<p>In this paper, we study the global boundedness of solutions to a two-species chemotaxis system with nonlinear resource consumption <Equation ID="Equ68"> <EquationSource Format="TEX">\(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla w),\,\,&amp; x\in \Omega ,\,\,t&gt;0, \\ v_t=\Delta v-\xi \nabla \cdot (v\nabla w)+\mu (v-v^l), &amp; x\in \Omega ,\,\,t&gt;0, \\ w_t=\Delta w-\frac{u+v}{(1+u+v)^\gamma }w, &amp; x\in \Omega ,\,\,t&gt;0, \end{array}\right. } \end{aligned}\)</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> <mi>χ</mi> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>t</mi> <mo>&gt;</mo> <mn>0</mn> <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> <mi>ξ</mi> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>v</mi> <mi>l</mi> </msup> <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> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>t</mi> <mo>&gt;</mo> <mn>0</mn> <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> <mfrac> <mrow> <mi>u</mi> <mo>+</mo> <mi>v</mi> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi>γ</mi> </msup> </mfrac> <mi>w</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>t</mi> <mo>&gt;</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in the smooth bounded domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Omega \subset \mathbb {R}^n\)</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> </math></EquationSource> </InlineEquation> with homogeneous Neumann boundary conditions, where the parameters <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\chi ,\,\xi ,\,l,\, \gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>ξ</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>l</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation> are positive constants. We prove that the system possesses a unique global bounded classical solution under a sufficient condition. Moreover, the large-time behavior of the solution is also investigated.</p>

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Global small-data solutions to a two-species chemotaxis system with nonlinear resource consumption

  • Chun Wu,
  • Jiajia Chen

摘要

In this paper, we study the global boundedness of solutions to a two-species chemotaxis system with nonlinear resource consumption \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla w),\,\,& x\in \Omega ,\,\,t>0, \\ v_t=\Delta v-\xi \nabla \cdot (v\nabla w)+\mu (v-v^l), & x\in \Omega ,\,\,t>0, \\ w_t=\Delta w-\frac{u+v}{(1+u+v)^\gamma }w, & x\in \Omega ,\,\,t>0, \end{array}\right. } \end{aligned}\) u t = Δ u - χ · ( u w ) , x Ω , t > 0 , v t = Δ v - ξ · ( v w ) + μ ( v - v l ) , x Ω , t > 0 , w t = Δ w - u + v ( 1 + u + v ) γ w , x Ω , t > 0 , in the smooth bounded domain \(\Omega \subset \mathbb {R}^n\) Ω R n with homogeneous Neumann boundary conditions, where the parameters \(\chi ,\,\xi ,\,l,\, \gamma \) χ , ξ , l , γ are positive constants. We prove that the system possesses a unique global bounded classical solution under a sufficient condition. Moreover, the large-time behavior of the solution is also investigated.