A functional analytic approach to uniqueness of solutions to some nonlinear Cauchy problems (Q1084257)

The author proves an abstract theorem on the uniqueness of strict solutions for nonlinear differential equations in Banach spaces and examines how to apply it to parabolic and hyperbolic systems of nonlinear partial differential equations which contain arbitrary nonlinearities in all spatial derivatives of the unknowns. The main results of the paper are Theorem 1 (Uniqueness Theorem), Proposition 3 and the Theorems 2-4. We mention here the following results: Theorem 2. Let M be a compact region in \({\mathbb{R}}^ n\) with smooth boundary, \(f: M\times {\mathbb{R}}^ m\to C^{\infty}{\mathbb{R}}^ m\) a strongly elliptic operator of order K, \(t>2\) \((n/p+k)\), \(Y=L^ p_ t(M,{\mathbb{R}}^ m)\cap L^ p_{k/2,0}(M,{\mathbb{R}}^ m)\), \(X=L^ p(M,{\mathbb{R}}^ m)\) F(.): \(Y\to X\) the vector field induced by f(.,.). Then there exists at most one maximally defined integral curve F(.) for each initial value in Y. Let \(f: {\mathbb{R}}^ n\times {\mathbb{R}}^{\tilde m}\to {\mathbb{R}}^ m\), \(\tilde m=m(2+2n+n(n+1)/2)\), be a map of certain type and consider the equation \[ \frac{\partial^ 2u}{\partial t^ 2}=f(t,u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial x_ 1},...,\frac{\partial u}{\partial x_ n},\frac{\partial^ 2u}{\partial t\partial x_ 1},...,\frac{\partial^ 2u}{\partial x^ 2_ n}),\quad u\in {\mathbb{R}}^ m. \] Theorem 4. Let \(t>n/2+4\), \(Y=H^ t({\mathbb{R}}^ n,{\mathbb{R}}^ m)\times H^{t-1}({\mathbb{R}}^ n,{\mathbb{R}}^ m)\), \(X=H^{t-1}({\mathbb{R}}^ n,{\mathbb{R}}^ m)\times H^{t-2}({\mathbb{R}}^ n,{\mathbb{R}}^ m)\). Let G(.): \(Y\to H^{t-2}({\mathbb{R}}^ n,{\mathbb{R}}^ m)\) be the map induced by f(.,.), and let F(.): \(Y\to X\) be the vector field defined by \(F(u,v)=(v,G(u,v))\). Then there exists at most one maximally defined integral curve for F(.) for each initial value in Y. The proofs are based on the Theorem 1, Proposition 3 and other auxiliary lemmas.