Blow-up for a degenerate parabolic equation with a nonlocal source. (Q1410256)

The authors consider the following non-local degenerate second order parabolic partial differential equation: \[ \begin{aligned} v_t(x, t) &= x^{\alpha} (v^m)_{xx} + \int_0^l v^p(x, t)dx - kv^q(x, t), \quad 0 < x < l,\;t > 0, \\ v(0, t) &= v(l, t) = 0,\quad t > 0,\\ v(x, 0) &= v_0(x), \quad 0 < x < l,\end{aligned} \] where \(0 < \alpha < 2,\) \(p\geq q > m > 1\). Under suitable conditions the existence and uniqueness of solution is established. Also, it is shown that the solution blows up in finite time, and the blow up is global. Moreover, estimates of the blow up rates as well as an example of a suitable initial value \(v_0(x)\), which satisfies the conditions for the blow up, are given.