Heteroclinic Solutions for Nonautonomous EFK Equations

and Applied Analysis 3 Lemma 3. Let q ∈ E. Then for any V > 0, r < s ∈ R such that q(t) ∉ B ε (V) and |q(t)| ⩽ V for any t ∈ [r, s], I (q) ⩾ √2σ ε,V 󵄨󵄨󵄨󵄨q (r) − q (s) 󵄨󵄨󵄨󵄨 . (16) In particular, if σ ε > 0, then for any r < s ∈ R such that q(t) ∉ B ε (V) for any t ∈ [r, s], I (q) ⩾ √2σ ε 󵄨󵄨󵄨󵄨q (r) − q (s) 󵄨󵄨󵄨󵄨 . (17) Proof. Denote l = |q(r) − q(s)| and τ = |r − s|. Then l = 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ∫ s r q 󸀠 (t) dt 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ⩽ ∫ s r 󵄨󵄨󵄨󵄨 q 󸀠 (t) 󵄨󵄨󵄨󵄨 dt ⩽ τ 1/2 (∫ s r 󵄨󵄨󵄨󵄨 q 󸀠 (t) 󵄨󵄨󵄨󵄨 2 dt) 1/2