BLOW-UP (May 31, 2000)

We now turn to the study of blow up. From several references [] blow-up does not occur in one dimension. Thus let us take the 2-dimensional equations. Since our moving mesh method is only designed for one dimension let us simplify the 2-dimensional model by requiring that the solutions is radially symmetric.

Several points have to be noticed. Only a radially symmetric profile can be incorporated. The disadvantages from our perspective is that we won't be able to observe metastability since the peak of the distribution is going to be centered at the origin (center of the domain). Thus there is not going to be any bias towards any point of the boundary and the peak will stay centered at the origin.

By considering a radially symmetric distribution it is possible to reduce the chemotactic equations (1) to the following system in polar coordinates

$$\begin{array}{rcl} u_t &=& d_1 u_{rr} - \chi (u\nabla v_r)_r... ... \mu v + \beta u + \displaystyle{1\over r} d_2 v_r, \end{array}$$ (4)

where the equation for the angle is omitted (radial symmetry) and now the spatial derivatives are with respect to the radius r. Notice the similarity between the equations of chemotaxis in one-dimension (2) and the new equations (4) for the radially symmetric case. In fact the latter are the same as the former with the extra terms terms proportional to 1/r. In fact there are this terms that are responsible for the blow-up since for small r they diverge to $+\infty$.

  

Figure: Typical plot of the maximum height of u(r,t) as a function of t

The numerical integrations of equations (4) lead to blow-up in finite time. The first suggestion of blow-up is given in figure 5 where we depict a typical run for the behaviour of the maximum of u(x,t) as time evolves. The figure suggests that the maximum of u blows to infinity (note the logarithmic scale on the vertical axis) as time approaches a critical time $t_c\approx 0.45$.

  

Figure: Maximum height of u(r,t) as a function of t_c-t. The fitted line corresponds to t_c=0.450207991, a = -1.238 and b = -6.815

The blow-up is made more evident when we plot max(u(x,t)) as a function of t_c-t in log-log coordinates. The fitted line shows that the blow up follows closely an expression of the form

$$\displaystyle\max(u(x,t))=\displaystyle{c\over (t_c-t)^\alpha}$$ (5)

where c is a constant and $\alpha\approx 1.238$.