Asymptotic Behaviour of Solutions of Parabolic Differential Inequalities

Extract

Let there be given a parabolic differential operator

where A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ R_n, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality

(1.1)

Under suitable restrictions on the functions c_i(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e_-λt, ∀ λ > 0, unless identically zero.