Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

<p style='text-indent:20px;'>In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ M:[0, \infty)\rightarrow (0, \infty) $\end{document}</tex-math></inline-formula> is a nondecreasing and continuous function, <inline-formula><tex-math id="M2">\begin{document}$ [u]_{\alpha, 2} $\end{document}</tex-math></inline-formula> is the Gagliardo-seminorm of <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> are the fractional Laplace operators, <inline-formula><tex-math id="M6">\begin{document}$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $\end{document}</tex-math></inline-formula> is a positive nonincreasing function and <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.</p>

General and optimal decay for a quasilinear parabolic viscoelastic system

<p style='text-indent:20px;'>In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying <inline-formula><tex-math id="M1">\begin{document}$ g'(t) \leq - \xi(t) H(g(t)) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ H $\end{document}</tex-math></inline-formula> is an increasing, convex function and <inline-formula><tex-math id="M3">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a nonincreasing function. Precisely, we establish a general and optimal decay result for a large class of relaxation functions which improves and generalizes several stability results in the literature. In particular, our result extends an earlier one in the literature, namely, the case of the polynomial rates when <inline-formula><tex-math id="M4">\begin{document}$ H(t) = t^p, \ t\geq 0, \forall p&gt;1 $\end{document}</tex-math></inline-formula>, instead the parameter <inline-formula><tex-math id="M5">\begin{document}$ p \in [1, \frac{3}{2}[ $\end{document}</tex-math></inline-formula>.</p>

Synchronization of Complex Dynamical Networks with Hybrid Time Delay under Event-Triggered Control: The Threshold Function Method

This paper investigates the synchronization of general complex dynamical networks (CDNs) with both internal delay and transmission delay. Event-triggered mechanism is applied for the feedback controllers, in which the triggered function is formed as a nonincreasing function. Both continuous feedback and sampled-data feedback methods are studied. According to Lyapunov stability theorem and generalized Halanay’s inequality, quasi-synchronization criteria are derived at first. The synchronization error is bounded with some parameters of the triggered function. Then, the completed synchronization can be guaranteed as a special case. Finally, coupled neural networks as numerical simulation examples are given to verify the theoretical results.

Blow-up results for a quasilinear von Karman equation of memory type

In this paper, we consider the blow-up result of solution for a quasilinear von Karman equation of memory type with nonpositive initial energy as well as positive initial energy. For nonincreasing function $g>0$ g > 0 and nondecreasing function f, we prove a finite time blow-up result under suitable condition on the initial data.

Individual Variability in Dispersal and Invasion Speed

We model the growth, dispersal and mutation of two phenotypes of a species using reaction–diffusion equations, focusing on the biologically realistic case of small mutation rates. Having verified that the addition of a small linear mutation term to a Lotka–Volterra system limits it to only two steady states in the case of weak competition, an unstable extinction state and a stable coexistence state, we exploit the fact that the spreading speed of the system is known to be linearly determinate to show that the spreading speed is a nonincreasing function of the mutation rate, so that greater mixing between phenotypes leads to slower propagation. We also find the ratio at which the phenotypes occur at the leading edge in the limit of vanishing mutation.

Sharp estimates of semistable radial solutions of k-Hessian equations

We consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

On the constancy of signs and order of magnitude of Fourier–Haar coefficients

In the paper, we investigate the relation between the properties of functions and their Fourier–Haar coefficients. We show that for some classes of functions Fourier–Haar coefficients have constant signs and order of magnitude. In 1964, Golubov proved in [B. I. Golubov, On Fourier series of continuous functions with respect to a Haar system (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 1964, 1271–1296] that if {f(x)\in C(0,1)}, then its Fourier–Haar coefficients have constant signs when {f(x)} is a nonincreasing function on {[0,1]}, and in some cases those coefficients have a certain order of magnitude. In the present paper, we continue to investigate the properties of functions which follow from the behavior of their Fourier–Haar coefficients.

NONINCREASING DEPTH FUNCTIONS OF MONOMIAL IDEALS

Given a nonincreasing function f : ℤ≥ 0 \{0} → ℤ≥ 0 such that (i) f(k) − f(k + 1) ≤ 1 for all k ≥ 1 and (ii) if a = f(1) and b = limk → ∞f(k), then |f−1(a)| ≤ |f−1(a − 1)| ≤ ··· ≤ |f−1(b + 1)|, a system of generators of a monomial ideal I ⊂ K[x1, . . ., xn] for which depth S/Ik = f(k) for all k ≥ 1 is explicitly described. Furthermore, we give a characterization of triplets of integers (n, d, r) with n > 0, d ≥ 0 and r > 0 with the properties that there exists a monomial ideal I ⊂ S = K[x1, . . ., xn] for which limk→∞ depth S/Ik = d and dstab(I) = r, where dstab(I) is the smallest integer k0 ≥ 1 with depth S/Ik0 = depth S/Ik0+1 = depth S/Ik0+2 = ···.

Speed of coming down from infinity for birth-and-death processes

We describe in detail the speed of `coming down from infinity' for birth-and-death processes which eventually become extinct. Under general assumptions on the birth-and-death rates, we firstly determine the behavior of the successive hitting times of large integers. We identify two different regimes depending on whether the mean time for the process to go from n+1 to n is negligible or not compared to the mean time to reach n from ∞. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is gradual and a law of large numbers and a central limit theorem for the hitting times sequence hold. By an inversion procedure, we deduce that the process is almost surely equivalent to a nonincreasing function when the time goes to 0. Our results are illustrated by several examples including applications to population dynamics and population genetics. The particular case where the death rate varies regularly is studied in detail.

Permutation Flow Shop Problem with Shortening Job Processing Times

In this paper, we consider a three-machine makespan minimization permutation flow shop scheduling problem with shortening job processing times. Shortening job processing times means that its processing time is a nonincreasing function of its execution start time. Optimal solutions are obtained for some special cases. For the general case, several dominance properties and two lower bounds are developed to construct a branch-and-bound (B&B) algorithm. Furthermore, we propose a heuristic algorithm to overcome the inefficiency of the branch-and-bound algorithm.