2013/05/27 by Vasilii V. Kurta, Kurta, Vasilii V.
Computer Science · Mathematics · #35k58 35b44 35b53 35d30 35r45 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35b44 #msc:35b53 #msc:35d30 #msc:35k58 #msc:35r45
paper · pdf · doi:10.48550/arxiv.1305.6251
arXiv admin note: text overlap with arXiv:1207.2500
arxiv created 2013/05/27 · openalex publication_date 2013/05/27 · arxiv updated 2013/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain a new Liouville comparison principle for weak solutions (u,v) of semilinear parabolic second-order partial differential inequalities of the form ut -\mathcal Lu- |u|q-1u≥ vt -\mathcal Lv- |v|q-1v (*) in the whole space \mathbb E = \mathbb R × \mathbb Rn. Here, n≥ 1, q>0 and \mathcal L=∑i,j=1n\frac∂∂xi [ aij(t, x) \frac∂∂xj], where aij(t,x), i,j=1,…,n, are functions that are defined, measurable and locally bounded in \mathbb E, and such that aij(t,x)=aji(t,x) and ∑i,j=1n aij(t,x)ξiξj≥ 0 for almost all (t,x)∈ \mathbb E and all ξ∈ \mathbb Rn. 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 ≡ v) weak solutions to (*) in the whole space \mathbb E, depend on the behavior of the coefficients of the operator \mathcal L at infinity and coincide with those obtained for solutions of (*) in the half-space \mathbb R+× \mathbb Rn. As direct corollaries we obtain new Liouville-type theorems for non-negative weak solutions u of the inequality (*) in the whole space \mathbb E in the case when v≡ 0. All the results obtained are new and sharp.