On a Liouville-type comparison principle for solutions of evolution inequalities

Abstract

The purpose of this work is to establish a Liouville-type comparison principle for solutions of higher order differential evolution inequalities of the form

in 𝕊 := {(t, x): t > 0, x ∈ ℝ ⁿ }, without any initial conditions on the solutions on the hyperplane t = 0 being involved, where L is a linear lth-order differential operator of the form

, a _α(t, x) are measurable bounded functions in 𝕊, |α| = α₁ + ··· + α _n , l ≥ 1 and n ≥ 1 are natural numbers, and q ≥ 1 is a real number. The principal examples of the operator L are the Laplacian Δ and the mth-order polyharmonic operator Δ ^m := Δ¹(Δ ^m−1), where Δ¹ := Δ and m ≥ 2 is a natural number.

Keywords:

• Liouville
• comparison principle
• evolution inequality

AMS Subject Classifications::

• Primary 35R45
• Secondary 35B53
• 35B51
• 35G05