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In this paper, we mainly study the asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half space. First, a narrow region principle for antisymmetric functions in unbounded domains is obtained, in which we remarkably weaken the decay condition u → 0 at infinity and only assume its growth rate does not exceed | x | γ (0 < γ < 2 s ) compared with (Adv. Math. 377:107463, 2021). Then we obtain asymptotic monotonicity of positive solutions of fractional parabolic equation on R + N × (0 , ∞) . [ABSTRACT FROM AUTHOR]

The stabilization properties of dissipative Timoshenko systems have been attracted the attention and efforts of researchers over the years. In the past 20 years, the studies in this scenario distinguished primarily by the nature of the coupling and the type or strength of damping. Particularly, under the premise that the Timoshenko beam model is a two-by-two system of hyperbolic equations, a large number of papers have been devoted to the study of the so-called partially damped Timoshenko systems by assuming damping effects acting only on the angle rotation or vertical displacement (Almeida Júnior et al. in Math Methods Appl Sci 36:1965–1976, 2013; in Z Angew Math Phys 65:1233–1249, 2014; Alves et al. in SIAM J Math Anal 51(6):4520–4543, 2019; Ammar-Khodja et al. in J Differ Equ 194:82–115, 2003; Muñoz Rivera and Racke in Discrete Contin Dyn Syst Ser B 9:1625–1639, 2003; J Math Anal Appl 341:1068–1083, 2008; Santos et al. in J Differ Equ 253(9):2715–2733, 2012). In these cases, the desired exponential decay property of the energy solutions is achieved when the non-physical equal wave speed assumption plays the role to stabilization according since the pioneering Soufyane's paper (C R Acad Sci Paris 328(8):731–734, 1999). Recent results due to Almeida Júnior et al. (Z Angew Math Phys 68(145):1–31, 2017; Z Angew Math Mech 98(8):1320–1333, 2018; IMA J Appl Math 84(4):763–796, 2019; Acta Mech 231:3565–3581, 2020) show that the second vibration mode or simply second spectrum of frequency and it's damaging consequences appears as a lost element in analysis of stabilization and now it's more clear that the damping importance into stabilization scenario of Timoshenko type systems. This paper considers a one-dimensional viscoelastic Timoshenko type system in the light of the second spectrum of frequency where the equal wave speed assumption is not needed for getting the exponential decay property. Precisely, we consider the so-called truncated version for the Timoshenko system according studies due to Elishakoff (Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010; ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11, 2015; Int J Solids Struct 109:143–151, 2017; J Sound Vib 435:409–430, 2017; Int J Eng Sci 116:58–73, 2017; Acta Mech 229:1649–1686, 2018; Z Angew Math Mech 98(8):1334–1368, 2018; Math Mech Solids, 2019) and we added a viscoelastic damping acting on shear force. We firstly prove the global well-posedness of the system by Faedo–Galerkin approximation. By assuming minimal conditions on the relaxation function, we establish an optimal explicit and energy decay rate for which exponential and polynomial rates are special cases. This result is new and substantially improves earlier results in the literature where the equal wave speeds plays the role for getting the stability properties. It is likely to open more research areas to Timoshenko system and probably others. [ABSTRACT FROM AUTHOR]

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: u tt - (1 + ϵ ‖ ∇ u ‖ 2 ) Δ u - σ (‖ ∇ u ‖ 2) Δ u t + f (u) = g (x , t) , where ϵ ∈ [ 0 , 1 ] is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: 1 ≤ p < p ∗ ∗ ≡ N + 4 (N - 4) + , (i) the related process has a pullback attractor A ϵ in natural energy space H = (H 0 1 ∩ L p + 1) × L 2 for each ϵ , and it is upper semicontinuous on the perturbation ϵ ; (ii) the related process has a partially strong pullback exponential attractor M ϵ for each ϵ , and it is Hölder continuous on ϵ ∈ [ 0 , 1 ] . These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case. [ABSTRACT FROM AUTHOR]

The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space and time dependent logistic source, 0.1 ∂ t u = Δ u - χ ∇ · (u ∇ v) + u (a (x , t) - b (x , t) u) , x ∈ R N , 0 = Δ v - λ v + μ u , x ∈ R N , where N ≥ 1 is a positive integer, χ , λ and μ are positive constants, and the functions a(x, t) and b(x, t) are positive and bounded. In the first of the series (Salako and Shen in Math Models Methods Appl Sci 28(11):2237–2273, 2018), we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initial data, and the asymptotic spreading for solutions with compactly supported or front like initial data. In the second of the series (Salako and Shen in J Math Anal Appl 464(1):883–910, 2018), we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In particular, in the case of space homogeneous logistic source (i.e. a (x , t) ≡ a (t) and b (x , t) ≡ b (t) ), we proved in Salako and Shen (J Math Anal Appl 464(1):883–910, 2018) that the unique spatially homogeneous strictly positive entire solution (u ∗ (t) , v ∗ (t)) of (0.1) is uniformly and exponentially stable with respect to strictly positive perturbations when 0 < 2 χ μ < inf t ∈ R b (t) . In the current part of the series, we discuss the existence of transition front solutions of (0.1) connecting (0, 0) and (u ∗ (t) , v ∗ (t)) in the case of space homogeneous logistic source. We show that for every χ > 0 with χ μ (1 + sup t ∈ R a (t) inf t ∈ R a (t) ) < inf t ∈ R b (t) , there is a positive constant c χ ∗ such that for every c ̲ > c χ ∗ and every unit vector ξ , (0.1) has a transition front solution of the form (u (x , t) , v (x , t)) = (U (x · ξ - C (t) , t) , V (x · ξ - C (t) , t)) satisfying that C ′ (t) = a (t) + κ 2 κ for some positive number κ , lim inf t - s → ∞ C (t) - C (s) t - s = c ̲ , and lim x → - ∞ sup t ∈ R | U (x , t) - u ∗ (t) | = 0 and lim x → ∞ sup t ∈ R | U (x , t) e - κ x - 1 | = 0. Furthermore, we prove that there is no transition front solution (u (x , t) , v (x , t)) = (U (x · ξ - C (t) , t) , V (x · ξ - C (t) , t)) of (0.1) connecting (0, 0) and (u ∗ (t) , v ∗ (t)) with least mean speed less than 2 a ̲ , where a ̲ = lim inf t - s → ∞ 1 t - s ∫ s t a (τ) d τ . [ABSTRACT FROM AUTHOR]

This work is concerned with a semilinear non-homogeneous Timoshenko system under the effect of two nonlinear localized frictional damping mechanisms. The main goal is to prove its uniform stability by imposing minimal amount of support for the damping and, as expected, without assuming any relation on the non-constant coefficients. This fact generalizes substantially the previous papers by Cavalcanti et al. (Z Angew Math Phys 65(6):1189–1206, 2014) and Santos et al. (Differ Integral Equ 27(1–2):1–26, 2014) at the levels of problem and method. It is worth mentioning that the methodologies of these latter cannot be applied to the semilinear case herein, namely when one considers the problem with nonlinear source terms. Thus, differently of Cavalcanti et al. (Z Angew Math Phys 65(6):1189–1206, 2014), Santos et al. (Differ Integral Equ 27(1–2):1–26, 2014), the proof of our main stability result relies on refined arguments of microlocal analysis due to Burq and Gérard (Contrôle Optimal des équations aux dérivées partielles, http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). As far as we know, it seems to be the first time that such a methodology has been employed to 1-D systems of Timoshenko type with nonlinear foundations. [ABSTRACT FROM AUTHOR]