A Liouville comparison principle for solutions of quasilinear singular parabolic inequalities

摘要

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) - |u|~(q-1)u ≥v_t - A(v) - |v|~(q-1)v on the set S_τ = (τ, +∞) × R~n, where q > 0, n > 1, r is a real number or r = -oo, and the differential operator A satisfies the a-monotonicity condition. Model examples of the operator A in our study are the well-known p-Laplacian operator defined by the relation ?_p(w) = div_x.(|?_xw|~(p-2)?_xw) and its well-known modification defined by △_p(w) = ∑_(i=0)~n ?/(?x_i) (|(?ω)/(?x_i)|~(p-2) (?ω)/(?x_i).