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 $$ Δ_{\mathcal{G}}=\dfrac{1}{2}\left(Δ_{x}+|x|^2Δ_{y}\right) $$ in Marcinkiewicz spaces. Furthermore, we provide sufficient conditions for the existence of positive, symmetric, and self-similar solutions.