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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 note we show finite time blow-up for a class of non-local active scalar equations on compact Riemannian manifolds. The strategy we follow was introduced by Silvestre and Vicol [Trans. Amer. Math. Soc. 368 (2016), pp. 6159–6188] to deal with the one dimensional Córdoba-Córdoba-Fontelos equation and might be regarded as an instance of De Giorgi's method. [ABSTRACT FROM AUTHOR]

The following equation is considered in this work: x'' + (−α + β cos (ρ t) + ƒ(t))x = 0, where the parameters α and β are real numbers, the frequency ρ is a positive real number, and ƒ: [0,∞) → R is a continuous bounded function, i.e., there exists a positive constant ƒ* such that |ƒ(t)| ≤ ƒ* for t ≥ 0. This equation is generally referred to as the Mathieu equation when ƒ* = 0. This work proposes a non-oscillation theorem that can be applied even if ƒ* ≠ 0. The required conditions are expressed by the parameters α, β, ρ, and a positive constant ƒ*. The results obtained herein include those by Sugie and Ishibashi [Appl. Math. Comput. 346 (2019), pp. 491–499]. Further, the result can be proved using the phase plane analysis proposed by Sugie [Monatsh. Math. 186 (2018), pp. 721–743]. Finally, the simple non-oscillation and oscillation conditions of the generalized Mathieu equation are summarized. [ABSTRACT FROM AUTHOR]