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The global smooth solutions of Cauchy problems for hyperbolic equation of Monge-Ampère type. (English) Zbl 0830.35082

The author studies the Cauchy problem for the hyperbolic Monge-Ampère equation

z xx z tt -z xt 2 =-k(x,t)in(x,t) 2 : t > 0,z(x,0)=ϕ(x)ε -1 ,z t (x,0)=ψ(x),

where all functions are smooth, k>0, and ε>0 is a parameter. Under certain technical conditions on ϕ,ψ and k, it is shown that there is a unique smooth global solution for all sufficiently small ε>0. In addition, the author gives a condition on ϕ and ψ which ensures that solutions with small ε blow up in finite time.

35L70Nonlinear second-order hyperbolic equations
35L15Second order hyperbolic equations, initial value problems