Borel summability of divergent solutions of the Cauchy problem to non-Kowalewskian equations (Q2759789)

The author considers the non-Kowalevskian Cauchy problem NEWLINE\[NEWLINED^p_t u(t,x)= D_x^qu(t,x), \quad 1\leq p<q,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0,x)=f(x), \quad D^j_t u(0,x)=0\quad \text{if}\quad 1\leq j<p,NEWLINE\]NEWLINE where \((t,x)\in \mathbb{C}^2\) and the Cauchy data \(f(x)\) are assumed to be holomorphic in a neighborhood of the origin \(x=0\). The problem has a unique formal solution. The author discusses in detail convergence and Borel summability of the solution under different assumptions on the behaviour of \(f(x)\) at infinity. Relevant reference is given by \textit{S. Ōuchi} [J. Math. Sci., Tokyo 2, 375-417 (1995; Zbl 0860.35018)], concerning the existence of the solutions in a sectorial domain for a similar problem.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].