Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source

Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

“…(i) Let a = 0 and 1 < p < p S . Then equation (1) has no nontrivial nonnegative radial solution in R N × R.…”

“…(i) Let a = 0, 1 < p < p S and let u = u(r, t) be a classical radial solution of (1) in R N × R with the number of sign-changes satisfying z (0,∞) (u(t)) ≤ M, ∀t ∈ R.…”

“…Theorem 1.1. (i) Let 1 < p < min{p B , p S (a)} and u be bounded nonnegative solution of equation (1) in R N × R. Then u ≡ 0. (ii) Let 1 < p < p S (a) and u be bounded nonnegative radial solution of equation (1) in R N × R. Then u ≡ 0.…”

“…The proofs of Theorem 1.1 and Theorem 1.2 follow the idea as in [3,1,30], which consists of three steps :…”

“…As the first topic, we are interested in the Liouville property -i.e. the nonexistence of solution of problem (1) in the entire space R N × R. We first recall its elliptic counterpart…”