Cauchy problem in function spaces with asymptotic expansions with respect to time variable

A system of nonlinear Cauchy problem ${\mathrm{\partial }}_t{u}_i=f_i(t,x,U,{\mathrm{\nabla }}_{x}U)$ $u_i(0,x)=u_{i,0}(x)$ is studied in function spaces with asymptotic expansion with respect to $t$. To be specific, it is discussed in Borel summable or multisummable function space.It is recognized that these functions are important classes in asymptotic analysis. We study equations under the condition $\{f_i(t,x,U,P){\}}_{i=1}^m$ are in these function spaces with respect to $t$ and show $\{u_i(t,x){\}}_{i=1}^m$ have also the same summability.