Local solvability of degenerate partial differential equations. (Russian) Zbl 0716.35047

Kuchko, L. P.

Teor. Funkts., Funkts. Anal. Prilozh. 52, 55-59 (1989).

The author considers the system of partial differential equations (1) \((V\phi)(x)=g(x,\phi (x))\), where \(x\in {\mathbb{R}}^ n\), \(\phi: {\mathbb{R}}^ n\to {\mathbb{R}}^ m\), V is a local smooth vector field and g: \({\mathbb{R}}^ n\times {\mathbb{R}}^ m\to {\mathbb{R}}^ m\) is a local \(C^{\infty}\)-mapping.
Let \({\hat \phi}{}_ 0\) be some formal solution of (1) and let \(\phi_ 0: {\mathbb{R}}^ n\to {\mathbb{R}}^ m\) be a \(C^{\infty}\)-mapping with Taylor series in the origin of coordinates. The substitution \(\phi \to \phi +\phi_ 0\) leads to the equation
(2) \((V\phi)(x)=\tilde g(x,\phi (x))\), where \(\tilde g(x,y)=g(x,y+\phi_ 0(x))-(V\phi_ 0)(x).\)
If equation (2) has a \(C^{\infty}\)-solution \(\phi\), the equation (1) should have a \(C^{\infty}\)-solution \(\phi+\phi_ 0\). In the case of \({\hat \phi}=0\) we can say that the formal solution of (1) is restored to the local \(C^{\infty}\)-solution of (1).
The author proves two theorems:
i) Let the vector field V be quasihyperbolic of order k and the Jacobi matrix \(Q(x)=(\partial g(x,y)/\partial y)|_{y=\phi_ 0}\) be such that \(Q(x)=O(\| x\|^ k)\). Then the formal solution \({\hat \phi}{}_ 0\) is restored to the local \(C^{\infty}\)-solution of (1); and
ii) Let in equation (1) the contraction of vector field V on its central manifold be quasihyperbolic of the order of k and formal solution \({\hat \phi}{}_ 0\) be such that \(Q(x)=(\partial g(x,y)/\partial y)|_{y=\phi_ 0}=O(\| x\|^ k)\), \(x\in L_ c\). Then the formal solution of (1) is restored to the local \(C^{\infty}\)-solution.

Reviewer: I.E.Tralle

MSC:

35L80

Degenerate hyperbolic equations

Keywords:

quasihyperbolic; central manifold