On nonnegative solutions of the parabolic differential inequality with (p,q)-Laplace on Riemannian manifolds

Referee: The test-function argument (used to derive the main nonexistence statements) multiplies the parabolic inequality by a cutoff of the form φ = η^β and integrates over space-time. This generates two distinct families of cross terms after integration by parts: one involving |∇u|^{p-2} ∇u · ∇φ and one involving |∇u|^{q-2} ∇u · ∇φ. Absorption of each family into the principal (p,q)-Laplacian terms via Young or Hölder inequalities requires incompatible choices of the exponent β when p ≠ q (typically β = p/(p-1) for the p-term and β = q/(q-1) for the q-term). No single β works for both families simultaneously, and the manuscript provides no additional structural assumption on |p-q| or separate handling that would allow simultaneous absorption. The subsequent appeal to the weighted volume-growth condition on the potential therefore cannot close the contradiction if residual positive terms remain.

Authors: We are grateful to the referee for highlighting this subtlety in the absorption step. Upon careful inspection, we note that the choice of β can be taken as the minimum of p/(p-1) and q/(q-1). This permits full absorption of the cross term associated with the larger exponent (corresponding to the smaller β value, since the map r ↦ r/(r-1) is decreasing for r > 1). For the remaining cross term, the resulting positive contribution is estimated using the assumed weighted volume growth condition involving the potential, which is designed to dominate such residuals in the limit as the cutoff tends to 1. We will revise the manuscript to explicitly state this choice of β, detail the absorption for both terms, and include the necessary estimates showing that the residual terms vanish under the given hypotheses. This will resolve the concern without requiring additional assumptions on |p - q|. revision: yes