The global structure of Robinson–Trautman space-times is studied. When the space-time topology is
R
+
x
R
x
S
2
it is shown that all Robinson–Trautman space-time can be
C
117
extended (in the vacuum Robinson–Trautman class of metrics) beyond the
r
= 2
m
'Schwarzschild-like' event horizon; evidence is given supporting the conjecture, that no smooth extensions beyond the
r
= 2
m
event horizon exist unless the metric is the Schwarzschild one. When the space-time topology is
R
+
x
R
x
2
M
, with 2
M
a higher genus surface, and the mass parameter
m
is negative, Schwarzchild-like event horizons are shown to occur. The Proofs of these results are based on the derivation of a detailed asymptotic expansion describing the long-time behaviour of the solutions of a nonlinear parabolic equation.
Long-time behaviour of doubly nonlinear parabolic equations