Semilinear Grushin heat inequalities with a Hardy-type potential outside the unit gauge ball

We investigate semilinear heat inequalities involving the Grushin operator and a Hardy-type potential in an exterior domain, supplemented with a Dirichlet-type boundary condition imposed in a weak sense. We identify a critical exponent that yields a complete dichotomy between the nonexistence and existence regimes. More precisely, in the subcritical and critical ranges, we prove the nonexistence of weak solutions under a natural positivity condition on the boundary datum. In the supercritical range, we construct explicit positive radial stationary solutions, which in turn generate nontrivial time-dependent solutions via a standard multiplicative procedure. As a direct consequence, we also obtain corresponding existence and nonexistence results for the associated stationary elliptic inequality in the same exterior domain. The analysis relies on suitable nonnegative admissible space–time test functions adapted to the anisotropic geometry induced by the Grushin operator, together with weighted estimates reflecting the Hardy singularity and the exterior-domain geometry.