Formal solutions of second order evolution equations (Q628819)

Summary: We study the initial value problem for a second order evolution equation \(\partial_tu= F(x,u,\nabla_xu,\nabla^2_xu)\), \(u|_{t=0}=u_0\), where \(F(x,u,p,q)\) is a polynomial function in variables \(u\in\mathbb R\), \(p\in\mathbb R^d\), \(q\in\mathbb R^{d^2}\) with coefficients analytic on a domain \(\Omega\subset\mathbb R^d\), \(d\geq 1\) and \(u_0\) is analytic on \(\Omega\). We construct a formal power series solution \(\widehat{u}(t,x)= \sum_{n=0}^\infty \varphi_n(x)t^n\) of the equation and prove that it satisfies Gevrey type estimates \(|\varphi_n(x)|\leq C^{n+1}n!\) for \(x\in K\Subset\Omega\) and \(n\in\mathbb N_0\), where \(C\) does not depend on \(n\). The proof is based on some combinatorial identities and estimates which may be of independent interest.