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Department of Mathematics and Statistics
of University of Helsinki
Formula 1. A semilinear equation with gradient blowup in finite time.
Figure 1. The Bratislava Castle.

This research project started during the FMS International Visitor
Program "Nonlinear parabolic problems" in Autumn 2005.
SOME RECENT ARTICLES

M.Fila, J.Taskinen, M.Winkler:
Convergence to a singular steady state of
a parabolic equation with gradient
blowup.
Appl.Math. Letters. 20 (2007), 578582.
It is known (from a paper by Ph.Souplet and J.L.Vazquez, 2006)
that the semilinear initialboundary problem presented in Formula 1
has a bounded solution u, global in time, which however ceases
to be a global classical solution: there exists a T, 0 < T < ∞ ,
such that first derivative u', or gradient, of u blows up at
the time T.
The blowup occurs spatially at one end of the interval.
On the other hand the equation has a singular steady state
v(x) = x ln x + x + C. Generalizing the solution u for times
t > T, a result from the above citation shows that u →
v as
t → ∞. In this paper we determine the rate of this convergence:
we show that  u v  ≤ exp(  (λ  ε) t )
where λ is the expect convergence exponent coming from a
linearization of the problem. (It comes from the first zero of
the 0th Bessel function.)
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