We are happy to announce our work on MMAS special issue “Integro-differential models of natural and anthropogenic processes and phenomena” is successfully completed, so please enjoy original and high-quality contributions related to the mathematical theory of such processes and phenomena including the models, computational algorithms and analysis and mathematical methods regarded as new and promising for the understanding the problems arise both in natural and anthropogenic conditions.

In studying the stability of B\’{e}nard problem we usually have to solve a variational problem to determine the critical Rayleigh number for linear or nonlinear stability. To solve the variational problem one usually transform it to an eigenvalue problem which is called Euler-Lagrange equations. An operator related to the Euler-Lagrange equations is usually referred to as Euler-Lagrange operator whose spectrum is investigated in this paper. We have shown that the operator possesses only the point spectrum consisting of real number, which forms a countable set. Moreover, it is found that the spectrum of the Euler-Lagrange operator depends on the thickness of the fluid layer.

We investigate Schwarz lemma in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. The bicomplex is a generalization of complex which has closed relation with Frac- tal geometry, Minkowski Space-Time, Maxwell’s equations, Schrödinger equation and Gaussian pulse wave. In this paper, we first construct a type of bicomplex Möbius transformation and obtain some results : the mapping properties on bicom- plex spheres and bicomplex ball, preserving the inverse points with respect to the bicomplex sphere (0,1). Then we obtain the Poisson integral formula in bicom- plex setting, and by using the Poisson integral formula, we give the Schwarz lemma for bicomplex holomorphic functions in bicomplex setting. Finally, we shall give the Schwarz lemma and the Schwarz-Pick type lemma for holomorphic functions in bicomplex analysis. These results may give new energy for the development of quan- tum mechanics.

In this research article, we construct a q-analogue of the operators defined by Betus and Usta (Numer. Methods Partial Differential Eq. 1-12, (2020)) and study approximation properties in a polynomial weighted space. Further, we modify these operators to study the approximation properties of differentiable functions in the same space and show that the mofidied operators give a better rate of convergence.

In this paper, we study the existence of global $L^2$-constrained minimizers related to the following Kirchhoff type equation: $$ -\left(a+b\ds\int_{\R^N}|\nabla u|^2\right)\Delta u-f(u)=\lambda u,~~~x\in \R^N,~\lambda\in\R,$$ where $N\leq3$, $a,$ $b>0$ are constants, $f(u)$ is a general $L^2$-subcritical nonlinearity. By using the concentration compactness principle, we prove the sharp existence and nonexistence of global $L^2$-constraint minimizers.

In this article, the generalized (3+1)-dimensional Korteweg-de Vries-Zakharov-Kuznetsov equation is investigated, which describes the influence of magnetic fields on weak ion-acoustic waves in plasma made up of cool and hot electrons and may be regarded as a nonlinear complicated physical model. To find out some new traveling waves solutions and other exact solutions, the improved $F$-expansion approach and the $\exp(-\phi (\zeta))$-expansion approach is applied to above mentioned nonlinear higher dimensional model. Several solutions have been found, including dark soliton, periodic type solitons, bell shaped solitons, single bell shaped solitons. We also show a graphical representation of a number of exact solutions to the equation, together with a description of their behaviour. The proposed techniques can also be used to solve a range of nonlinear evolution problems in mathematical physics and plasma physics.

This work brings a one-dimensional logistic harvesting model with Allee effect to the time-varying framework. This new framework is more sober than the autonomous version of the system because it; the framework, permits all environment-dependent coefficients to depend on time. Based on these coefficients, we derive sets of conditions that drive population to “mathematical” extinction. More precisely, we investigate various local and global stability notions including uniform stability, attractivity, asymptotic stability and the (uniform) exponential stability.

We consider the following X-ray free electron lasers Schr\”{o}dinger equation \begin{equation*} (i\nabla-A)^2u+V(x)u-\frac{\mu}{|x|} u=\left(\frac{1}{|x|}*|u|^2\right) u-K(x)|u|^{q-2} u, \,\, x\in \mathbb{R}^3, \end{equation*} where $A\in L_{loc}^2(\mathbb{R}^3,\mathbb{R}^3)$ denotes the magnetic potential such that the magnetic field $B=\text{curl} \, A$ is $\mathbb{Z}^{3}$-periodic, $\mu\in \mathbb{R}$, $K \in L^{\infty}\left(\mathbb{R}^3\right)$ is $\mathbb{Z}^{3}$ -periodic and non-negative, $q\in(2,4)$. Using the variational method, based on a profile decomposition of the Cerami sequence in $H^1_A\left(\mathbb{R}^3\right)$, we obtain the existence of the ground state solution for suitable $\mu\geq0$. When $\mu<0$ is small, we also obtain the non-existence. Furthermore, we give a description for the asymptotic behaviour of the ground states as $\mu \to 0^+$.

In this paper, a backward problem for a nonlinear time-fractional diffusion equation in an axis-symmetric cylinder has been considered. Under some assumptions, we prove the existence and uniqueness of the solution to the nonlinear problem. The ill-posedness of the backward problem is established and we obtain the error estimates by a generalized quasi-boundary value regularization method.

We present Wong-type oscillation criteria for nonlinear impulsive differential equations having discontinuous solutions and involving both negative and positive coefficients. We use a technique that involves the use of a nonprincipal solution of the associated linear homogeneous equation. The existence of principal and nonpricipal solutions was recently obtained by the present authors in [J. Math. Anal. Appl. 503 (2021) 125311]. As special cases, we have superlinear and sublinear Emden-Fowler equations under impulse effects. It is shown that the oscillation behavior changes due to impulses, in particular impulses acting on the solution itself, not on its derivative. An example is also given to illustrate the importance of the results.

The generalized MHD-Boussinesq equations are studied in this paper. The well-posedness of the strong solutions for the generalized MHD-Boussinesq equations is proved. Asymptotic regularity for the generalized MHD-Boussinesq equations is proved in $H^{\frac{9}{2}}\times H^{\frac{9}{2}}\times H^{\frac{9}{2}}$. The higher regularity for the generalized MHD-Boussinesq equations is proved in $H^{a}\times H^{a}\times H^{a}$ for $a\geq\frac{9}{2}$.

In this paper, we make an effort to establish Lieb's and Lions' type theorems on Heisenberg Group, and then apply them to study the existence of solution for variational problem on Heisenberg group.

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A kind of nonlinear non-instantaneous impulsive equation with state-dependent delay is studied here. By ultilizing suitable fixed point theorem and the theory of semigroup in Banach space, the uniqueness and existence results of S-asymptotically w-periodic mild solutions are obtained, respectively. In the end, two examples are presented to demonstrate the validity of the obtained results.

An initial-boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann-Dirichlet type is investigated in this work. From a physical point of view, the initial-boundary value problem considered here is a mathematical model of quasi-stationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. In this article, by the Galerkin method to prove the existence of a weak solution to the initial-boundary value problem for a pseudoparabolic equation in a bounded domain. On the basis of a priori estimates, we prove a local existence theoremand uniqueness for a weak generalized solution of the initial-boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann-Dirichlet boundary condition are obtained.

Recently, an infinitesimal approach for finding reciprocal transformations has been proposed. The method uses the group analysis approach and consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). Similar to the classical group analysis this approach can be also applied for finding all reciprocal transformations (not only composing a group) of the equations under study. The present paper provides this algorithm. As an illustration, the method is applied to the two-dimensional stationary gas dynamics equations. Equivalence group, group of reciprocal transformations and completeness of all discrete reciprocal transformations are presented in the paper. The results are stated in form of a theorem.

The low Mach number limit of the nonisentropic compressible Hookean elastodynamic equations is rigorously proved with respect to well-prepared initial data. We introduce certain suitable seminorms to obtain the uniform estimate of solutions, for which the critical point is to cancel the higher order derivate terms caused by the coupling of velocity and deformation gradient.

Abstract. In this manuscript we consider the internal control problem for the fifth order KdV type equation, commonly called the Kawahara equation, on unbounded domains. Precisely, under certain hypotheses over the initial and boundary data, we are able to prove that there exists an internal control input such that solutions of the Kawahara equation satisfies an integral overdetermination condition. This condition is satisfied when the domain of the Kawahara equation is posed in the real line, left half-line and right half-line. Moreover, we are also able to prove that there exists a minimal time in which the integral overdetermination condition is satisfied. Finally, we show a type of exact controllability associated with the “mass” of the Kawahara equation posed in the half-line.