Existence and Nonexistence of Nontrivial Doubly Periodic Solutions of Nonlinear Telegraph Equations

Abstract

Aims/ Objectives: We discuss the existence and nonexistence of nontrivial nonnegative doubly periodic solutions for nonlinear telegraph equations
 utt-uxx+cut+a(t,x)u=λf (t,x,u) ,
 where c > 0 is a constant, λ > 0 is a positive parameter, a ∈ C(R2,R+), f ∈ C(R2 × R+,R+), and a, f are 2π-periodic in t and x. The proof is based on a known xed point theorem due to Schauder. In previous articles, a single telegraph equation or telegraph system have been widely studied, but there is relatively little research on nonlinear telegraph equations with a parameter and the nonlinearities are nonnegative. We would like do some research on this topic. We give new conclusions on the existence and nonexistence of nontrivial nonnegative doubly periodic solutions for nonlinear telegraph equations under sublinear assumptions. Study Design: Study on the existence and nonexistence of nontrivial nonnegative doubly periodic solutions. Place and Duration of Study: School of Applied Science, Beijing Information Science & Technology University, September 2020 to present.Methodology: We prove the existence and nonexistence of nontrivial nonnegative doubly periodic solutions by the results of Schauder's xed point theorem. Results: We give new conclusions of existence and nonexistence of nontrivial nonnegative doubly periodic solutions for the equations. Conclusion: We prove the existence and nonexistence of nontrivial nonnegative doubly periodic solutions for nonlinear telegraph equations utt − uxx + cut + a(t, x)u = λf (t, x, u), and give new conclusions.