A Cauchy–Kovalevskaja type theory in the Gevrey class for PDEs with shrinkings

This article concerns the Cauchy–Kovalevskaja type theorem for the partial differential equation∂1lu(t,x)=f(t,x,u(t,x),∂2ku(t,x),∂2pu(α(t)t,x),∂2qu(t,β(t)x)).(*)In (*) ff is continuous in (t,x,u1,…,u4)(t,x,u1,…,u4) and Gevrey of order λ(>1)λ(>1) in (x,u1,…,u4)(x,u1,…,u4). It is assumed that kk satisfies 0⩽λk⩽l0⩽λk⩽l. p,qp,q denote arbitrary fixed positive integers. α(t)α(t) and β(t)β(t) in (**) are called shrinkings, since they satisfy the conditions 0