Bounded solutions of nonlinear hyperbolic equations with time delay

• Main Authors: Allaberen Ashyralyev, Deniz Agirseven
• Format: Article
• Language: English
• Published: Texas State University 2018-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Nonlinear hyperbolic equation; time delay; bounded solution
• Online Access: http://ejde.math.txstate.edu/Volumes/2018/21/abstr.html

We consider the initial value problem $$\displaylines{ \frac{d^{2}u}{dt^{2}}+Au(t)=f(u(t),u(t-w)), \quad t>0, \cr u(t)=\varphi (t),\quad -w\leq t\leq 0 }$$ for a nonlinear hyperbolic equation with time delay in a Hilbert space with the self adjoint positive definite operator A. We establish the existence and uniqueness of a bounded solution, and show application of the main theorem for four nonlinear partial differential equations with time delay. We present first and second order accuracy difference schemes for the solution of one dimensional nonlinear hyperbolic equation with time delay. Numerical results are also given.