<p>We consider a parabolic-ODE-parabolic chemotaxis system with radially symmetric initial data in a two-dimensional disk under the 0-Neumann boundary condition. Although our system shares similar mathematical structures as the Keller–Segel system, the remarkable characteristic of the system we consider is that its solutions cannot blow up in finite time. In this paper, fousing on blowup solutions in infinite time, we confirm concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system have a singularity like a Dirac delta function in infinite time. This means that there exist a time sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{t_k\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, a weight <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m \ge 8\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>8</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>, and a nonnegative function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f \in L^1(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ87"> <EquationSource Format="TEX">$\begin{aligned} u(\cdot ,t_k) {\mathop {\rightharpoonup }\limits ^{*}} m \delta (0) + f\ \textrm{as}\ t_k \rightarrow \infty . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mover> <mo>⇀</mo> <mrow> <mrow /> <mo>∗</mo> </mrow> </mover> <mi>m</mi> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mspace width="4pt" /> <mtext>as</mtext> <mspace width="4pt" /> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo stretchy="false">→</mo> <mi>∞</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>We highlight this result is obtained by showing uniform-in-time boundedness of some energy functional. Moreover, we study whether <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m = 8\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>8</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m > 8\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>8</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>, which is an open problem in the Keller–Segel system. It is proved that the weight <i>m</i> of a delta function singularity is larger than <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$8\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>8</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> under a specific assumption associated with a Lyapunov functional. This finding suggests the relationship between solutions blowing up in infinite time and an unboundedness of a Lyapunov functional, which contrasts with the Keller–Segel system.</p>

Concentration phenomena to a chemotaxis system with indirect signal production

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2025年10月10日

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Yuri Soga

摘要

We consider a parabolic-ODE-parabolic chemotaxis system with radially symmetric initial data in a two-dimensional disk under the 0-Neumann boundary condition. Although our system shares similar mathematical structures as the Keller–Segel system, the remarkable characteristic of the system we consider is that its solutions cannot blow up in finite time. In this paper, fousing on blowup solutions in infinite time, we confirm concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system have a singularity like a Dirac delta function in infinite time. This means that there exist a time sequence $\{t_k\}$ ${t_{k}}$ , a weight $m \ge 8\pi $ $m \geq 8 π$ , and a nonnegative function $f \in L^1(\Omega )$ $f \in L^{1} (Ω)$ such that $\begin{aligned} u(\cdot ,t_k) {\mathop {\rightharpoonup }\limits ^{*}} m \delta (0) + f\ \textrm{as}\ t_k \rightarrow \infty . \end{aligned}$ $\begin{matrix} u (\cdot, t_{k}) \overset{*}{⇀} m δ (0) + f as t_{k} \to \infty . \end{matrix}$ We highlight this result is obtained by showing uniform-in-time boundedness of some energy functional. Moreover, we study whether $m = 8\pi $ $m = 8 π$ or $m > 8\pi $ $m > 8 π$ , which is an open problem in the Keller–Segel system. It is proved that the weight m of a delta function singularity is larger than $8\pi $ $8 π$ under a specific assumption associated with a Lyapunov functional. This finding suggests the relationship between solutions blowing up in infinite time and an unboundedness of a Lyapunov functional, which contrasts with the Keller–Segel system.