  We present in this paper a proof of well-posedness and convergence for the
parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional
semilinear heat equation. Since the equation we study is an evolution one, each
subproblem at each step has its own local existence time, we then determine a
common existence time for every problem in any subdomain at any step. We also
introduce a new technique: Exponential Decay Error Estimates, to prove the
convergence of the Schwarz Methods, with multisubdomains, and then apply it to
our problem.
