J Austral Math Soc Ser B 33 pp149--163, 1991.

Determination of a control parameter in a parabolic partial differential equation

J. R. Cannon, Yanping Lin and Shingmin Wang

(Received 25 November 1988; revised 22 June 1989)

Abstract

The authors consider in this paper the inverse problem of finding a pair of functions (u, p) such that

ut = uxx + pu + F(x, t),       0 < x < 1, 0 < t £ T,
u(x, 0) = f(x),       0 £ x £ 1,
a1(t) ux(0, t) + b1(t) u(0, t) + g1(t) u(1, t) = g1(t),       0 < t £ T,
a2(t) ux(1, t) + b2(t) u(0, t) + g1(t) u(1, t) = g2(t),       0 < t £ T,
ós(t)                                                                      
ô   u(x, t) dx = E(t),       0 £ t £ T,   0 < s(t) £ 1.
õ0                                                                        

where F, f, E, s, ai, bi, gi, gi, i = 1, 2, are given functions.
  The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.

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Authors

J. R. Cannon
Department of Mathematics, Lamar University, Beaumont, TX 77710.
Shingmin Wang
Department of Applied Mathematics, University of Waterloo, Waterloo, Canada N2L 3G1.
Yanping Lin
Division of Mathematics and Computer Science, Northeast Missouri State University, Kirksville, MO 63501.