Senba, Takasi

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Title Blowup of solutions to some systems related to Keller-Segelsystem (Theory of Bio-Mathematics and It's Applications)Author(s)Senba, TakasiCitation数理解析研究所講究録 (2005), 1432: 43-48Issue Date2005-05URL http://hdl.handle.net/2433/47413RightType Departmental Bulletin PaperTextversionpublisherKyoto University43Blowup of solutionsto some systems related to Keller-Segelsystem(Keller-Segel )(Takasi Senba)Faculty of Engineering, University of MiyazakiKeller and Segel introduced a parabolic system to describe the aggrega-tion of cellular slime molds.Keller and Segel introduced a parabolic system to describe the aggrega-tion of cellular slime molds. They introduce the system according to thefollowing hypothesis.Cells sense the gradient of chemical concentration and move toward higherconcentration. We refer to the phenomenon as chemotaxis. Cells producethe chem ical substance.Then, the chemical substance is an attractant.The following system is so called Keller-Segel system.(KS) $\{$$\frac{vu_{t}\partial^{t}u}{\partial\nu}=\frac{\partial vv}{\partial\nu}=0\mathrm{o}\mathrm{n}\partial\Omega \mathrm{x}[0,\infty)=\Delta-v+u\mathrm{i}\mathrm{n}\Omega \mathrm{x}[0,\infty)=\nabla\cdot(\nabla u-u\nabla v)\mathrm{i}\mathrm{n}\Omega \mathrm{x}[,0,’\infty)$,$u(\cdot,0)=u_{0},v(\cdot,0)=v_{0}$ in $\Omega$ .Here, $\Omega\subseteq \mathrm{R}^{2}$ is a bounded domain with smooth boundary $\partial\Omega$ and $u\mathrm{o}(\not\equiv 0)$and $v_{0}$ are smooth and nonnegative in 0.$u(x, t)$ represents the density of cells at $(x, t)$ and $v(x, t)$ represents thechemical concentration at $(x, t)$ .In this system, cells sense the gradient of chemical substance, move to-ward higher concentration and produce the chemical substance. Then, thedirection of flow due to chemotaxis is almost opposite to one of diffusion.If we can neglect the diffusion of cells and chemical substance. At a place,some cells exist and produce some chemical substance. Then, some cells sensethe gradient of chemical concentration, and move toward the place. Then1432 2005 43-4844at the place the density of cells increases and much chemical substance isproduced. By repeating this story, the aggregation of cells occurs.However, in this explanation the diffusion is neglected. Then, when theintensity of chemotaxis is more strong than one of diffusion the aggrega-tion occurs. I think that the blowup of solutions to mathematical modelcorresponds to the aggregation.The followings are fundamental mathematical results.Proposition 1 The system (KS) has the unique classical solution (u, v)for given initial conditions $u_{0}$ and $v_{0}$ in $\Omega\cross$ $(0, T_{\max})$ and the solution satisfiesthat$u(x, t)>0$ and $v(x, t)>0$ for $(x, t)\in\overline{\Omega}\mathrm{x}$ $(0, T_{\max})$and that$\oint_{\Omega}u(x, t)dx=\int_{\Omega}u_{0}(x)dx\equiv$ A.If $T_{\max}<\infty$ , it holds that$\lim_{tarrow\max}\mathrm{m}\mathrm{a}_{\frac{\mathrm{x}}{\Omega}}u(x, t)=\lim_{tx\inarrow T_{\max}}\mathrm{m}\mathrm{a}_{\frac{\mathrm{x}}{\Omega}}v(x, t)=\infty x\in$.$Here_{f}T_{\max}$ denotes the maximal existence time.If $\lim_{tarrow T}(x\mathrm{m}\mathrm{a}_{\frac{\mathrm{x}}{\Omega}}u(x, t))\in=\infty$ , we say thatthe solution blows uP at the time $T$ andthat $T$ is the blowup time.If there exist two sequences $\{q_{n}\}\subset$$\overline{\Omega}$ and $\{t_{n}\}$ $\subset$ $(0, T)$ such that$\lim_{narrow\infty}(q_{n}, t_{n})=(q, T)$ and $\lim_{\tau\iotaarrow\infty}u(q_{n}, t_{n})=$+00, we say that the point $q$ is a blowuppoint. Moreover, 5 denotes the set ofblowup points.Before the explanation of our mathe-matical results, I shall describe our conjec-ture of blowup solutions. The conjectureis one of our goal. And our mathematicalresults mentioned late are evidences that (Balls representsthis conjecture is true. delta functions $4\pi\delta_{q}$ or $8\pi\delta_{q}.$)Our Conjecture: Suppose that thesolution blows up in finite time $T_{\max}$ .Figure 1: Image of Our Conjectur$\mathrm{e}$45Then, the number of blowup points is fi-nite, and the solution satisfies the follow-ing.$u( \cdot, t)arrow\sum_{q\in B}m_{*}(q)\delta_{q}+f$in $\mathcal{M}(\overline{\Omega})$as $t\prec T_{\max}$ .Here, $m_{*}(q)=\{$the point $q$ and$8\pi$ if $q\in\Omega$ , , $\delta_{q}$ is the delta function whose suppose is$4\pi$ if $q\in\partial\Omega$ ,$f$ is a nonnegative function belonging in $L^{1}(\Omega)\cap C(\overline{\Omega}\backslash B)$For our conjecture, I can get the following results.Theorem 1 (Herrero-Velazquez) Let $\Omega=$ {x $\in \mathrm{R}^{2}||x|<L\}$ and$L\in(0, \infty)$ . Then, there exists a radial solution to (KS) satisfying$u(\cdot, t)arrow \mathrm{S}\pi\delta_{0}+f$ in $\mathcal{M}(\overline{\Omega})$ ,as $tarrow T_{\max}(<\infty)$ ,where $f$ is a nonnegative and radial function belonging in $L^{1}(\Omega)\cap C(\overline{\Omega}\backslash \{0\})$ .Theorem 1 says only the existence of a blowup solution having the deltafunction singularity. Then, we do not know whether all blowup solutionshave such a delta function singularities or not.Theorem 2 (Nagai-Senba-Suzuki) Suppose that the $solu_{b}^{l}ion$ to (KS)blows uP in the finite time and that the blowup points are finite. Then, it holdsthat$u( \cdot, t)arrow\sum_{q\in B}m(q)\delta_{q}+f$in $\mathcal{M}(\overline{\Omega})$ as $tarrow T_{\max}$ ,where $m(q)\geqq m_{*}(q)$ and $f$ is a nonnegative function belonging in $L^{1}(\Omega)\cap$ $C(\overline{\Omega}\backslash B)$ .Theorem 2 say that all blowup solutions have delta function singularities.However, if our conjecture is true, the assumption of finiteness of blowuppoints is not necessary, and the constants $m(q)$ must be equal to $8\pi$ or $4\pi$ .In order to consider the finiteness of blowup points and the decision ofthe quantity $m(q)$ , we consider the following system.(N) $\{$$u_{t}=\nabla$ . (Vu $-u\nabla v$ ) in $\Omega \mathrm{x}[0, \infty)$ ,$0=\triangle v-v+u$ in $\Omega \mathrm{x}$ $[0, T_{\max})$ ,$\frac{\partial u}{\partial\iota\prime}=\frac{\partial v}{\partial\nu}=0$ on $\partial\Omega \mathrm{x}[0, T_{\max})$ ,$u(\cdot, 0)=u_{0}$ in $\Omega$ .48This system is introduced as a simplified system of Keller-Segel systemby Professor Nagai, Then, we refer this system as Nagai system. The differ-ence between Nagai system and Keller-Segel system is the second equation.Since the second equation of Nagai system is an elliptic equation, the initialcondition $v_{0}$ is not necessary.The analysis of solutions to Nagai system is more easy than one of Keller-Segel system. And we believe that the structure of solutions to Nagai systemis similar as one of Keller-Segel system. Then, we investigate the blowupsolutions to Nagai system.Theorem 3 (Suzuki) If the solution to (N) blows up in the finite time,then the solution satisfies$u( \cdot, t)arrow\sum_{q\in \mathcal{B}}m_{*}(q)\delta_{q}+f$in At $(\overline{\Omega})$ as $tarrow T_{\max}$where $f$ is a nonnegative function belonging in $L^{1}(\Omega)\cap C(\overline{\Omega}\backslash B)$ .We consider the behavior of blowup solutions until now. Next, we shallconsider the condition of blowup of solutions for Nagai system.Theorem 4(Suppose the following (i) or (ii).(i) $\oint_{\Omega}u_{0}(x)dx>8\pi$ and $\int_{\Omega}u_{0}(x)|x-q|^{2}dx\ll 1$ for some $q\in\Omega$ .(ii) $\oint_{\Omega}u_{0}(x)dx>4\pi$ and $\oint_{\Omega}u_{0}(x)|x$ $-q|^{2}dx\ll 1$ for some $q$ {: $\partial\Omega$ , and theboundary is line in the neighbourhood of the point $q$ .Then, the solution to (N) blows up in the finite time.By Theorem 3, we obtain that the blowup solutions found by Theorem 4have delta function singularities.The conclusion about blowup condition is as follows.If total mass is less than $4\pi$ , the solution can not blowup. In the radialcase, if total mass is less than $8\pi$ , the solution can not blowup.If more than $4\pi$ mass concentrate near a point on the boundary, thesolution blows up. If more than $8\pi$ mass concentrate near a point in thedomain, the solution blows up.If the solution blows up in finite time, the delta function appears at eachblowup point.47Next, we describe the relation between the structure of solutions and theassumption of the intensity of chemotaxis. In order to describe the relation,we consider the following general case.$(N)_{\varphi}\{$$u_{t}=\nabla$ . (Vu $-u\nabla\varphi(v)$ ) in $\Omega\cross$ $[0, T_{\max})$ ,$0=\Delta v-v+u$ in $\Omega \mathrm{x}[0, T_{\max})$ ,$\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0$ on $\partial\Omega \mathrm{x}[0, T_{\max})$ ,$u(\cdot, 0)=u_{0}$ in $\Omega$ .In the case where $\varphi(v)=v$ , the system is Nagai system.The first equation represents the change of density of cells. The term $\nabla u$represents the flow due to diffusion of cells, and the term $u\nabla\varphi(v)$ representsthe flow due to chemotaxis. Then, we assume that the flow due to chemotaxisis expressed by the term $u\nabla\varphi(v)$ , by using a function $\varphi$ .We refer to the function $\varphi$ as the sensitivity function. Since the chemicalsubstance is the attractant, then the differential of $\varphi$ must be positive.The typical examples of sensitivity functions are $\log v$ , $v^{p}(p>0)7 \frac{v}{1+v}$and so on.Then, in order to investigate the relation between the intensity of chemo-taxis and the structure of blowup solutions, we investigate the relation be-tween the sensitivity function $\varphi$ and the structure of blowup solutions.Here, we treat only the radial case.Theorem 5 (Nagai-Senba) Let $\Omega=\{x\in \mathrm{R}^{2}||x|<L\}(0<L<\infty)$and $u_{0}$ be positive and radial in $\overline{\Omega}$ .(i) Suppose that $\varphi(v)=\log v$ or $v^{p}(0<p<1)$ . Then, the radial solution to$(N)_{\varphi}$ exists globally in time and satisfies$\sup\{u(x, t) |(x, t)\in\overline{\Omega}\mathrm{x}[0, \infty)\}<\infty$ .(ii) Suppose that $\varphi(v)=v^{p}(1<p)$ . if $u_{0}$ satisfies$\oint_{\Omega}u_{0}(x)dx>0$ and $\int_{\Omega}|x|^{2}u_{0}(x)dx\ll 1$ ,the radial solution blows up in finite time.Then, the relation between the sensitivity function and blowup of solu-tions is as follows.In the case where $\varphi(v)=v$ , it holds that the differential of $\varphi(v)$ is equalto 1. Then, if the total mass is less than $4\pi$ , the blowup can not occur. Ifthe total mass is more than $4\pi$ , the blowup can occur.48In the case where $\varphi(v)=\log v$ or $v^{p}(0<p<1)$ , it holds that $\lim_{varrow\infty}\varphi’(v)=$$0$ . Then, the intensity of chemotaxis is more weak than one of $\varphi(v)=v$ , when$v$ is large. In this case, Theorem 5 says that blowup can not occur,In the case where $\varphi(v)=v^{p}(p>1)$ , it holds that $\lim_{varrow\infty}\varphi’(v)=\infty$.Then, the intensity of chemotaxis is more strong than one of $\varphi(v)=v$ , when$v$ is large. In this case, Theorem 5 says that blowup can occur, even if thetotal mass is small.That is to say, the structure of solutions change, when the system changes

Keller-Segel系に関係する方程式の解の爆発について (生物数学の理論とその応用)

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Keller-Segel系に関係する方程式の解の爆発について (生物数学の理論とその応用)

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