I nt. J. Mh. Math. Sci. Vol. No. 3 (7980] 599-603 599 ON THE BACKWARD HEAT PROBLEM: EVALUATION OF THE NORM OF YVES BIOLLAY Department of Mathematics Cornell University Ithaca, New York 14850 U.S.A. (Received July 19, 1979) (o) BSTRCT. W how .n h. pape ha II ,u II II ull bounded one imposes on u (solution of the backward heat equation) the condition }I u<x,t> II M. A HIder type of inequality is also given if one supposes {l ut(x,T) l{ - K. <Tif KEY WORDS AD PHRASES: Pabolic equations, Impropy posed problems. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODES: 35R0, 35R25. i. INTRODUCTION. A lot of authors have dealt the backward heat problem and considered equations of various kind. It is known that this problem is an improperly posed problem and the dependence of the solution as function of the initial data is an important aspect of it. The a priori inequalities (see Sigillito Ill) give immediately sev- eral informations. Among the methods of investigation, that of the logarithmic convexity is relatively simple when one is able to define- the difficulty is
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II II II ull - Hindawi Publishing Corporationdownloads.hindawi.com/journals/ijmms/1980/361874.pdftions to boundary value problems, Research Notes in Mathe-matics 13, Pitman (1977).
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Int. J. Mh. Math. Sci.Vol. No. 3 (7980] 599-603
599
ON THE BACKWARD HEAT PROBLEM:
EVALUATION OF THE NORM OF
YVES BIOLLAYDepartment of Mathematics
Cornell UniversityIthaca, New York 14850 U.S.A.
(Received July 19, 1979)
(o)BSTRCT. W how .n h. pape ha II ,u II II ull bounded
one imposes on u (solution of the backward heat equation) the condition
}I u<x,t> II M. A HIder type of inequality is also given if one supposes
{l ut(x,T) l{ - K.
<Tif
KEY WORDS AD PHRASES: Pabolic equations, Impropy posed problems.