Mathematics and Computer Science, Volume 1

4Blow Up of the Higher-Order Kirchhoff-Type System With Logarithmic Nonlinearities

Nazlı Irkil* and Erhan Pişkin

Department of Mathematics, Dicle University, Diyarbakır, Turkey

Abstract

This contribution investigates the solution of high-order Kirchhoff-type system with logarithmic nonlineraities. Under the appropriate assumptions, we establish the global nonexistence of the solution at low initial energy level E (0) < d.

Keywords: Blow up, system of higher-order, Kirchhoff type equation

4.1 Introduction

This article deals with the following nonlinear higher-order Kirchhoff equations with logarithmic nonlinearities and with nonlinear damping term

(4.1)

with initial data for x ∈ Ω

and boundary value for x ∈ ∂Ω, t ≥ 0 and k = 0, 1, 2, ... m − 1,

where p ≥ 2γ + 2 are real numbers and m ≥ 1 are nonnegative integers. The Kirchhoff term M (s) = β₁ + β₂s^γ, γ > 0, β₁ ≥ 1, β₂ ≥ 0. We will take β₁ = β₂ = 1 for simplificity. Ω ⊂ Rⁿ is a regular and bounded domain with smooth boundary ∂Ω. r indicates the unit outward normal vector on ∂Ω, and shows the kth order normal derivation. D is the ...