A New Critical Phenomenon for Semilinear Parabolic Problems

Abstract

We study the Cauchy problem of the inhomogeneous semilinear parabolic equations Δu + u^p − u_t + w = 0 onMⁿ × (0, ∞) with initial valueu₀ ≥ 0, whereMⁿis a Riemannian manifold with possibly nonnegative Ricci curvature. There is an exponentp* which is critical in the following sense. When 1 < p ≤ p*, the above problem has no global positive solution for any nonnegativew = w(x) not identically zero and for anyu₀ ≥ 0; whenp > p* the problem has a global positive solution for somew = w(x) > 0 and someu₀ ≥ 0.