Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

In this article, we focus on stability and ψ-algebraic decay (algebraic decay in the sense of ψ-function) of the equilibrium to the nonlinear ψ-fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ-Caputo fractional derivative, another with ψ-Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

Partial-limit Solutions and Rational Solutions with Parameter for the Fokas-Lenells Equation

A partial-limit procedure is applied to soliton solutions of the Fokas-Lenells equation. Multiple-pole solutions related to real repeated eigenvalues are obtained. For the envelop | u | 2 , the simplest solution corresponds to a real double eigenvalue, showing a solitary wave with algebraic decay. Two such solitons allow elastic scattering but asymptotically with zero phase shift. Single eigenvalue with higher multiplicity gives rise to rational solutions which contain an intrinsic parameter, live on a zero background, and have slowly-changing amplitudes.

Censored stable subordinators and fractional derivatives

Based on the popular Caputo fractional derivative of order β in (0, 1), we define the censored fractional derivative on the positive half-line ℝ+. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing β-stable process in ℝ+. We provide a series representation for the inverse of this censored fractional derivative. We are then able to prove that this censored process hits the boundary in a finite time τ ∞, whose expectation is proportional to that of the first passage time of the β-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of τ ∞. This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.

Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in ℝ 3

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small H N N ≥ 3 solution; in particular, we only require that the H 4 norms of the initial data be small when N ≥ 5 . Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.

Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel

This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski’s coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel K K which can be written as K = 2 + ε W K=2+\varepsilon W . The perturbation is assumed to have homogeneity zero and might also be singular both at zero and at infinity. Under further regularity assumptions on W W , we will show that for sufficiently small ε \varepsilon there exists, up to normalisation of the tail behaviour at infinity, at most one self-similar profile. Establishing uniqueness of self-similar profiles for Smoluchowski’s coagulation equation is generally considered to be a difficult problem which is still essentially open. Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel.

Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model

<p style='text-indent:20px;'>This work deals with a Neumann initial-boundary value problem for a two-species predator-prey chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\chi\nabla\cdot(u\nabla w)+u(\lambda-u+av),\quad &amp;x\in \Omega,\quad t&gt;0,\\ v_t = d_2\Delta v+\xi\nabla\cdot(v\nabla w)+v(\mu-v-bu),\quad &amp;x\in \Omega,\quad t&gt;0,\\ 0 = d_3\Delta w-\alpha w+\beta_1 u+ \beta_2 v,\quad &amp;x\in\Omega,\quad t&gt;0,\\ \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n \,\,(n = 2,3) $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where the parameters <inline-formula><tex-math id="M3">\begin{document}$ d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b $\end{document}</tex-math></inline-formula> are positive. It is shown that for any appropriate regular initial date <inline-formula><tex-math id="M4">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ v_0 $\end{document}</tex-math></inline-formula>, the corresponding system possesses a global bounded classical solution in <inline-formula><tex-math id="M6">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, and also in <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> being sufficiently small. Moreover, by constructing some suitable functionals, it is proved that if <inline-formula><tex-math id="M9">\begin{document}$ b\lambda&lt;\mu $\end{document}</tex-math></inline-formula> and the parameters <inline-formula><tex-math id="M10">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> are sufficiently small, then the solution <inline-formula><tex-math id="M12">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> of this system converges to <inline-formula><tex-math id="M13">\begin{document}$ (\frac{\lambda+a\mu}{1+ab}, \frac{\mu-b\lambda}{1+ab}, \frac{\beta_1(\lambda+a\mu)+\beta_2(\mu-b\lambda)}{\alpha(1+ab)}) $\end{document}</tex-math></inline-formula> exponentially as <inline-formula><tex-math id="M14">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula>; if <inline-formula><tex-math id="M15">\begin{document}$ b\lambda\geq \mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> is sufficiently small and <inline-formula><tex-math id="M17">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is arbitrary, then the solution <inline-formula><tex-math id="M18">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> converges to <inline-formula><tex-math id="M19">\begin{document}$ (\lambda,0,\frac{\beta_1\lambda}{\alpha}) $\end{document}</tex-math></inline-formula> with exponential decay when <inline-formula><tex-math id="M20">\begin{document}$ b\lambda&gt; \mu $\end{document}</tex-math></inline-formula>, and with algebraic decay when <inline-formula><tex-math id="M21">\begin{document}$ b\lambda = \mu $\end{document}</tex-math></inline-formula>.</p>

A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

On Global Well-Posedness and Temporal Decay for 3D Magnetic Induction Equations with Hall Effect

The main purpose of this paper is to study the global existence and uniqueness of solutions for three-dimensional incompressible magnetic induction equations with Hall effect provided that ∥u0∥H32+ε+∥b0∥H2(0<ε<1) is sufficiently small. Moreover, using the Fourier splitting method and the properties of decay character r*, one also shows the algebraic decay rate of a higher order derivative of solutions to magnetic induction equations with the Hall effect.

A Supersymmetric Hierarchical Model for Weakly Disordered 3d Semimetals

In this paper, we study a hierarchical supersymmetric model for a class of gapless, three-dimensional, weakly disordered quantum systems, displaying pointlike Fermi surface and conical intersections of the energy bands in the absence of disorder. We use rigorous renormalization group methods and supersymmetry to compute the correlation functions of the system. We prove algebraic decay of the two-point correlation function, compatible with delocalization. A main technical ingredient is the multiscale analysis of massless bosonic Gaussian integrations with purely imaginary covariances, performed via iterative stationary phase expansions.