Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations propagating over a zero background with a finite velocity. Such effects are known to be atypical for parabolic equations and appear as a consequence of the degeneration of the equation at the points where the desired function vanishes. Previously, we have constructed it, but here the case of power nonlinearity is considered. It allows for conducting a more detailed analysis. We prove a new theorem for the existence of solutions of this type in the class of piecewise analytical functions, which generalizes and specifies the earlier statements. We find and study exact solutions having the diffusion wave type, the construction of which is reduced to the second-order Cauchy problem for an ordinary differential equation (ODE) that inherits singularities from the original formulation. Statements that ensure the existence of global continuously differentiable solutions are proved for the Cauchy problems. The properties of the constructed solutions are studied by the methods of the qualitative theory of differential equations. Phase portraits are obtained, and quantitative estimates are determined by constructing and analyzing finite difference schemes. The most significant result is that we have shown that all the special cases for incomplete equations take place for the complete equation, and other configurations of diffusion waves do not arise.

CHARACTERISTIC FUNCTION OF THE SYSTEM D–EQUATIO

This paper is devoted to the problems of studying the multiperiodic solution of some evolutionary equations. The article also discusses the existence and uniqueness of a multiperiodic solution with respect to vector functions for an evolutionary reduced equation. Studies have been conducted on the characteristic function of a certain system of the evolutionary equation. Some properties of the vector function are proved. They can be used in the further study of oscillatory bounded solutions of evolutionary equations. Based on the argumentation of the theorem on the existence and uniqueness of an almost multiperiodic solution of the specified system, considered using the method of shortening the characteristic function. All estimates of the characteristic function are based on the enhanced Lipschitz condition, first introduced by academician K. P. Persidskiy. The results will also be useful in the study of periodic solutions of evolutionary equations of mathematical physics

SIMULATION ANALYSIS OF THE EVOLUTIONARY MECHANISM OF LOW-CARBON RURAL LOGISTICS

To promote the low-carbonization of rural logistics, the embeddedness among low-carbon rural logistics potential (LRLP), low-carbon rural logistics strength (LRLS) and environmental regulations (ERs) was clarified by using the analytical paradigm of impetus–basis–target. According to the embeddedness, an evolutionary framework of the low-carbon rural logistics system was constructed to analyze the evolutionary process of this system. Then, a three-dimensional evolutionary equation of the low-carbon rural logistics system was proposed based on the evolutionary framework, and the evolutionary threshold criterion of the system was obtained through the stability analysis of the evolutionary equation. The interaction mechanism within the evolutionary process of the low-carbon rural logistics system in China was verified through data simulation in MATLAB. The results show that in an environment with lower ERs, the effect of simply relying on LRLS to promote the sustainable development of low-carbon rural logistics is not obvious, and the overall effect is not significant. In the long term, under the influence of different external control variables, LRLP plays a pivotal role in evolution, and improving LRLP is conducive to promoting the sustainable development process of low-carbon rural logistics. Keywords: environmental regulations; low-carbon rural logistics potential; low-carbon rural logistics strength; the evolutionary framework; three-dimensional evolutionary equation

A regularized isothermal phase-field model of two-phase solid–fluid mixture and its spatial dissipative discretization equations

The present paper is devoted to a model describing a two-phase isothermal mixture, in which one of the phases obeys solid-like (namely, elastic) rheology. A fully Eulerian description is considered. To describe the stress–strain behaviour of the solid phase the elastic energy term is added to the Helmholtz free energy. The term depends on Almansi strain tensor. In its turn, the strain tensor is defined as the solution of the corresponding evolutionary equation. Considered model belongs to the phase field family. Formally it describes two-component mixture and uses mass densities of the components as order parameters. A distinctive feature of the considered model is its preliminary regularization according to the quasi-hydrodynamic framework. The dissipativity in total energy is proved when periodic boundary conditions are imposed. A spatial dissipative semi-discrete (continuous in time and discrete in space) scheme based on staggered grids is suggested. The theoretical results remain valid in the absence of the regularization. The results of a numerical study in a 2D setting are presented.

Integration of the Harry Dym equation with an integral type source

In the work, we deduce the evolution of scattering data for a spectral problem associated with the nonlinear evolutionary equation of Harry Dym with a self-consistent source of integral type. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Harry Dym equation with an integral type source.

On totally global solvability of evolutionary equation with unbounded operator

Let $X$ be a Hilbert space, $U$ be a Banach space, $G\colon X\to X$ be a linear operator such that the operator $B_\lambda=\lambda I-G$ is maximal monotone with some (arbitrary given) $\lambda\in\mathbb{R}$. For the Cauchy problem associated with controlled semilinear evolutionary equation as follows \[x^\prime(t)=Gx(t)+f\bigl( t,x(t),u(t)\bigr),\quad t\in[0;T];\quad x(0)=x_0\in X,\] where $u=u(t)\colon[0;T]\to U$ is a control, $x(t)$ is unknown function with values in $X$, we prove the totally (with respect to a set of admissible controls) global solvability subject to global solvability of the Cauchy problem associated with some ordinary differential equation in the space $\mathbb{R}$. Solution $x$ is treated in weak sense and is sought in the space $\mathbb{C}_w\bigl([0;T];X\bigr)$ of weakly continuous functions. In fact, we generalize a similar result having been proved by the author formerly for the case of bounded operator $G$. The essence of this generalization consists in that postulated properties of the operator $B_\lambda$ give us the possibility to construct Yosida approximations for it by bounded linear operators and thus to extend required estimates from “bounded” to “unbounded” case. As examples, we consider initial boundary value problems associated with the heat equation and the wave equation.