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On approximating properties of solutions of the heat equation. (English) Zbl 1089.35012

Imanuvilov, Oleg (ed.) et al., Control theory of partial differential equations. Papers of the conference, Washington, D. C., USA, May 30 – June 1, 2003. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-8247-2546-8/pbk). Lecture Notes in Pure and Applied Mathematics 242, 43-50 (2005).
Summary: By the maximum principle for the heat equation, a solution corresponding to zero initial data and produced by a positive Dirichlet boundary control is positive (i.e., belongs to the cone of positive functions). The notice is devoted to the question: Is the set of such solutions dense in the cone? The answer turns out to be negative: in 1D case we construct an explicit example of a positive function separated from this set by a positive \(L_2\)-distance.
For the entire collection see [Zbl 1066.93001].

MSC:

35B50 Maximum principles in context of PDEs
35K05 Heat equation
35K20 Initial-boundary value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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