Non-local problem with an integral condition for a parabolic equation with a Bessel operator

For the parabolic equation with the Bessel operator
$$ \frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}+\frac{k}{x}\frac{\partial u}{\partial x} $$
in the rectangular domain $0 < x < l,$ $0 < t\leq T,$ we consider a boundary value problem with the non-local integral condition of the first kind
$$\int\limits_0^l u(x,t)\,x\,dx=0,\ \ 0\leq t \leq T.$$
This problem is reduced to an equivalent boundary value problem with mixed boundary conditions of the first and third kind. It is shown that the homogeneous equivalent boundary value problem has only a trivial zero solution, and hence the original inhomogeneous problem cannot have more than one solution. This proof uses Gronwall's lemma. Then, by the method of spectral analysis, the existence theorem for a solution to an equivalent problem is proved. This solution is defined explicitly in the form of a Dini series. Sufficient conditions with respect to the initial condition are obtained. These conditions guarantee the convergence of the constructed series in the class of regular solutions.