Global existence and asymptotic behavior of solutions to a semilinear parabolic equation on Carnot groups

In this paper we consider the semilinear parabolic equation Sigma(m)(ij=1) a(ij)X(j)X(j)u - partial derivative(left perpendicular)u + Vu(p) = 0 with a general class of potentials V = V(xi, t), where A = {a(ij)}(i,j) is a positive definite symmetric matrix and the X-j's denotes a system of left-invariant vector fields on a Carnot group G. Based on a fixed point argument and by establishing some new estimates involving the heat kernel, we study the existence and large-time behavior of global positive solutions to the preceding equation.