end{document}</tex-math></inline-formula> and the function <inline-formula><tex-math id="M6">\begin{document}$ f\in C^{1}([0,\infty)) $\end{document}</tex-math></inline-formula> is a generalized growth term.</p><p style='text-indent:20px;'><inline-formula><tex-math id="M7">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> When <inline-formula><tex-math id="M8">\begin{document}$ f\equiv0, $\end{document}</tex-math></inline-formula> it is shown that the solution of the above system is global and uniformly bounded for all <inline-formula><tex-math id="M9">\begin{document}$ \chi,\beta>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ m,\alpha\in\mathbb{R} $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'><inline-formula><tex-math id="M11">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> When <inline-formula><tex-math id="M12">\begin{document}$ f\not\equiv0 $\end{document}</tex-math></inline-formula> and assume that <inline-formula><tex-math id="M13">\begin{document}$ f(u)\leq ku-bu^{\gamma+1} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ k,b,\gamma>0, $\end{document}</tex-math></inline-formula> it is proved that the solution of the above system is also global and uniformly bounded for all <inline-formula><tex-math id="M15">\begin{document}$ \chi,\beta>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ m,\alpha\in\mathbb{R}. $\end{document}</tex-math></inline-formula></p>

Boundedness and Asymptotic Stability in a Chemotaxis Model with Signal-Dependent Motility and Nonlinear Signal Secretion

“…(1.1b) by u k (k>0). When k ≥ 1, Galakhov et al [9] considered the global dynamics of solutions, thereinto, by assuming that σ>k+1 or σ=k+1, µ>((nk−2)/nk)χ, they obtained the global boundedness result; and this boundedness result was extended to the borderline case σ = k+1, µ = ((nk−2)/nk)χ, n ≥ 3 by Hu and Tao [13]; then Xiang [33] removed the restrictions k≥1 and n≥3, under the condition k+1<max{σ,1+2/n} or k+1 = σ, µ ≥ ((nk−2)/nk)χ, he proved that the solution is globally bounded; afterwards, Xiang et al [30] further extended the result in [33] to the case with nonlinear diffusion function D(u) and nonlinear sensitivity function S(u); lately, considering the chemo-repulsion case, Hu et al [15] established the global bound-edness of solutions for the quasilinear case without any conditions on parameters. As for the parabolic-parabolic case, one can refer to the literatures [14,24,26] and the reference thereinto.…”