Abstract: The following nonlinear hyperbolic equation is discussed in this paper:u_n+A²u+M(x,||A^2/3u||₂²)Au=0, where A=Δ+I and x∈Rⁿ. The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.

Key words: nonlinear hyperbolic equations, unbounded domain, energy estimation, fixed point method