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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]

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]

In this paper we consider the existence problem for the elliptic equation $ \Delta u+K(x)e^{2u}=0$ on $B^2=\{x \in R^2 \mid |x|<1\}$, which arises in the study of conformal deformation of the hyperbolic disc. We prove an existence result for the above equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions of a one-parameter family of logistic equations on $R_+$ or on $R$. These equations are stationary versions of the Fisher equations and the KPP equations. We also study the blow-up region of a sequence of the solutions when the parameter approaches a critical value and the non-existence of positive solutions beyond the critical value. We use the direct method and the sub and super solution method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the problem of generation of evolution operators associated with linear evolution equations in a general Banach space. The stability condition is proposed from the viewpoint of finite difference approximations. It is shown that linear evolution operators can be generated even if the stability condition given here is assumed instead of Kato's stability condition. [ABSTRACT FROM AUTHOR]

We give a complete description of the Morita equivalence classes of blocks with elementary abelian defect groups of order 8 and of the derived equivalences between them. A consequence is the verification of Broué's abelian defect group conjecture for these blocks. It also completes the classification of Morita and derived equivalence classes of 2-blocks of defect at most three defined over a suitable field. [ABSTRACT FROM AUTHOR]

We prove an infinite-dimensional KAM theorem with dense normal frequencies. In this theorem, we relax the separation condition on normal frequencies which is required by the KAM theorem. [ABSTRACT FROM AUTHOR]

We prove a differential Harnack inequality for the Endangered Species Equation, which is a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle. As an application of this inequality, we will show that positive solutions to this equation must blow up in finite time. [ABSTRACT FROM AUTHOR]