Dissipativity and stability for semilinear anomalous diffusion equations involving delays

Abstract

We analyze the dissipativity and stability of solutions to a class of semilinear anomalous diffusion equations involving delays. The existence of absorbing set, the stability, and weak stability will be shown under suitable assumptions on the nonlinearity. Our analysis is based on new Halanay‐type inequality, local estimates, and fixed‐point arguments.

An improved fractional Halanay inequality with distributed delays

“…We are now in a position to improve the stability analysis given in Ke and Thuy [28,Section 4]. Let Ω ⊂ R d be a bounded domain with smooth boundary 𝜕Ω.…”

“…We are now in a position to improve the stability analysis given in Ke and Thuy [28, Section 4]. Let $$$ \Omega \subset {\mathrm{\mathbb{R}}}^d $$$ be a bounded domain with smooth boundary $$$ \mathrm{\partial \Omega } $$$.…”

“…Utilizing the argument in Ke and Thuy [28, Theorem 4.1], we gain the following strict Mittag–Leffler stability.…”

“…Proof The existence of the unique solution follows from Ke and Thuy [28, Theorem 4.1]. Let $$$ {u}_1\left(\cdotp, {\varphi}_1\right) $$$ and $$$ {u}_2\left(\cdotp, {\varphi}_2\right) $$$ be solutions of ()–() with initial data $$$ {\varphi}_1 $$$ and $$$ {\varphi}_2 $$$, respectively.…”

“…Thus, (46) holds with 𝜆 = 𝜇 and M = 𝛼 (−𝜇h 𝛼 ) . (b) Many authors use the following Mittag-Leffler stability (see Li et al[8] and their followers): There exists 𝜆 > 0, Because the Mittag-Leffler function admits the asymptotic behavior as in Lemma 1, it means that their Mittag-Leffler stability requires only a certain power decaying rate; meanwhile, our strict Mittag-Leffler stability needs the decay rate to be at least O(t −𝛼 ) where 𝛼 is the order of fractional derivative.Utilizing the argument in Ke and Thuy[28, Theorem 4.1], we gain the following strict Mittag-Leffler stability. Assume that the nonlinearity 𝑓 satisfies the Lipschitz condition||𝑓 (t, v 1 , w 1 ) − 𝑓 (t, v 2 , w 2 )|| ≤ 𝛽||v 1 − v 2 || + 𝜅||w 1 − w 2 ||, for all t ≥ 0, v i , w i ∈ L 2 (Ω), i ∈ {1, 2}, where 𝛽 ≥ 0, 𝜅 > 0 such that 𝛽 + 𝜅 < 𝜆 𝛾 1 ,where 𝜆 1 is the first eigenvalue of the Dirichlet Laplacian.…”