Nonlinear evolution PDEs in ${\mathbb{R}}^{+}\times {\mathbb{C}}^d$: existence and uniqueness of solutions, asymptotic and Borel summability properties

Abstract

We consider a system of $n$-th order nonlinear quasilinear partial differential equations of the form

with $\mathbf{u}∊{\mathbb{C}}^r$, for $t∊(0,T)$ and large $|\mathbf{x}|$ in a poly-sector $S$ in ${\mathbb{C}}^d$ (${\partial }_{\mathbf{x}}^{\mathbf{j}}\equiv {\partial }_{x_1}^{j_1}{\partial }_{x_2}^{j_2}\cdots {\partial }_{x_d}^{j_d}$ and $j_1+\cdots +j_d⩽n$). The principal part of the constant coefficient $n$-th order differential operator $\mathcal{P}$ is subject to a cone condition. The nonlinearity $\mathbf{g}$ and the functions ${\mathbf{u}}_{\mathbf{I}}$ and $\mathbf{u}$ 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 $|\mathbf{x}|$.

Under further regularity conditions on $\mathbf{g}$ and ${\mathbf{u}}_{\mathbf{I}}$ which ensure the existence of a formal asymptotic series solution for large $|\mathbf{x}|$ to the problem, we prove its Borel summability to the actual solution $\mathbf{u}$.

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.

These results are instrumental in constructive analysis of singularity formation in nonlinear PDEs with prescribed initial data, an application referred to in the paper.

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.