Critical Exponents for Semilinear PDEs with Bounded Potentials

Abstract

Using heat kernel estimates obtained in [18] and the Feynman-Kac formula, we investigate finite-time blow-up and stability of semilinear partial differential equations of the form ∂wt/∂t (x) = Δwt(x) − V (x)wt(x) + vt(x)G(wt(x)), w 0(x) ≥ 0, x ∈ ℝd, where v and G are positive measurable functions subject to certain growth conditions, and V is a positive bounded potential. We recover the results of [19] and [14] by probabilistic arguments and in the quadratic decay case V (x) ∼+∞ a(1 +|x|2)−1, a > 0, we find two critical exponents β*(a), β*(a) with 0 < β*(a) ≤ β*(a) < 2/d, such that any nontrivial positive solution blows up in finite time if 0 < β < β*(a), whereas if β*(a) < β, then nontrivial positive global solutions may exist.