<p>This paper examines the inhomogeneous Neumann boundary value problem for a high-dimensional chemotaxis-consumption model with a logistic source, under the nonlinear boundary condition <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mi>p</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$|u|^{p}$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$1 < p < \frac{3}{2}$</EquationSource> </InlineEquation>. The initial data <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$u_{0}, v_{0}$</EquationSource> </InlineEquation> are assumed to be nonnegative and satisfy <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msubsup> <mrow> <mo stretchy="false">∥</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo stretchy="false">∥</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi mathvariant="normal">∞</mi> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </msup> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mrow> <mo stretchy="false">∥</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo stretchy="false">∥</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi mathvariant="normal">∞</mi> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>8</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msubsup> <mrow> <mo stretchy="false">∥</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo stretchy="false">∥</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi mathvariant="normal">∞</mi> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mn>4</mn> </msubsup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </msup> <mo><</mo> <mi>μ</mi> </math></EquationSource> <EquationSource Format="TEX">$\frac{k(k-1)}{2(k+1)}\left ( \frac{4(k-1)(4k^{2}+n)\|v_{0}\|^{2}_{L^{\infty }(\Omega )}}{k+1} \right )^{\frac{1}{k}} + \frac{2(k + n -1)\|v_{0}\|_{L^{\infty }(\Omega )}^{2}}{k+1} \left ( \frac{8(k-1)(k+n-1)(4k^{2} + n)\|v_{0}\|_{L^{\infty }(\Omega )}^{4}}{k+1} \right )^{\frac{k-1}{2}} < \mu $</EquationSource> </InlineEquation> for some <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>k</mi> <mo>></mo> <mo movablelimits="false">max</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> <mo stretchy="false">}</mo> </math></EquationSource> <EquationSource Format="TEX">$k > \max \{1,\frac{n}{2}\}$</EquationSource> </InlineEquation>. We prove the existence of classical bounded global solutions for the system in bounded convex domains with a smooth boundary.</p>

Global Existence and Uniform Boundedness in a Chemotaxis System with Signal Consumption and Logistic Growth Under Nonlinear Boundary Conditions

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发表日期：

2026年4月14日

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Bui Le Trong Thanh,
Vo Hoang Nhat

摘要

This paper examines the inhomogeneous Neumann boundary value problem for a high-dimensional chemotaxis-consumption model with a logistic source, under the nonlinear boundary condition $| u |^{p}$ $|u|^{p}$ , where $1 < p < \frac{3}{2}$ $1 < p < \frac{3}{2}$ . The initial data $u_{0}, v_{0}$ $u_{0}, v_{0}$ are assumed to be nonnegative and satisfy $\frac{k (k - 1)}{2 (k + 1)} {(\frac{4 (k - 1) (4 k^{2} + n) {∥ v_{0} ∥}_{L^{\infty} (Ω)}^{2}}{k + 1})}^{\frac{1}{k}} + \frac{2 (k + n - 1) {∥ v_{0} ∥}_{L^{\infty} (Ω)}^{2}}{k + 1} {(\frac{8 (k - 1) (k + n - 1) (4 k^{2} + n) {∥ v_{0} ∥}_{L^{\infty} (Ω)}^{4}}{k + 1})}^{\frac{k - 1}{2}} < μ$ $\frac{k(k-1)}{2(k+1)}\left ( \frac{4(k-1)(4k^{2}+n)\|v_{0}\|^{2}_{L^{\infty }(\Omega )}}{k+1} \right )^{\frac{1}{k}} + \frac{2(k + n -1)\|v_{0}\|_{L^{\infty }(\Omega )}^{2}}{k+1} \left ( \frac{8(k-1)(k+n-1)(4k^{2} + n)\|v_{0}\|_{L^{\infty }(\Omega )}^{4}}{k+1} \right )^{\frac{k-1}{2}} < \mu $ for some $k > max {1, \frac{n}{2}}$ $k > \max \{1,\frac{n}{2}\}$ . We prove the existence of classical bounded global solutions for the system in bounded convex domains with a smooth boundary.