Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source

Abstract

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source \begin{document}$ \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), {v_t = \Delta v- v +u}, {w_t = - vw}, \quad {\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0}, x\in \partial\Omega, t>0, {u(x, 0) = u_0(x)}, v(x, 0) = v_0(x), w(x, 0) = w_0(x), x\in \Omega, \end{array}\right. ~~~~~~~~~~~~~~~~~(0.1)$ \end{document} in a smooth bounded domain \begin{document}$ \mathbb{R}^N(N\geq1) $\end{document} , with parameter \begin{document}$ r>1 $\end{document} . the parameters \begin{document}$ a\in \mathbb{R}, \mu>0, \chi>0 $\end{document} . It is shown that when \begin{document}$ r>2 $\end{document} , or \begin{document}$ \begin{equation*} \mu>\mu^{*} = \begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}, if r = 2, \end{array} \end{equation*} $\end{document} the considered problem possesses a global classical solution which is bounded, where \begin{document}$ C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1} $\end{document} is a positive constant which is corresponding to the maximal sobolev regularity. Here \begin{document}$ C_{\beta} $\end{document} is a positive constant which depends on \begin{document}$ \xi $\end{document} , \begin{document}$ \|u_0\|_{C(\bar{\Omega})}, \|v_0\|_{W^{1, \infty}(\Omega)} $\end{document} and \begin{document}$ \|w_0\|_{L^\infty(\Omega)} $\end{document} . This result improves or extends previous results of several authors.