On the semilinear heat equation with the Grushin operator

In this work, we study the heat equation with Grushin’s operator. We present an expression for its heat kernel, prove its decay in $L^p$ spaces, and that it is an approximation of the identity. As a consequence, the heat semigroup associated with Grushin’s operator using this heat kernel is estimated in Lebesgue spaces. Next, we use the results to prove the existence, uniqueness, continuous dependence, and blowup alternative of mild solutions of a nonlinear Cauchy problem associated with Grushin’s operator. A global existence result is also presented.