Mathematics and Computer Science, Volume 1

2Blow Up and Decay of Solutions for a Klein-Gordon Equation With Delay and Variable Exponents

Hazal Yüksekkaya* and Erhan Pişkin

Department of Mathematics, Dicle University, Diyarbakır, Turkey

Abstract

In this article, we deal with a Klein-Gordon equation with delay and variable exponents. Under appropriate conditions, we establish the blow up of solutions in a finite time. Also, we get the decay results utilizing the Komornik integral inequality.

Keywords: Blow up, decay, delay term, Klein-Gordon equation, variable exponents

2.1 Introduction

In this article, we deal with the Klein-Gordon equation with delay and variable-exponents as follows:

(2.1)

where Ω is bounded domain in Rⁿ, n ≥ 1, with ∂Ω. τ > 0 is time delay term, b ≥ 0 is a constant, m ≠ 0 is a real constant, μ₁ > 0 is a constant and μ₂ is a real number. The functions u₀, u₁, f₀ are the initial data to be specified later.

r (.) and p (.) are variable exponents which given as measurable functions on satisfy:

(2.2)

Where

and

The Klein-Gordon equation appears in many scientific applications such as quantum field theory, nonlinear ...