New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

For the spatially-distributed Fermi–Pasta–Ulam (FPU) equation, irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes. The main result of this paper is the construction of families of special nonlinear systems of the Schrödinger type—quasinormal forms—whose nonlocal dynamics determines the local behavior of solutions to the original problem, as t→∞. On their basis, results are obtained on the asymptotics in the main solution of the FPU equation and on the interaction of waves moving in opposite directions. The problem of “perturbing” the number of N elements of a chain is considered. In this case, instead of the differential operator, with respect to one spatial variable, a special differential operator, with respect to two spatial variables appears. This leads to a complication of the structure of an irregular solution.

The Cauchy problem for inhomogeneous parabolic Shilov equations

In this paper, we consider the Cauchy problem for parabolic Shilov equations with continuous bounded coefficients. In these equations, the inhomogeneities are continuous exponentially decreasing functions, which have a certain degree of smoothness by the spatial variable. The properties of the fundamental solution of this problem are described without using the kind of equation. The corresponding volume potential, which is a partial solution of the original equation, is investigated. For this Cauchy problem the correct solvability in the class of generalized initial data, which are the Gelfand and Shilov distributions, is determined.

Phragm\’{e}n-Lindel\”{o}f alternative results in time-dependent double-diffusive Darcy plane flow

This paper investigates the spatial behavior of the solutions of the double-diffusive Darcy plane flow in a semi-infinite channel. Using the energy estimate method and the differential inequality technology, a differential inequality about the solutions is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variable. In the case of decay, we obtain the upper bound for the total energy. We also give some remarks to generalize the results of this paper.

Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

We propose a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for the proposed wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then the proposed viscoacoustic wave equation can be directly solved using the pseudospectral method (PSM). We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of the proposed approximate dispersion term and approximate dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results show that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q < 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we demonstrate the accuracy of HPSFDM in heterogeneous media by several numerical examples.

Stability of constant steady states of a chemotaxis model

The Cauchy problem for the parabolic–elliptic Keller–Segel system in the whole n-dimensional space is studied. For this model, every constant $$A \in {\mathbb {R}}$$ A ∈ R is a stationary solution. The main goal of this work is to show that $$A < 1$$ A < 1 is a stable steady state while $$A > 1$$ A > 1 is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.

A Spectral Collocation Technique for Riesz Fractional Chen-Lee-Liu Equation

This paper discusses the study of optical solitons that are modeled by Riesz fractional Chen-Lee-Liu model, one of the versions of the famous nonlinear Schrödinger equation. This model is solved by the assistance of consecutive spectral collocation technique with two independent approaches. The first is the approach of the spatial variable, while the other is the approach of the temporal variable. It is concluded that the method of the current paper is far more efficient and credible for the proposed problem. Numerical results illustrate the performance efficiency of the algorithm. The results also point out that the scheme can lead to spectral accuracy of the studied model.

An evolutionary Haar-Rado type theorem

In this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ f ( D u ) ) + D u g ( x , u ) = 0 , where u attains time-independent boundary values $$u_0$$ u 0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values $$u_0$$ u 0 admit a modulus of continuity $$\omega $$ ω and the estimate $$|u(x,t)-u_0(\gamma )| \le \omega (|x-\gamma |)$$ | u ( x , t ) - u 0 ( γ ) | ≤ ω ( | x - γ | ) holds, then u admits the same modulus of continuity in the spatial variable.

Properties of single layer potentials for a pseudo- differential equation related to a linear transformation of a rotationally invariant stable stochastic process

This article is aimed at determining existence conditions of single layer potentials for pseudo-differential equations related to some linear transformations of a rotationally invariant stable stochastic process in a multidimensional Euclidean space and investigating their properties as well. The carrier surface of the potential is smooth enough. In this article, we consider two main cases: the first, when this surface is bounded and closed; the second, when it is unbounded, but could be presented by an explicit equation in some coordinate system. The density of this potential is a continuous function. It is bounded with respect to the spatial variable and, probably, has an integrable singularity with respect to the time variable at zero. Classic properties of this potential, including a jump theorem of the action result of some operator (an analog of the co-normal differential) at its surface points, considered. A rotationally invariant $\alpha$-stable stochastic process in $\mathbb{R}^d$ is a L\'{e}vy process with the characte\-ristic function of its value in the moment of time $t>0$ defined by the expression $\exp\{-tc|\xi|^\alpha\}$, $\xi\in\mathbb{R}^d$, where $\alpha\in(0,2]$, $c>0$ are some constants. If $\alpha=2$ and $c=1/2$, we get Brownian motion and classic theory of potential. There are many different results in this case. The situation of $\alpha\in(1,2)$ is considered in this paper. We study constant and invertible linear transformations of the rotationally invariant $\alpha$-stable stochastic process. The related pseudo-differential equation is the parabolic equation of the order $\alpha$ of the ``heat'' type in which the operator with respect to the spatial variable is the process generator. The single layer potential is constructed in the same way as the single layer potential for the heat equation in the classical theory of potentials. That is, we use the fundamental solution of the equation, which is the transition probability density of the related process. In our theory, the role of the gradient operator is performed by some vector pseudo-differential operator of the order $\alpha-1$. We have already studied the following main properties of the single layer potentials: the single layer potential is a solution of the relating equation outside of the carrier surface and the jump theorem is held. These properties can be useful to solving initial boundary value problems for the considered equations.