On third order hyperbolic equations

Abstract

We shall consider the third order hyperbolic equation in [0,T] ×Rx $\left\{ {_{u\left( {0,x} \right) = u_0 \left( x \right), ut\left( {0,x} \right) = u_1 \left( x \right), u_{tt} \left( {0,x} \right) = u_2 \left( x \right),}^{u_{ttt} - t^\alpha u_{txx} + t^\beta u_{ttx} + t^\eta u_{tx} + t^\sigma u_{tt} + t^\mu u_x + t^\omega u_t + t^\theta u = 0} } \right.$ where α≥ 2, Β ≥ 1, η ≥ 0, λ ≥ 0, Σ≥ 0, Μ ≥ 0, Ω ≥ 0 and θ≥ 0 are integers. We prove that the Cauchy problem (1) is Gevrey well-posed.