On behavior of solutions of nonlinear differential equations in Hilbert space. II (Q1105110)

[Part I appeared in the North-Texas State University (1981).] The author studies equations of the type (1) \(t(du/dt)=B(,u(t))\) and \[ (2)\quad t\frac{\partial u}{\partial t}=F(t,x,u,\frac{\partial u}{\partial x},...,\frac{\partial^ mu}{\partial x^ m}) \] and obtains some results concerning quasiuniqueness of solutions, that is uniqueness up to the class of ``flat'' functions, which turn out to be the familiar test functions such that \(t^{-K}f(t)\to 0\) \((\forall K>0)\) as \(t\to 0\), (or \(\tau^ K\tilde f(\tau)\to 0\) as \(\tau\to \infty\), where \(\tilde f(\tau)=f(t)\), \(\tau =1/t.)\) Under special assumptions the author extends Agmon-Nirenberg results for linear systems to some nonlinear differential equations. The reviewer tried some physical interpretations of the class of equations discussed here and had difficulties checking the validity of a priori inequalities of the type \(\| \dot B\| \leq \gamma (t)\cdot \| B\| +\beta (t)\cdot \| u\|\) assumed by the author.