<p>In this paper, we investigate the global existence of classical solutions to the following parabolic–parabolic–parabolic two-species chemotaxis-competition system with singular sensitivity and nonlinear productions <Equation ID="Equ59"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi _1\nabla \cdot (\frac{u}{w}\nabla w)+u(a_1-b_1u^{\alpha _1}-c_1v^{\beta _2}), & t>0,~x\in \Omega ,\\ v_t=\Delta v-\chi _2\nabla \cdot (\frac{v}{w}\nabla w)+v(a_2-b_2v^{\alpha _2}-c_2u^{\beta _1}), & t>0,~x\in \Omega ,\\ w_t=\Delta w-w+u^{\gamma _1}+v^{\gamma _2}, & t>0,~x\in \Omega , \quad \quad \quad \quad \quad \quad \quad \quad {\mathrm{(0.1)}}\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }=0, & t>0,~x\in \partial \Omega ,\\ u(0,x)=u_0(x),v(0,x)=v_0(x),w(0,x)=w_0(x), & x\in \Omega , \end{array}\right. } \end{aligned}$</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> <mi>w</mi> </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> <msup> <mi>u</mi> <msub> <mi>α</mi> <mn>1</mn> </msub> </msup> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <msup> <mi>v</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> </msup> <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> <mi>w</mi> </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> <msup> <mi>v</mi> <msub> <mi>α</mi> <mn>2</mn> </msub> </msup> <mo>-</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <msup> <mi>u</mi> <msub> <mi>β</mi> <mn>1</mn> </msub> </msup> <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>w</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>w</mi> <mo>-</mo> <mi>w</mi> <mo>+</mo> <msup> <mi>u</mi> <msub> <mi>γ</mi> <mn>1</mn> </msub> </msup> <mo>+</mo> <msup> <mi>v</mi> <msub> <mi>γ</mi> <mn>2</mn> </msub> </msup> <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> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mn>0.1</mn> <mo stretchy="false">)</mo> </mrow> </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>w</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <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> </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 convex smooth domain, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\chi _i,b_i,c_i,\alpha _i,\beta _i>0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</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> <mo>,</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a_i\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1\le \frac{8(\chi _1+\chi _2)}{8(\chi _1+\chi _2)-(\chi _1-\chi _2)^2}\le \gamma _i(i=1,2).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mfrac> <mrow> <mn>8</mn> <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> <mrow> <mn>8</mn> <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> <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> </mrow> </mfrac> <mo>≤</mo> <msub> <mi>γ</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> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> If <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma _1<\alpha _1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma _2<\min \{\alpha _2,\beta _2\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma _2<\alpha _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma _1<\min \{\alpha _1,\beta _1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <Equation ID="Equ60"> <EquationSource Format="TEX">\(\begin{aligned} \chi _1+\chi _2\ge {\left\{ \begin{array}{ll} \frac{N\chi _1^2\gamma ^2_1}{2}=\min \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \}, & \text {if}~\frac{\chi _1}{\chi _2}\gamma _1<\gamma _2<\min \{\alpha _2,\beta _2\},\\ \frac{N\chi _2^2\gamma ^2_2}{2}=\min \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \}, & \text {if}~\frac{\chi _2}{\chi _1}\gamma _2<\gamma _1<\min \{\alpha _1,\beta _1\},\\ \frac{N\chi _1\chi _2\gamma _1\gamma _2}{2}, & \text {if}~\gamma _1<\alpha _1~\text {and}~\gamma _2<\alpha _2,\\ \frac{N\chi _1\chi _2\gamma _1\gamma _2}{2}\in \Big (\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big ), & \text {if}~\frac{\chi _1}{\chi _2}\gamma _1<\gamma _2<\alpha _2,\\ \frac{N\chi _1\chi _2\gamma _1\gamma _2}{2}\in \Big (\frac{N\chi _2^2\gamma ^2_2}{2},\frac{N\chi _1^2\gamma ^2_1}{2}\Big ), & \text {if}~\frac{\chi _2}{\chi _1}\gamma _2<\gamma _1<\alpha _1,\\ \max \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \},\\ \end{array}\right. } \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo>≥</mo> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mo movablelimits="true">min</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>if</mtext> <mspace width="3.33333pt" /> <mfrac> <msub> <mi>χ</mi> <mn>1</mn> </msub> <msub> <mi>χ</mi> <mn>2</mn> </msub> </mfrac> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mo movablelimits="true">min</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>if</mtext> <mspace width="3.33333pt" /> <mfrac> <msub> <mi>χ</mi> <mn>2</mn> </msub> <msub> <mi>χ</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mfrac> <mrow> <mi>N</mi> <msub> <mi>χ</mi> <mn>1</mn> </msub> <msub> <mi>χ</mi> <mn>2</mn> </msub> <msub> <mi>γ</mi> <mn>1</mn> </msub> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>if</mtext> <mspace width="3.33333pt" /> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mspace width="3.33333pt" /> <mtext>and</mtext> <mspace width="3.33333pt" /> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mfrac> <mrow> <mi>N</mi> <msub> <mi>χ</mi> <mn>1</mn> </msub> <msub> <mi>χ</mi> <mn>2</mn> </msub> <msub> <mi>γ</mi> <mn>1</mn> </msub> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>∈</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>if</mtext> <mspace width="3.33333pt" /> <mfrac> <msub> <mi>χ</mi> <mn>1</mn> </msub> <msub> <mi>χ</mi> <mn>2</mn> </msub> </mfrac> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mfrac> <mrow> <mi>N</mi> <msub> <mi>χ</mi> <mn>1</mn> </msub> <msub> <mi>χ</mi> <mn>2</mn> </msub> <msub> <mi>γ</mi> <mn>1</mn> </msub> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>∈</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>if</mtext> <mspace width="3.33333pt" /> <mfrac> <msub> <mi>χ</mi> <mn>2</mn> </msub> <msub> <mi>χ</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mo movablelimits="true">max</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>we prove that for any given nonnegative initial functions <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$u_0,v_0\in C^0(\overline{\Omega })$</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> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$w_0\in W^{1,\infty }(\Omega ),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> <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> problem (0.1) admits a global classical solution. Moreover, if <Equation ID="Equ61"> <EquationSource Format="TEX">$\begin{aligned} \chi _1+\chi _2\ge \max \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \}, a_1<-\frac{1}{\gamma _1}~\text {and}~a_2<-\frac{1}{\gamma _2}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo>≥</mo> <mo movablelimits="true">max</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>N</mi> <msubsup> <mi>χ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi>γ</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> <mo>,</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo><</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>γ</mi> <mn>1</mn> </msub> </mfrac> <mspace width="3.33333pt" /> <mtext>and</mtext> <mspace width="3.33333pt" /> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo><</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mfrac> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>we prove that the problem (0.1) has a global bounded classical solution.</p>

Global existence of classical solutions to a fully parabolic two-species chemotaxis-competition model with singular sensitivity and nonlinear productions

期刊论文
发表日期：

2026年3月17日

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Weiyi Zhang,
Zuhan Liu

摘要

In this paper, we investigate the global existence of classical solutions to the following parabolic–parabolic–parabolic two-species chemotaxis-competition system with singular sensitivity and nonlinear productions $\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi _1\nabla \cdot (\frac{u}{w}\nabla w)+u(a_1-b_1u^{\alpha _1}-c_1v^{\beta _2}), & t>0,~x\in \Omega ,\\ v_t=\Delta v-\chi _2\nabla \cdot (\frac{v}{w}\nabla w)+v(a_2-b_2v^{\alpha _2}-c_2u^{\beta _1}), & t>0,~x\in \Omega ,\\ w_t=\Delta w-w+u^{\gamma _1}+v^{\gamma _2}, & t>0,~x\in \Omega , \quad \quad \quad \quad \quad \quad \quad \quad {\mathrm{(0.1)}}\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }=0, & t>0,~x\in \partial \Omega ,\\ u(0,x)=u_0(x),~~v(0,x)=v_0(x),~~w(0,x)=w_0(x), & x\in \Omega , \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u - χ_{1} \nabla \cdot (\frac{u}{w} \nabla w) + u (a_{1} - b_{1} u^{α_{1}} - c_{1} v^{β_{2}}), & t > 0, x \in Ω, \\ v_{t} = Δ v - χ_{2} \nabla \cdot (\frac{v}{w} \nabla w) + v (a_{2} - b_{2} v^{α_{2}} - c_{2} u^{β_{1}}), & t > 0, x \in Ω, \\ w_{t} = Δ w - w + u^{γ_{1}} + v^{γ_{2}}, & t > 0, x \in Ω, (0.1) \\ \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = \frac{\partial w}{\partial ν} = 0, & t > 0, x \in \partial Ω, \\ u (0, x) = u_{0} (x), v (0, x) = v_{0} (x), w (0, x) = w_{0} (x), & x \in Ω, \end{matrix}) \end{matrix}$ where $\Omega \subset \mathbb {R}^N(N\ge 1)$ $Ω \subset R^{N} (N \geq 1)$ is a bounded convex smooth domain, $\chi _i,b_i,c_i,\alpha _i,\beta _i>0,$ $χ_{i}, b_{i}, c_{i}, α_{i}, β_{i} > 0,$ $a_i\in \mathbb {R}$ $a_{i} \in R$ and $1\le \frac{8(\chi _1+\chi _2)}{8(\chi _1+\chi _2)-(\chi _1-\chi _2)^2}\le \gamma _i(i=1,2).$ $1 \leq \frac{8 (χ_{1} + χ_{2})}{8 (χ_{1} + χ_{2}) - {(χ_{1} - χ_{2})}^{2}} \leq γ_{i} (i = 1, 2) .$ If $\gamma _1<\alpha _1$ $γ_{1} < α_{1}$ and $\gamma _2<\min \{\alpha _2,\beta _2\}$ $γ_{2} < min {α_{2}, β_{2}}$ or $\gamma _2<\alpha _2$ $γ_{2} < α_{2}$ and $\gamma _1<\min \{\alpha _1,\beta _1\}$ $γ_{1} < min {α_{1}, β_{1}}$ or \(\begin{aligned} \chi _1+\chi _2\ge {\left\{ \begin{array}{ll} \frac{N\chi _1^2\gamma ^2_1}{2}=\min \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \}, & \text {if}~\frac{\chi _1}{\chi _2}\gamma _1<\gamma _2<\min \{\alpha _2,\beta _2\},\\ \frac{N\chi _2^2\gamma ^2_2}{2}=\min \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \}, & \text {if}~\frac{\chi _2}{\chi _1}\gamma _2<\gamma _1<\min \{\alpha _1,\beta _1\},\\ \frac{N\chi _1\chi _2\gamma _1\gamma _2}{2}, & \text {if}~\gamma _1<\alpha _1~\text {and}~\gamma _2<\alpha _2,\\ \frac{N\chi _1\chi _2\gamma _1\gamma _2}{2}\in \Big (\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big ), & \text {if}~\frac{\chi _1}{\chi _2}\gamma _1<\gamma _2<\alpha _2,\\ \frac{N\chi _1\chi _2\gamma _1\gamma _2}{2}\in \Big (\frac{N\chi _2^2\gamma ^2_2}{2},\frac{N\chi _1^2\gamma ^2_1}{2}\Big ), & \text {if}~\frac{\chi _2}{\chi _1}\gamma _2<\gamma _1<\alpha _1,\\ \max \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \},\\ \end{array}\right. } \end{aligned}\) $\begin{matrix} χ_{1} + χ_{2} \geq \{\begin{matrix} \frac{N χ_{1}^{2} γ_{1}^{2}}{2} = min {\frac{N χ_{1}^{2} γ_{1}^{2}}{2}, \frac{N χ_{2}^{2} γ_{2}^{2}}{2}}, & if \frac{χ_{1}}{χ_{2}} γ_{1} < γ_{2} < min {α_{2}, β_{2}}, \\ \frac{N χ_{2}^{2} γ_{2}^{2}}{2} = min {\frac{N χ_{1}^{2} γ_{1}^{2}}{2}, \frac{N χ_{2}^{2} γ_{2}^{2}}{2}}, & if \frac{χ_{2}}{χ_{1}} γ_{2} < γ_{1} < min {α_{1}, β_{1}}, \\ \frac{N χ_{1} χ_{2} γ_{1} γ_{2}}{2}, & if γ_{1} < α_{1} and γ_{2} < α_{2}, \\ \frac{N χ_{1} χ_{2} γ_{1} γ_{2}}{2} \in (\frac{N χ_{1}^{2} γ_{1}^{2}}{2}, \frac{N χ_{2}^{2} γ_{2}^{2}}{2}), & if \frac{χ_{1}}{χ_{2}} γ_{1} < γ_{2} < α_{2}, \\ \frac{N χ_{1} χ_{2} γ_{1} γ_{2}}{2} \in (\frac{N χ_{2}^{2} γ_{2}^{2}}{2}, \frac{N χ_{1}^{2} γ_{1}^{2}}{2}), & if \frac{χ_{2}}{χ_{1}} γ_{2} < γ_{1} < α_{1}, \\ max {\frac{N χ_{1}^{2} γ_{1}^{2}}{2}, \frac{N χ_{2}^{2} γ_{2}^{2}}{2}}, \end{matrix}) \end{matrix}$ we prove that for any given nonnegative initial functions $u_0,v_0\in C^0(\overline{\Omega })$ $u_{0}, v_{0} \in C^{0} (\bar{Ω})$ and $w_0\in W^{1,\infty }(\Omega ),$ $w_{0} \in W^{1, \infty} (Ω),$ problem (0.1) admits a global classical solution. Moreover, if $\begin{aligned} \chi _1+\chi _2\ge \max \Big \{\frac{N\chi _1^2\gamma ^2_1}{2},\frac{N\chi _2^2\gamma ^2_2}{2}\Big \}, a_1<-\frac{1}{\gamma _1}~\text {and}~a_2<-\frac{1}{\gamma _2}, \end{aligned}$ $\begin{matrix} χ_{1} + χ_{2} \geq max {\frac{N χ_{1}^{2} γ_{1}^{2}}{2}, \frac{N χ_{2}^{2} γ_{2}^{2}}{2}}, a_{1} < - \frac{1}{γ_{1}} and a_{2} < - \frac{1}{γ_{2}}, \end{matrix}$ we prove that the problem (0.1) has a global bounded classical solution.