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.