<p>In this paper, we investigate the global existence of classical solution to the following two-species chemotaxis-competition model with weak singular sensitivity and Lotka-Volterra competitive kinetics <Equation ID="Equ1"> <EquationNumber>0.1</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi _1\nabla \cdot (\frac{u}{w^\alpha }\nabla w)+u(a_1-b_1u-c_1v), & t>0,~x\in \Omega ,\\ v_t=\Delta v-\chi _2\nabla \cdot (\frac{v}{w^\beta }\nabla w)+v(a_2-b_2v-c_2u), & t>0,~x\in \Omega ,\\ \tau w_t=\Delta w- w+ u+ v, & t>0,~x\in \Omega ,\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }=0, & t>0,~x\in \partial \Omega ,\\ u(0,x)=u_0(x),v(0,x)=v_0(x),\tau w(0,x)=\tau w_0(x), & x\in \Omega , \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>-</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mfrac> <mi>u</mi> <msup> <mi>w</mi> <mi>α</mi> </msup> </mfrac> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mi>u</mi> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>-</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mfrac> <mi>v</mi> <msup> <mi>w</mi> <mi>β</mi> </msup> </mfrac> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mi>v</mi> <mo>-</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>τ</mi> <msub> <mi>w</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>w</mi> <mo>-</mo> <mi>w</mi> <mo>+</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mfrac> <mrow> <mi>∂</mi> <mi>u</mi> </mrow> <mrow> <mi>∂</mi> <mi>ν</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>∂</mi> <mi>v</mi> </mrow> <mrow> <mi>∂</mi> <mi>ν</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>∂</mi> <mi>w</mi> </mrow> <mrow> <mi>∂</mi> <mi>ν</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>τ</mi> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>τ</mi> <msub> <mi>w</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^N(N\ge 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a bounded smooth domain, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\tau \in \{0,1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and the parameters <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha ,\beta ,\chi _i,a_i,b_i,c_i(i=1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <msub> <mi>χ</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>c</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are all positive constants. When <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\tau =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha =\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>, we prove that there exist <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M^>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mo>∗</mo> </msup> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C^>0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mo>∗</mo> </msup> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> if <Equation ID="Equ142"> <EquationSource Format="TEX">$\begin{aligned} \alpha \in (0,1),~\min \{b_1,b_2,c_1,c_2\}>\chi M^(N,\alpha ,\chi ),\text {when}~\chi _1=\chi _2=:\chi , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>α</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo stretchy="false">}</mo> </mrow> <mo>></mo> <mi>χ</mi> <msup> <mi>M</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>χ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mtext>when</mtext> <mspace width="3.33333pt" /> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>:</mo> <mi>χ</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>or <Equation ID="Equ143"> <EquationSource Format="TEX">$\begin{aligned} \begin{aligned} \alpha \in \left( \frac{1}{2},1\right) ,~\min \{b_1,b_2&,c_1,c_2\}>\min \{\chi _1M^(N,\alpha ,2\chi _1)+(\chi _2-\chi _1)^2C^(N,\alpha ,\chi _2,\chi _1),\\&\chi _2M^(N,\alpha ,2\chi _2)+(\chi _1-\chi _2)^2C^*(N,\alpha ,\chi _1,\chi _2)\},~\text {when}~\chi _1\ne \chi _2 \end{aligned} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> <mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> </mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>,</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">}</mo> <mo>></mo> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> </mrow> <msub> <mi>χ</mi> <mn>1</mn> </msub> <msup> <mi>M</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mn>2</mn> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>C</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>χ</mi> <mn>2</mn> </msub> <msup> <mi>M</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mn>2</mn> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>C</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">}</mo> <mo>,</mo> <mspace width="3.33333pt" /> <mtext>when</mtext> <mspace width="3.33333pt" /> </mrow> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>≠</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>then for any given nonnegative initial data <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$u_0, v_0\in C^0(\overline{\Omega }),~\text {and}~\int _{\Omega }u_0(x)+v_0(x){\text {d}}x>0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>∈</mo> <msup> <mi>C</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mover> <mi mathvariant="normal">Ω</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mtext>and</mtext> <mspace width="3.33333pt" /> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mtext>d</mtext> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the problem (<InternalRef RefID="Equ1">0.1</InternalRef>) possesses a unique globally defined classical solution. Moreover, the solutions are shown to be uniformly bounded of solution under the additional assumption <Equation ID="Equ144"> <EquationSource Format="TEX">$\begin{aligned}\alpha \in \left( \frac{1}{2},\frac{1}{2}+\frac{1}{N}\right) \text {with}N\ge 2. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mfenced> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mtext>with</mtext> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>N</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>When <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\tau =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha ,\beta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove that if <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\min \{b_1,b_2,c_1,c_2\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is sufficiently large, then for any given nonnegative initial data <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$u_0, v_0\in C^0(\overline{\Omega }),~\text {and}~0<w_0(x)\in W^{1,\infty }(\Omega ),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>∈</mo> <msup> <mi>C</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mover> <mi mathvariant="normal">Ω</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mtext>and</mtext> <mspace width="3.33333pt" /> <mn>0</mn> <mo><</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the problem (<InternalRef RefID="Equ1">0.1</InternalRef>) possesses a unique global classical solution.</p>

Global existence, boundedness of classical solutions of two-species chemotaxis-competition model with weak singular sensitivity

期刊论文
发表日期：

2025年10月10日

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Weiyi Zhang,
Zuhan Liu

摘要

In this paper, we investigate the global existence of classical solution to the following two-species chemotaxis-competition model with weak singular sensitivity and Lotka-Volterra competitive kinetics 0.1 $\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi _1\nabla \cdot (\frac{u}{w^\alpha }\nabla w)+u(a_1-b_1u-c_1v), & t>0,~x\in \Omega ,\\ v_t=\Delta v-\chi _2\nabla \cdot (\frac{v}{w^\beta }\nabla w)+v(a_2-b_2v-c_2u), & t>0,~x\in \Omega ,\\ \tau w_t=\Delta w- w+ u+ v, & t>0,~x\in \Omega ,\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }=0, & t>0,~x\in \partial \Omega ,\\ u(0,x)=u_0(x),~~v(0,x)=v_0(x),~~\tau w(0,x)=\tau w_0(x), & x\in \Omega , \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u - χ_{1} \nabla \cdot (\frac{u}{w^{α}} \nabla w) + u (a_{1} - b_{1} u - c_{1} v), & t > 0, x \in Ω, \\ v_{t} = Δ v - χ_{2} \nabla \cdot (\frac{v}{w^{β}} \nabla w) + v (a_{2} - b_{2} v - c_{2} u), & t > 0, x \in Ω, \\ τ w_{t} = Δ w - w + u + v, & t > 0, x \in Ω, \\ \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = \frac{\partial w}{\partial ν} = 0, & t > 0, x \in \partial Ω, \\ u (0, x) = u_{0} (x), v (0, x) = v_{0} (x), τ w (0, x) = τ w_{0} (x), & x \in Ω, \end{matrix}) \end{matrix}$ where $\Omega \subset \mathbb {R}^N(N\ge 1)$ $Ω \subset R^{N} (N \geq 1)$ is a bounded smooth domain, $\tau \in \{0,1\}$ $τ \in {0, 1}$ and the parameters $\alpha ,\beta ,\chi _i,a_i,b_i,c_i(i=1,2)$ $α, β, χ_{i}, a_{i}, b_{i}, c_{i} (i = 1, 2)$ are all positive constants. When $\tau =0$ $τ = 0$ and $\alpha =\beta $ $α = β$ , we prove that there exist $M^*>0$ $M^{*} > 0$ and $C^*>0,$ $C^{*} > 0,$ if $\begin{aligned} \alpha \in (0,1),~~~\min \{b_1,b_2,c_1,c_2\}>\chi M^*(N,\alpha ,\chi ),~~\text {when}~\chi _1=\chi _2=:\chi , \end{aligned}$ $\begin{matrix} α \in (0, 1), min {b_{1}, b_{2}, c_{1}, c_{2}} > χ M^{*} (N, α, χ), when χ_{1} = χ_{2} = : χ, \end{matrix}$ or $\begin{aligned} \begin{aligned} \alpha \in \left( \frac{1}{2},1\right) ,~\min \{b_1,b_2&,c_1,c_2\}>\min \{\chi _1M^*(N,\alpha ,2\chi _1)+(\chi _2-\chi _1)^2C^*(N,\alpha ,\chi _2,\chi _1),\\&\chi _2M^*(N,\alpha ,2\chi _2)+(\chi _1-\chi _2)^2C^*(N,\alpha ,\chi _1,\chi _2)\},~\text {when}~\chi _1\ne \chi _2 \end{aligned} \end{aligned}$ $\begin{matrix} \begin{matrix} α \in (\frac{1}{2}, 1), min {b_{1}, b_{2} & , c_{1}, c_{2}} > min {χ_{1} M^{*} (N, α, 2 χ_{1}) + {(χ_{2} - χ_{1})}^{2} C^{*} (N, α, χ_{2}, χ_{1}), \\ χ_{2} M^{*} (N, α, 2 χ_{2}) + {(χ_{1} - χ_{2})}^{2} C^{*} (N, α, χ_{1}, χ_{2})}, when χ_{1} \neq χ_{2} \end{matrix} \end{matrix}$ then for any given nonnegative initial data $u_0, v_0\in C^0(\overline{\Omega }),~\text {and}~\int _{\Omega }u_0(x)+v_0(x){\text {d}}x>0,$ $u_{0}, v_{0} \in C^{0} (\bar{Ω}), and \int_{Ω} u_{0} (x) + v_{0} (x) d x > 0,$ the problem (0.1) possesses a unique globally defined classical solution. Moreover, the solutions are shown to be uniformly bounded of solution under the additional assumption $\begin{aligned}\alpha \in \left( \frac{1}{2},\frac{1}{2}+\frac{1}{N}\right) ~~\text {with}~~N\ge 2. \end{aligned}$ $\begin{matrix} α \in (\frac{1}{2}, \frac{1}{2} + \frac{1}{N}) with N \geq 2 . \end{matrix}$ When $\tau =1$ $τ = 1$ and $\alpha ,\beta >0$ $α, β > 0$ , we prove that if $\min \{b_1,b_2,c_1,c_2\}$ $min {b_{1}, b_{2}, c_{1}, c_{2}}$ is sufficiently large, then for any given nonnegative initial data $u_0, v_0\in C^0(\overline{\Omega }),~\text {and}~0<w_0(x)\in W^{1,\infty }(\Omega ),$ $u_{0}, v_{0} \in C^{0} (\bar{Ω}), and 0 < w_{0} (x) \in W^{1, \infty} (Ω),$ the problem (0.1) possesses a unique global classical solution.