Nonlinear evolution PDEs in R+ × C: existence and uniqueness of solutions, asymptotic and Borel summability properties

We consider a system of n-th order nonlinear quasilinear partial differential equations of the form ut + P(∂ x)u+ g ( x, t, {∂ xu} ) = 0; u(x, 0) = uI(x) with u ∈ Cr, for t ∈ (0, T ) and large |x| in a poly-sector S in Cd (∂ x ≡ ∂1 x1∂ j2 x2 ...∂ jd xd and j1 + ... + jd ≤ n). The principal part of the constant coefficient n-th order differential operator P is subject to a cone condition. The nonlinearity g and the functions uI and u satisfy analyticity and decay assumptions in S. The paper shows existence and uniqueness of the solution of this problem and finds its asymptotic behavior for large |x|. Under further regularity conditions on g and uI which ensure the existence of a formal asymptotic series solution for large |x| to the problem, we prove its Borel summability to the actual solution u. The structure of the nonlinearity and the complex plane setting preclude standard methods. We use a new approach, based on Borel-Laplace regularization and Écalle acceleration techniques to control the equation. In special cases motivated by applications we show how the method can be adapted to obtain short-time existence, uniqueness and asymptotic behavior for small t, of sectorially analytic solutions, without size restriction on the space variable. Correspondence to: O. Costin, Mathematics Department, Rutgers University, Busch Campus, Hill Center, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA 2 O. Costin and S. Tanveer