Geodesic
Exponential solubility classes in a problem for the heat equation with a non-local condition for the time averages
Sbornik. Mathematics, Tome 196 (2005) no. 9, pp. 1319-1348 
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. Sbornik. Mathematics, Tome 196 (2005) no. 9, pp. 1319-1348. http://geodesic.mathdoc.fr/item/SM_2005_196_9_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\sigma<\sqrt{\pi/T}$ .

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