Advances in Nonlinear Analysis (May 2014)
A Liouville comparison principle for solutions of semilinear parabolic inequalities in the whole space
Abstract
We obtain a new Liouville comparison principle for 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(*)$u_t -{\mathcal {L}}u- |u|^{q-1}u\ge v_t -{\mathcal {L}}v- |v|^{q-1}v\qquad (*)$ in the whole space ℝ×ℝn${{\mathbb {R}} \times \mathbb {R}^n}$. Here, n≥1${n\ge 1}$, q>1${q>1}$ and ℒ=∑i,j=1n∂∂xiaij(t,x)∂∂xj,$ {\mathcal {L}}=\sum _{i,j=1}^n\frac{\partial }{{\partial }x_i}\biggl [ a_{ij}(t, x) \frac{\partial }{{\partial }x_j}\biggr ],$ where aij(t,x)${a_{ij}(t,x)}$, i,j=1,...,n${i,j=1,\ldots ,n}$, are functions that are defined measurable and locally bounded in ℝ×ℝn${{\mathbb {R}} \times \mathbb {R}^n}$, and such that aij(t,x)=aji(t,x)${a_{ij}(t,x)=a_{ji}(t,x)}$ and ∑i,j=1naij(t,x)ξiξj≥0$ \sum _{i,j=1}^n a_{ij}(t,x)\xi _i\xi _j\ge 0 $ for almost all (t,x)∈ℝ×ℝn${(t,x)\in {\mathbb {R}} \times \mathbb {R}^n}$ and all ξ∈ℝn${\xi \in \mathbb {R}^n}$. 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¬≡v${{u\lnot \equiv v}}$) weak solutions to (*) in the whole space ℝ×ℝn${{\mathbb {R}} \times \mathbb {R}^n}$, depend on the behavior of the coefficients of the operator ℒ${\mathcal {L}}$ at infinity and coincide with those obtained for solutions of (*) in the half-space ℝ+×ℝn${{\mathbb {R}}_+\times {\mathbb {R}}^n}$. As direct corollaries we obtain new Liouville-type theorems for non-negative weak solutions u of inequality (*) in the whole space ℝ×ℝn${{\mathbb {R}} \times \mathbb {R}^n}$ in the case when v≡0${v\equiv 0}$. All the results obtained are new and sharp.
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