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

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 $\sigma<\sqrt{\pi/T}$ .