A linear recursive scheme associated with the love equation

Abstract

This paper shows the existence of a unique weak solution of the following Dirichlet problem for a nonlinear Love equation $$ \left\{ \begin{aligned} &u_{tt}-u_{xx}-\varepsilon u_{xxtt}=f(x,t,u,u_{x},u_{t},u_{xt}), \quad 0<x<L,~0<t<T, \\ &u(0,t)=u(L,t)=0, \\ &u(x,0)=\tilde{u}_{0}(x),\qquad u_{t}(x,0)= \tilde{u}_{1}(x), \end{aligned} \right. $$ where e>0 is a constant and \(\tilde{u}_{0}\), \(\tilde{u}_{1}\), f are given functions. This is done by combining the linearization method for a nonlinear term, the Faedo–Galerkin method and the weak compactness method.