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Mat. Sb., 2005, Volume 196, Number 9, Pages 71–102 (Mi msb1421)  

This article is cited in 1 scientific paper (total in 1 paper)

Exponential solubility classes in a problem for the heat equation with a non-local condition for the time averages

A. Yu. Popova, I. V. Tikhonovb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Moscow Engineering Physics Institute (State University)

Abstract: A non-local problem (with respect to time) for the heat equation is considered for $x\in\mathbb R^n$, $ 0\leqslant t\leqslant T$: find a function $u(x,t)$ such that
$$ \frac{\partial u}{\partial t}=\Delta u,\qquad \frac1T\int_0^Tu(x,t) dt=\varphi(x). $$
An explicit formula for the solution is found. The question of its applicability is discussed. A description of well-posedness classes is presented. The main conjecture is as follows: as $|x|\to\infty$, the solution $u(x,t)$ grows no more rapidly than $\exp(\sigma|x|)$ with $\sigma<\sqrt{\pi/T}$ .

DOI: https://doi.org/10.4213/sm1421

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English version:
Sbornik: Mathematics, 2005, 196:9, 1319–1348

Bibliographic databases:

UDC: 517.956
MSC: 35K05
Received: 14.10.2004

Citation: A. Yu. Popov, I. V. Tikhonov, “Exponential solubility classes in a problem for the heat equation with a non-local condition for the time averages”, Mat. Sb., 196:9 (2005), 71–102; Sb. Math., 196:9 (2005), 1319–1348

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. I. V. Tikhonov, Yu. V. Gavris, T. Z. Adzhieva, “Struktura reshenii modelnoi obratnoi zadachi teploprovodnosti v klassakh funktsii eksponentsialnogo rosta”, Chelyab. fiz.-matem. zhurn., 1:3 (2016), 37–62  mathnet
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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