On the Heat equation involving a Grushin operator in Marcinkiewcz spaces

In this work, we give sufficient conditions for the existence and uniqueness of the heat equation involving the operator $$\Delta_{\mathcal{G}}=\dfrac{1}{2}\left(\Delta_{x}+|x|^2\Delta_{y}\right)$$ in Marcinkiewicz spaces. Furthermore, we provide sufficient conditions for the existence of positive, symmetric, and self-similar solutions.