This paper investigates the properties of the solutions of the equation of heat conduction with dissipation, which is associated with a harmonic oscillator - the operator $-d^2/dx^2 + x^2$, $x\in \mathbb{R}$ (non-negative and self-adjoint in $L_2(\mathbb{R})$).
An explicit form of the function is given, which is analogous to the fundamental solution of the Cauchy problem for the heat conduction equation.
A formula that describes all infinitely differentiable (with respect to the variable $x$) solutions of such an equation was found, well-posedness of the Cauchy problem for the heat conduction equation with dissipation with the initial function, which is an element of the space of generalized functions $(S_{1/2}^{1/2})'$, is established.
It is established that $(S_{1/2}^{1/2})'$ is the "maximum" space of initial data of the Cauchy problem, for which the solutions are infinite
functions differentiable by spatial variable. The main means of research are formal Hermite series, which are identified with linear continuous functionals defined on $S_{1/2}^{1/2}$.

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