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

Abstract

We consider a system of $n$-th order nonlinear quasilinear partial differential equations of the form $${\bf u}_t + \mathcal{P}(\partial_{\bf x}^{\bf j}){\bf u}+{\bf g} \left( {\bf x}, t, \{\partial_{\bf x}^{\bf j} {\bf u}\}) =0; {\bf {u}}({\bf x}, 0) = {\bf {u}}_I({\bf x})$$ with $\mathbf{u}\in\CC^{r}$, for $t\in (0,T)$ and large $|\mathbf{x}|$ in a poly-sector $S$ in ${\mathbb{C}}^d$ (${\partial }_{\mathbf{x}}^{\mathbf{j}}\equiv {\partial }_{x_1}^{j_1}{\partial }_{x_2}^{j_2}...{\partial }_{x_d}^{j_d}$ and $j_1+...+j_d\le n$). 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 $\mb u_I$ and $\mb u$ 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 $\mb g$ and $\mb u_I$ which ensure the existence of a formal asymptotic series solution for large $|\mb x|$ to the problem, we prove its Borel summability (and automatically its asymptoticity) to an actual solution $\mb u$.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 $t$,without size restriction on the space variable.

References in corpus (2)

• On the formation of singularities of solutions of nonlinear differential systems in antistokes directions
• Complex Singularity Analysis for a nonlinear PDE

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