Affiliations: Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, MI 48107-8604, USA E-mails: [email protected]; [email protected]
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 ut−ℒu−|u|q−1u≥vt−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=Σi,j=1n ∂/∂xi [aij(t,x) ∂/∂xj]. We assume that the coefficients aij(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 aij(t,x)=aji(t,x), i,j=1,…,n, for almost all (t,x)∈S and that Σi,j=1naij(t,x)ξiξj≥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.
