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This paper provides a refinement of the study of asymptotic behaviour for a class of nonlocal in time telegraph equations with positively singular kernels. Based on fundamental properties of relaxation functions and recent representation of the fundamental solution in [Nonlinear Anal. 193 (2020), 111411], we establish the asymptotic expansions of the variance of the stochastic process for both long-time and short-time, which sharply improves the main result in [Proc. Amer. Math. Soc. 149 (2021), 2067–2080] by removing their technical conditions on the regularly varying behaviours and reformulating the asymptotic expansion in a more natural form. By analysing a new noncommutative operation on a subclass of completely positive functions, we provide a new way to construct finitely many ultraslow subdiffusion processes that are rapidly slower than a given ultraslow kernel. Consequently, we show that for a given completely monotonic ultraslow kernel, there is an induced kernel whose corresponding mean square displacement is logarithmic. [ABSTRACT FROM AUTHOR]

We study the fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic operators with nonnegative potentials in the reverse Hölder class. The main aim of the paper is to give pointwise estimates of the heat kernel of the operators above, which improve and generalize the main results by Kurata [J. London Math. Soc. (2) 62 (2000), pp. 885–903]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the long time behavior of the damped forced generalized Ostrovsky equation below the energy space. First, by using Fourier restriction norm method and Tao's [k,Z]-multiplier method, we establish the multi-linear estimates, including the bilinear and trilinear estimates on the Bourgain space Xs,b. Then, combining the multi-linear estimates with the contraction mapping principle as well as L2 energy method, we establish the global well-posedness and existence of the bounded absorbing sets in L2. Finally, we show the existence of global attractor in L2 and its compactness in H5 by means of the high-low frequency decomposition method, cut-off function, tail estimate together with Kuratowski α-measure in order to overcome the non-compactness of the classical Sobolev embedding. This result improves earlier ones in the literatures, such as Goubet and Rosa [J. Differential Equations 185 (2002), no. 1, 25-53], Moise and Rosa [Adv. Differential Equations 2 (1997), no. 2, 251-296], Wang et al. [J. Math. Anal. Appl. 390 (2012), no. 1, 136-150], Wang [Discrete Contin. Dyn. Syst. 35 (2015), no. 8, 3799-3825], and Guo and Huo [J. Math. Anal. App. 329 (2007), no. 1, 392-407]. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Lazer-McKenna conjecture for the suspension bridge model in higher dimension. We also discuss some properties of the limiting problem related to the Swift-Hohenberg equation. [ABSTRACT FROM AUTHOR]