Existence and non-existence of global solutions for a higher-order semilinear parabolic system

Abstract

In this paper, we study the higher-order semilinear parabolic system {ut + (-Δ)mu= |v|p, (t,x) ∈ ℝ+ 1 × ℝN, vt + (-Δ)mv = |u|q, (t,x) ∈ ℝ+ 1 × ℝN, u(0,x) = u0(x), v(0,x) = v0(x), x ∈ ℝN, where m >1, p, q ≥ 1 and pq > 1. We prove that if N/2m > max{1 + p/pq - 1, 1 + q/pq - 1}, then solutions with small initial data exist globally in time. If the exponents p, q meet some additional conditions, we can derive decay estimates

u(t)

∞ C(1 + t)-σ′,

v(t)

∞ ≤ C(1 + t)-σ″, where σ′ and σ″ are positive constants. On the other hand, if N/2m < max{1 + p/pq - 1, 1 + q/pq - 1}, then every solution with initial data having positive average value does not exist globally in time. Indiana University Mathematics Journal ©.