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Optimal Liouville-type theorems for a system of parabolic inequalities

Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

E-mail Address: [email protected]

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and

Quoc Hung Phan

Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S0219199719500433Cited by:8 (Source: Crossref)

Abstract

We establish optimal Liouville-type theorems for the system of parabolic inequalities

\{\begin{matrix} u_{t} - Δ u \geq v^{p}, \\ v_{t} - Δ v \geq u^{q} \end{matrix}

and for the scalar inequality

w_{t} - Δ w \geq w^{p}

in the whole space

ℝ^{N} \times ℝ

and in

ℝ^{N} \times (0, \infty)

. Our optimal Liouville-type theorems are proved for two different classes of solutions: the nontrivial nonnegative and the positive.

Keywords:

AMSC: Primary 35B53, Primary 35R45, Primary 35K55, Secondary 35B33

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