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 $${\mathrm{\Delta }}_{\mathcal{G}}={\displaystyle \frac{1}{2}}({\mathrm{\Delta }}_x+\mathrm{\mid }x{\mathrm{\mid }}^2{\mathrm{\Delta }}_y)$$ in Marcinkiewicz spaces. Furthermore, we provide sufficient conditions for the existence of positive, symmetric, and self-similar solutions.