Summability of the formal power series solutions of a certain class of inhomogeneous nonlinear partial differential equations with a single level
The work is devoted to the study of a family of inhomogeneous nonlinear Cauchy problems of the form \[\partial_{t}^{\kappa}u-\sum_{i\in\mathcal{K}}\sum_{q\in Q_i}P_{i,q}(t,x,u)\partial_t^i\partial_x^qu=\tilde{f}(t,x),\] under initial conditions \(\partial_t^ju(0,x)=\varphi_j(x)\), for \(0\le j\le \kappa-1\), for some \(\kappa\ge1\), and where \(Q_i\) (resp. \(\mathcal{K}\)) is a nonempty subset of \(\mathbb{N}\) for all \(i\) (resp. \(\{0,\ldots,\kappa-1\}\)). \(P_{i,q}\) stands for a polynomial in its last variable with analytic coefficients in a product of discs at the origin. The inhomogeneity \(\tilde{f}\) is a formal power series in \(t\) with coefficients in a common neighborhood of the origin, which coincides with the domain of holomorphy of the initial condition. Under the assumption of the existence of a single slope of the Newton polygon associated to the problem, the author proves summability of the unique formal solution of the main problem under study, say \(\tilde{u}(t,x)\), iff the inhomogeneity and the formal series \(\partial_x^n\tilde{u}(t,0)\) are so along the same direction. The sum satisfies the same equation with the inhomogeneity substituted by the corresponding sum (Theorem 3.8). The proof of the result is based on tight upper estimates and technical results applied to an auxiliary equation leading to the conclusion.
inhomogeneous partial differential equation
scientific article; zbMATH DE number 7471739
scientific article