A Liouville comparison principle for solutions of semilinear parabolic second-order partial differential inequalities

Abstract

We obtain a new Liouville comparison principle for entire weak solutions (u,v) of semilinear parabolic second-order partial differential inequalities of the form

u_t−ℒu−|u|^q−1u≥v_t−ℒv−|v|^q−1v  (*)

in the half-space S=R₊×Rⁿ, where n≥1 is a natural number, q>0 is a real number and

ℒ=Σ_i,j=1ⁿ ∂/∂x_i [a_ij(t,x) ∂/∂x_j].

We assume that the coefficients a_ij(t,x), i,j=1,…,n, of the operator ℒ are functions that are defined, measurable and locally bounded in S. We also assume that a_ij(t,x)=a_ji(t,x), i,j=1,…,n, for almost all (t,x)∈S and that

Σ_i,j=1ⁿa_ij(t,x)ξ_iξ_j≥0

for almost all (t,x)∈S and all ξ∈Rⁿ. 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\equiv v) entire weak solutions to (*) in S, depend on the behavior of the coefficients of the operator ℒ at infinity. As direct corollaries we obtain a new Fujita comparison principle for entire weak solutions (u,v) of the Cauchy problem for the inequality (*), as well as new Liouville-type and Fujita-type theorems for non-negative entire weak solutions u of the inequality (*) in the case when v≡0. All the results obtained are new and sharp.