A Liouville comparison principle for solutions of semilinear parabolic inequalities in the whole space

We obtain a new Liouville comparison principle for weak solutions (u, v) of semilinear parabolic second-order partial differential inequalities of the form u(t) - Lu - vertical bar u vertical bar(q-1)u >= v(t) - Lv - vertical bar v vertical bar(q-1)v in the whole space R x R-n. Here, n >= 1 , q > 1 and L = Sigma(i,) (n)(j=1) partial derivative/partial derivative x(i) [a(ij)(t, x)partial derivative/partial derivative x(j)], where a(ij)(t, x), i, j = 1, ... , n , are functions that are defined, measurable and locally bounded in R x R-n, and such that a(ij)(t, x) = a(ji)(t, x) and Sigma(n)(i, j=1) a(ij)(t, x)xi(i)xi(j) >= 0 for almost all (t, x) is an element of R x R-n and all xi is an element of R-n. We show that the critical exponents in the Liouville comparison principle obtained, which are responsible for the non-existence of non-trivial (i.e., such that u not equivalent to v ) weak solutions to (*) in the whole space R x R-n, depend on the behavior of the coefficients of the operator L at infinity and coincide with those obtained for solutions of (*) in the half-space R+ x R-n . As direct corollaries we obtain new Liouville-type theorems for non-negative weak solutions u of the inequality (*) in the whole space R x R-n in the case when v equivalent to 0 . All the results obtained are new and sharp.