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Blow-up in a chemotaxis model without symmetry assumptions. (English) Zbl 1017.92006

From the introduction: Since 1970 when E. F. Keller and L. A. Segel [J. Theor. Biol. 26, 399-415 (1970)] introduced their model for the aggregaton of Dictyostelium discoideum, which is given in a simplified version by the equations \[ a_t=\nabla (\nabla a-\widetilde\chi a\nabla c),\quad c_t=k_c\Delta c-\gamma c+\widetilde\alpha a,\;x\in\Omega,\;t>0,\tag{1} \]
\[ \partial a/\partial n=\partial c/\partial n=0,\;x\in\partial \Omega,\;t>0, \quad a(0,x)=a_0(x),\;c(0,x)=c_0(x),\;x\in\Omega, \] many authors have studied the possible blow-up of the solution of this system. The function \(a(x,t)\) represents the Dictyostelium myxamoeba density in point \(x\in\Omega\) at time \(t\), and the function \(c(x,t)\) stands for the cAMP density, which attracts the myxamoebae to move positive chemotactically in the direction of a higher cAMP concentration. \(\widetilde\alpha,\widetilde\chi,k_c\) and \(\gamma\) denote positive constants. Here \(n\) denotes the outer normal vector field on \(\partial\Omega\). If one uses the transformation \[ A(t,x) =|\Omega |a(t,x)/ \int_\Omega a_0(x)dx,\;C(t,x)=\widetilde\chi \Bigl(c (t,x)-|\Omega|^{-1} \int_\Omega c(t,x) dx\Bigr),\tag{2} \] and the notation \(\alpha\chi\) instead of \(\widetilde\alpha \widetilde\chi \int_\Omega a(x,t)dx/ |\Omega|\), we get a transformed version of the Keller-Segel model. This transformed system is given by \[ A_t=\nabla \cdot (\nabla A-A\nabla C),\quad C_t=k_c\Delta C-\gamma C+\alpha \chi(A-1),\;x\in\Omega,\;t>0,\tag{3} \]
\[ \partial A/ \partial n=\partial C\partial n=0,\;x\in\partial \Omega,\;t>0,\quad A(0,x)=A_0(x) >0,\;C(0,x)= C_0(x),\;x\in\Omega, \]
\[ \int_\Omega A_0(x)dx= |\Omega|,\quad \int_\Omega C(t,x)=0,\quad t\geq 0. \] We study the possibility that solutions of the system (3) might blow up (which would imply blow-up for solutions of the system (1)). For clarity, we give the definition of solutions of the system (3), which we will refer to as blow-up solutions.
Definition: A solution of (3) blows up or is a blow-up solution provided there is a time \(T_{\max} \leq\infty\) such that \[ \limsup_{t\to T_{\max}} \bigl\|A(x,t)\bigr \|_{L^\infty (\Omega)}= \infty \quad\text{or}\quad \limsup_{t\to T_{\max}} \bigl\|C^+(x,t)\bigr \|_{L^\infty (\Omega)}=\infty, \] where \(C^+(x,t)\) denotes the positive part of the function \(C(x,t)\). If \(T_{\max}< \infty\) we say that the solution of (3) blows up in finite time, and if \(T_{\max}=\infty\) we will call it blow-up in infinite time.
We prove the existence of blow-up solutions of (3) for a smooth domain \(\Omega\subset \mathbb{R}^2\), provided \(4\pi k_c<\alpha \chi|\Omega |\) and \(\alpha\chi |\Omega |/k_c\neq 4\pi m\), where \(m\in \mathbb{N}\).

MSC:

92C17 Cell movement (chemotaxis, etc.)
35A99 General topics in partial differential equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B99 Qualitative properties of solutions to partial differential equations
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