  The key tool of this paper is a new Carleman estimate for an arbitrary
parabolic operator of the second order for the case of reversed time data. This
estimate works on an arbitrary time interval. On the other hand, the previously
known Carleman estimate for the reversed time case works only on a sufficiently
small time interval. First, a stability estimate is proven. Next, the
quasi-reversibility numerical method is proposed for an arbitrary time interval
for the linear case. This is unlike a sufficiently small time interval in the
previous work. The convergence rate for the quasi-reversibility method is
established. Finally, the quasilinear parabolic equation with reversed time is
considered. A weighted globally strictly convex Tikhonov-like functional is
constructed. The weight is the Carleman Weight Function which is involved in
that Carleman estimate. The global convergence of the gradient projection
method to the exact solution is proved for this functional.
