Fixed Point Theorem Based Solvability of 2-Dimensional Dissipative Cubic Nonlinear Klein-Gordon Equation

Md Asaduzzaman,Adem Kilicman,Siti Hasana Sapar,Md Zulfikar Ali

doi:10.3390/math8071103

Md Asaduzzaman, Adem Kilicman + Show 2 more

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https://doi.org/10.3390/math8071103

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Abstract

The purpose of this article is to establish the solvability of the 2-Dimensional dissipative cubic nonlinear Klein-Gordon equation (2DDCNLKGE) through periodic boundary value conditions (PBVCs). The analysis of this study is founded on the Galerkin’s method (GLK) and the Leray-Schauder’s fixed point theorem (LS). First, the GLK method is used to construct some uniform priori estimates of approximate solution to the corresponding equation of 2DDCNLKGE. Finally, the LS fixed point theorem is applied to obtain the efficient and straightforward existence and uniqueness criteria of time periodic solution to the 2DDCNLKGE.

Highlights

The nonlinear Klein-Gordon equation (NLKGE for short) has been obtained by a modification of nonlinear Schrödinger equation i∂t ψ = − 12 ∂x 2 ψ + k ψ ψ, where ψ(x, t) is a complex field
Certain nonlinear physical systems expressed with nonlinear partial differential equations (NLPDEs) may be transformed into nonlinear ordinary differential equations by using traveling wave transformations, and the travelling wave solutions of these NLPDEs is analogous to the exact solutions of corresponding nonlinear ordinary differential equations
There is a certain focus on the uniqueness of the time-periodic solution of 2DDCNLKGE given by Equation (5), applying the Galerkin’s method (GLK) method and the Leray-Schauder’s fixed point theorem (LS) fixed point theorem

Summary

Introduction

In 2004, Gao and Guo [10] established solvability of the time-periodic solution of a 2D dissipative quadratic NLKG equation given by (2) with time periodic boundary value conditions using the GLK method [11,12] and the LS fixed point theorem [13]. There is a certain focus on the uniqueness of the time-periodic solution of 2DDCNLKGE given by Equation (5), applying the GLK method and the LS fixed point theorem. Inspired by the above-mentioned works in this paper, we establish a solvability for the following 2DDCNLKGE with PBVCs applying the GLK method and the LS fixed point theorem: utt − ∆u + αut + βu + γ|u|3 u = f (x, t), x ∈ Ω, u(x + L, t) = u(x, t), x ∈ Ω,.

Preliminary Notes

Existence of Uniform Priori Estimates for the Solution of 2DDCNLKGE

Solvability of Periodic Solution to the 2DDCNLKGE

Conclusions