We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A typical example of energy functional we consider is the one given by the nonparametric area integrand f(x,ξ)=√1+∥ξ∥2, which corresponds with the time-dependent minimal surface equation. We also study the asymptotic behaviour of the solutions.

The first and third authors have been partially supported by the Spanish DGICYT, Project PB98-1442. The second author acknowledges partial support by the TMR European Project “Viscosity Solutions and their Applications”, reference FMRX-CT98-0234 and the PNPGC, Project BFM 2000-0962-C02-01.