A Liouville comparison principle for solutions of quasilinear singular parabolic inequalities

We obtain a Liouville comparison principle for entire weak solutions (u, v) of quasilinear singular parabolic second-order partial differential inequalities of the form u(t) - A(u) - vertical bar u vertical bar(q-1) u >= v(t) - A(v) - vertical bar v vertical bar(q-1) v on the set S-tau = (tau, +infinity) x R-n, where q > 0, n >= 1, tau is a real number or tau = -infinity, and the differential operator A satisfies the alpha-monotonicity condition. Model examples of the operator A in our study are the well-known p-Laplacian operator defined by the relation Delta(p)(w) = div(x)(vertical bar del(x)w vertical bar(p-2)del(x)w) and its well-known modification defined by (Delta) over bar (p)(w) = Sigma(n)(i=1) partial derivative/partial derivative x(i) (vertical bar partial derivative w/partial derivative x(i vertical bar)(p-2) partial derivative/partial derivative x(i)).