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Lower Bounds for Solutions of Parabolic Differential Inequalities

Published online by Cambridge University Press:  20 November 2018

Hajimu Ogawa
Hajimu Ogawa*
Affiliation:
University of California, Riverside
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Let P be the parabolic differential operator

where E is a linear elliptic operator of second order on D × [0, ∞), D being a bounded domain in Rn. The asymptotic behaviour of solutions u(x, t) of differential inequalities of the form

1

has been investigated by Protter (4). He found conditions on the functions ƒ and g under which solutions of (1), vanishing on the boundary of D and tending to zero with sufficient rapidity as t → ∞, vanish identically for all t ⩾ 0. Similar results have been found by Lees (1) for parabolic differential inequalities in Hilbert space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 667 - 672
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Lees, M., Asymptotic behaviour of solutions of parabolic differential inequalities, Can. J. Math., 14 (1962), 626–631.Google Scholar
2. Lions, J. L. and Malgrange, B., Sur l'unicité rétrograde dans les problèmes mixtes paraboliques, Math. Scand., 8 (1960), 277–286.Google Scholar
3. Ogawa, H., Lower bounds for solutions of differential inequalities in Hilbert space, Proc. Amer. Math. Soc., 16 (1965), 1241–1243.Google Scholar
4. Protter, M. H., Properties of solutions of parabolic equations and inequalities, Can. J. Math., 13 (1961), 331–345.Google Scholar