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A linear recursive scheme associated with the Love equation. (English) Zbl 1310.35174

Summary: This paper shows the existence of a unique weak solution of the following Dirichlet problem for a nonlinear Love equation \[ \begin{cases} u_{tt}-u_{xx}-\varepsilon u_{xxtt}=f(x,t,u,u_{x},u_{t},u_{xt}), \quad 0<x<L,~0<t<T, \\ u(0,t)=u(L,t)=0, \\ u(x,0)=\tilde{u}_{0}(x),\qquad u_{t}(x,0)= \tilde{u}_{1}(x), \end{cases} \] where \(\varepsilon>0\) 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.

MSC:

35L76 Higher-order semilinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35D30 Weak solutions to PDEs
35Q74 PDEs in connection with mechanics of deformable solids
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