Abstract: | In this work, we study the quasilinear initial-boundary value problem $sumlimits_{i, j = 1}^m {X_i^ * A_{i j} (x, t, u)X _j u} + sumlimits_{j = 1}^m {B_j (x, t, u)X _j u + C} (x, t, u) = 0$ , in ]0,T[×Ω;u| t=0 =φ, onΩ;u=ψ, on [0,T[×?Ω, whereX={X 1, ...,X m } is a system of real smooth vector fields which is defined on an open domainM of R n , and satisfies the Hörmander's condition, Ω?M. Assume that ?Ω is non characteristic for the systemX 1,...,X m . Under some hypothesis for the boundary of domain and the elliptic structure condition for nonlinear coefficientsA ij ,B j ,C, (i, j=1, ...,m), we have proved that the existence and regularity of solution for aboveinitial-boudary value problems. |