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Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients. (English) Zbl 1036.35051
Singular first order linear partial differential equation with polynomial coefficients \[ (a+bx+c(x,y))yu_x(x,y)+(d+e(x,y))y^2u_y(x,y)+u(x,y)=f(x,y),\tag{*} \] where \(x,y\in\mathbb C\), \(a,b\) and \(d\) are complex constants, \(c(x,y)\) and \(e(x,y)\) are polynomials of at least first degree with respect to \(y\), and \(f(x,y)\) is a holomorphic function at the origin is considered. The Borel summability of the formal solutions to (*) is studied: \(u(x,y)\) is a Borel summable in \(\theta\in\mathbb R\) if there exists a holomorphic function \(U(x,y)\) such that there exists some positive constants \(C\) and \(K\) for which \(\max_{| x| \leq r}| U(x,y)-\sum_{k=0}^{n-1}u_k(x)y^k| \leq CK^nn!| y| ^n\) for \(y\in\{y: | y-Te^{i\theta}| <T\}\). The conditions under which the formal solutions to (*) are Borel summable are given by analytic continuation properties and growth estimates for the coefficients of (*).

35C20 Asymptotic expansions of solutions to PDEs
35C10 Series solutions to PDEs
35F05 Linear first-order PDEs