Logarithmic convexity, first order differential inequalities and some applications

Abstract:

Let, for $t \in [0,T)(T < \infty ),D(t)$ be a dense linear subspace of a Hilbert space $H$, and let $M(t)$ and $N(t)$ be linear operators (possibly unbounded) mapping $D(t)$ into $H$. Let $f:[0,T) \times H \to H$. We give sufficient conditions on $M,N$ and $f$ in order to insure uniqueness and stability of solutions to \[ (1)\quad M(t)du/dt = N(t)u + f(t,u),\quad u(0)\;\text {given}.\] This problem is not in general well posed in the sense of Hadamard. We cite some examples of (1) from the literature. We also give some examples of the problem \[ (2)\quad M(t)\frac {{{d^2}u}}{{d{t^2}}} = N(t)u + f\left ( {t,u,\frac {{du}}{{dt}}} \right ),\quad u(0),\frac {{du}}{{du}}(0)\;\text {prescribed},\] for which questions of uniqueness and stability were discussed in a previous paper.