Global existence of classical solutions of a two-species chemotaxis-competition system with consumption or linear signal production on RN

In this paper, we investigate the global existence of classical solutions for the following two-species chemotaxis system with Lotka-Volterra competitive kinetics on $R^N$(0.1)$\{\begin{array}{cc}{u}_t=\Delta u-{\chi }_1\nabla \cdot \left(u\nabla w\right)+u(a_1-b_{1}u-c_{1}v),& t>0,x\in R^N,\\ v_t=\Delta v-{\chi }_2\nabla \cdot \left(v\nabla w\right)+v(a_2-b_{2}v-c_{2}u),& t>0,x\in R^N,\\ \tau w_t=\Delta w+g(u,v,w),& t>0,x\in R^N,\\ u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right),w(0,x)=w_0\left(x\right),& x\in R^N,\end{array}$ in the following cases: (i) $\tau >0$, $g(u,v,w)=-(u+v)w$, (ii) $\tau >0$, $g(u,v,w)=\alpha u+\beta v-\lambda w$ and (iii) $\tau =0$, $g(u,v,w)=\alpha u+\beta v-\lambda w$, where $N\ge 1$ is a positive integer, ${\chi }_i$ are nonzero numbers, $a_i,b_i,c_i(i=1,2)$ and $\lambda ,\alpha ,\beta$ are positive constants. We first prove that (0.1) has a unique nonnegative classical solution $(u,v,w)$ on the maximal interval $(0,T_{\max })$. Next, we prove that if there exists $p>\max \{1,\frac{N}{2}\}$ such that$\underset{t\to T_{\max }-}{\lim \sup }\underset{x_0\in R^N}{\sup }\underset{B(x_0,1)}{\int }{(u+v)}^p(t,x)dx<\infty ,$ then $T_{\max }=\infty$ and ${\lim \sup }_{t\to \infty }({\Vert u(t,\cdot )\Vert }_{\infty }+{\Vert v(t,\cdot )\Vert }_{\infty })<\infty$. Finally, we provide sufficient conditions for the global existence and boundedness of classical solutions for three different models of (0.1). It follows that nonnegative classical solution of the three different models of (0.1) exists globally and stays bounded in one- and two-dimensional settings for any chemotaxis sensitivity ${\chi }_i(i=1,2)$.