We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f
(u)dx)2,
with Robin boundary conditions. It is known that there exists a critical
value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and
the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary
solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using
comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2
for 0 < (λ−λ*) [Lt ] 1,
where tu is a constant. A numerical estimate is obtained
using a Crank-Nicolson scheme.
The method of non-local transformations: Applications to blow-up problems
