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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Logarithmic convexity, first order differential inequalities and some applications

Author: Howard Allen Levine
Journal: Trans. Amer. Math. Soc. 152 (1970), 299-320
MSC: Primary 35.95
MathSciNet review: 0274988
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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

$\displaystyle (1)\quad M(t)du/dt = N(t)u + f(t,u),\quad u(0)\;$given$\displaystyle .$

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

$\displaystyle (2)\quad M(t)\frac{{{d^2}u}}{{d{t^2}}} = N(t)u + f\left( {t,u,\frac{{du}}{{dt}}} \right),\quad u(0),\frac{{du}}{{du}}(0)\;$prescribed$\displaystyle ,$

for which questions of uniqueness and stability were discussed in a previous paper.

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Keywords: Logarithmic convexity, differential inequalities, uniqueness and stability, problems of mixed type, non-well-posed problems, Chaplygin equation, linear elasticity, heat equation, Cauchy problem
Article copyright: © Copyright 1970 American Mathematical Society