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It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs. [ABSTRACT FROM AUTHOR]

This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries. [ABSTRACT FROM AUTHOR]

In this paper, we would like to consider the Cauchy problem for semilinear σ$$ \sigma $$‐evolution equations with time‐dependent damping for any σ≥1$$ \sigma \ge 1 $$. Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from σ=1$$ \sigma =1 $$ to σ>1$$ \sigma >1 $$ and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ$$ \sigma $$ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the multiplicity results to a class of p$$ p $$‐Kirchhoff type elliptic equation with the homogeneous Neumann boundary conditions by the abstract linking lemma due to Brézis and Nirenberg. We obtain the twofold results in subcritical and critical cases, which is a meaningful addition and completeness to the known results about Kirchhoff equation. At the same time, this paper also gives a method to deal with p$$ p $$‐Laplacian, Kirchhoff equation, and some Kirchhoff type equation in a unified variational framework. [ABSTRACT FROM AUTHOR]

This paper considers the following fractional Kirchhoff–Schrödinger–Poisson equation: a+b∫ℝ3(−Δ)α2u2dx(−Δ)αu+ϕu=h(u)+λuinℝ3,(−Δ)βϕ=u2inℝ3,$$ \left\{\begin{array}{lll}& \left(a&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;3}{\left&#x0007C;{\left(-\Delta \right)}&#x0005E;{\frac{\alpha }{2}}u\right&#x0007C;}&#x0005E;2\mathrm{d}x\right){\left(-\Delta \right)}&#x0005E;{\alpha }u&#x0002B;\phi u&#x0003D;h(u)&#x0002B;\lambda u& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\\ {}& {\left(-\Delta \right)}&#x0005E;{\beta}\phi &#x0003D;{u}&#x0005E;2& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\end{array}\right. $$where a,b>0,0<α,β<1,2α+2β>3,λ∈ℝ,h∈C(ℝ,ℝ)$$ a,b>0,0<\alpha, \beta <1,2\alpha &#x0002B;2\beta >3,\lambda \in \mathrm{\mathbb{R}},h\in C\left(\mathrm{\mathbb{R}},\mathrm{\mathbb{R}}\right) $$. For the case of h(u)∼|u|p−2u$$ h(u)\sim {\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u $$ with p∈(8α+4β−3α+β,8α3+2)∪(8α3+2,2α∗),2α∗=63−2α$$ p\in \left(\frac{8\alpha &#x0002B;4\beta -3}{\alpha &#x0002B;\beta },\frac{8\alpha }{3}&#x0002B;2\right)\cup \left(\frac{8\alpha }{3}&#x0002B;2,{2}_{\alpha}&#x0005E;{\ast}\right),{2}_{\alpha}&#x0005E;{\ast }&#x0003D;\frac{6}{3-2\alpha } $$, the nonlinear term h$$ h $$ satisfies more general conditions, we obtain the existence of normalized solutions for the above system by using Pohozaev manifold and variational method. [ABSTRACT FROM AUTHOR]

In this paper, we consider two string–beam–string boundary transmission problems with localized frictional damping and delay. The first transmission problem is that the frictional damping and delay are in the left string equation and the right string equation is thermoelastic. The second transmission problem is that the damping and the delay are both only acting on the middle beam equation. Using the semigroup theory, we get the existence and the uniqueness of solution under suitable assumptions on the weight of the damping and the weight of the delay. Furthermore, we prove that the first transmission problem is exponentially stable and the second transmission problem is polynomially stable using the frequency domain method and the multiplier techniques. [ABSTRACT FROM AUTHOR]

Massless field equations are fundamental in particle physics. In Clifford analysis, the Euclidean version of these equations has been dealt with, but it is not clear, even in dimension 4, what should be the right analogue of massless field equations for fields with values in a general irreducible Spin(4)‐module. The main aim of the paper is to explain that a good possibility is to take the so‐called generalized Cauchy–Riemann equations proposed a long time ago by Stein and Weiss. For this choice of the equations, we show that their polynomial solutions form different irreducible Spin(4)‐modules. This is an important step in developing the corresponding function theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the critical p$$ p $$‐Laplacian equation of Kirchhoff type −M∫ℝN|∇u|pdxΔpu+V(x)|u|p−2u=λK(x)f(u)+|u|p∗−2u,x∈ℝN,$$ -M\left({\int}_{{\mathrm{\mathbb{R}}}&#x0005E;N}{\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;p dx\right){\Delta}_pu&#x0002B;V(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u&#x0003D;\lambda K(x)f(u)&#x0002B;{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p&#x0005E;{\ast }-2}u,x\in {\mathrm{\mathbb{R}}}&#x0005E;N, $$ where M$$ M $$ is the Kirchhoff function, Δpu=div(|∇u|p−2∇u)$$ {\Delta}_pu&#x0003D;\operatorname{div}\left({\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;{p-2}\nabla u\right) $$ with N20$$ N\ge 2,\lambda >0 $$ is a parameter, p∗=Np/(N−p)$$ {p}&#x0005E;{\ast }&#x0003D; Np/\left(N-p\right) $$, and the potentials V$$ V $$ and K$$ K $$ vanish at infinity. Under some suitable assumptions on f$$ f $$, by using the constraint minimization approach, we obtain a least energy sign‐changing solution uλ$$ {u}_{\lambda } $$ to this problem if λ$$ \lambda $$ is large enough and show the energy of uλ$$ {u}_{\lambda } $$ is strictly larger than twice that of the ground state solutions. Moreover, by considering a wider class of M$$ M $$ and f$$ f $$, we exploit the truncation argument to find a nontrivial solution if λ$$ \lambda $$ is sufficiently large via some analytic skills. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D incompressible magnetohydrodynamic equations and establish several logarithmically improved regularity criteria in the homogeneous Besov space. These criteria are based on certain assumptions made in the velocity field and pressure, respectively. The first result solely based on the velocity field indicates that the velocity field plays a dominant role in determining the regularity of weak solutions, when compared with the magnetic field. [ABSTRACT FROM AUTHOR]

In this paper, we study the Neumann problem for a class of nonlocal dispersal models with inhomogeneous kernel function. We investigate the existence, uniqueness, and limit of solutions when the inhomogeneous diffusion kernel is rescaled. Our result exhibits that the inhomogeneous nonlocal dispersal equation is analogous to local problem without convection in the limit case. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the long time behavior of the extensible beams equation with the nonlocal weak damping utt + Δ²u - m(||∇u||²)Δu + ||ut||put + ƒ(u) = h. In the case of critical growth of the nonlinearity, the polynomial attractor which can quantitatively describe the attractive rate of attractors for the equation is studied. The work generalizes and improves previous results for the extensible beams equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well-posedness and asymptotic behavior for a singular epitaxial thin-film growth equation with logarithmic nonlinearity under the Navier boundary condition. Based on the technique of cut-off and combining with Hardy-Sobolev inequality, the technique of Faedo-Galerkin, and multiplier, we establish the local solvability. Meantime, by virtue of the family of potential wells, we obtain the threshold between the existence and nonexistence of the global solution (including the critical case) and give the upper bound of lifespan and the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and the lifespan are derived. [ABSTRACT FROM AUTHOR]

In this paper, we consider the blowup mechanism of regular solutions to the spherically symmetry relativistic Euler-Poisson equations with repulsive force and damping. By means of a weighted function and the structured functional, we show the regular solution will blow up in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we establish iterative Hille-type criteria for advanced functional half-linear dynamic equations of the second order. These results extend and improve recent criteria established by multiple authors for the same equation and encompass classical criteria. We provide an example to demonstrate the significance of the results obtained. [ABSTRACT FROM AUTHOR]

This article extracts analytical solutions of the Fokas-Lenells equation describing pulse propagation in optical fibers and presents a comprehensive analysis of the analytical solutions. This paper focuses on deriving analytical, non-perturbative solutions for the Fokas-Lenells equation by reducing it to a system of ordinary differential equations using the Lie symmetry method. Three generators for the Lie algebra are obtained, and three reductions are explored by combining vector generators. These algebraic structures provide a powerful framework for simplifying complex mathematical and physical problems by revealing underlying symmetries and patterns. The graphical representations of obtained solutions are rendered. Besides, the influence of parameters within the equation on the behavior of the solutions has been demonstrated in the figures. These results contribute to a deeper understanding of the Fokas-Lenells equation and have implications in various fields of physics and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we study several problems of the fifth‐order Camassa‐Holm type (FOCHT) equation. The local well‐posedness and blow‐up scenario are established at first. Then we prove the global existence under some conditions and analyze the large‐time behavior of the support of momentum density. Finally, we discuss the persistence property in Sobolev space. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear damped beam equation with logarithmic source and memory term. By using Galerkin method, logarithmic Sobolev inequality, and some compactness arguments, the global existence of weak solutions is obtained. Furthermore, we show the decay estimate of related energy under the smallness assumption on the initial data. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an uncoupled and linearized compact finite difference scheme for the generalized dissipative symmetric regularized long‐wave (GDSRLW) equations. The unique solvability and some a priori estimates of the proposed difference scheme are rigorously proved based on the mathematical induction method. To obtain the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm estimation of numerical solutions, the discrete energy method is used to prove the convergence and stability of the difference scheme. The proposed difference scheme preserves the original energy dissipation property, and the convergence of the scheme is proved to be fourth‐order in space and second‐order in time in the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm for both u$$ u $$ and ρ$$ \rho $$. Some numerical experiments are given to verify the theoretical analysis and the reliability of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

This paper provides some information on the eigenvalues and eigenfunctions of some left‐definite system of first‐order differential equations subject to eigenparameter‐dependent boundary conditions. Namely, we show that the pair of solutions of the system of equations satisfying some initial conditions exists and is unique, and this pair is analytic in the spectral parameter of order 1/2. We also introduce Lagrange's formula for the left‐definite equation. Using some Prüfer angels, we investigate oscillation of zeros of eigenfunctions and asymptotics equations for the eigenvalues of the problem. Moreover, we share some ordinary and Fréchet derivatives of eigenvalues and eigenfunctions with respect to some elements of data. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global smooth solution of the 3D generalized magneto‐micropolar equations. When the velocity dissipation is logarithmically hyperdissipative and the magnetic diffusion is fractional Laplacian, based on some new observations on the nonlinear structure for the magneto‐micropolar equations, it is examined the system has a unique global smooth solution (u,w,b)∈C([0,T];Hs(ℝ3))$$ \left(u,w,b\right)\in C\left(\left[0,T\right];{H}&#x0005E;s\left({\mathrm{\mathbb{R}}}&#x0005E;3\right)\right) $$. [ABSTRACT FROM AUTHOR]

This paper studies the fractional quasi‐geostrophic equation with modified dissipation term. We first prove the existence and uniqueness of smooth solution of the fractional quasi‐geostrophic equation by using the regularization method. Then, utilizing the Fourier‐splitting method, we obtain the long‐time behavior of the solution. More precisely, we show that under suitable assumptions on the initial data, for any multi‐index γ,m≥4$$ \gamma, m\ge 4 $$ and α∈12,1$$ \alpha \in \left(\frac{1}{2},1\right) $$, the solution ψ$$ \psi $$ satisfies C1(1+t)−1+|γ|2+2α≤∇γψ(·,t)L2≤C2(1+t)−1+|γ|2+2α,|γ|=0,1,...,m−1,t≥1,$$ {C}_1{\left(1&#x0002B;t\right)}&#x0005E;{-\frac{1&#x0002B;\mid \gamma \mid }{2&#x0002B;2\alpha }}\le {\left\Vert {\nabla}&#x0005E;{\gamma}\psi \left(\cdotp, t\right)\right\Vert}_{L&#x0005E;2}\le {C}_2{\left(1&#x0002B;t\right)}&#x0005E;{-\frac{1&#x0002B;\mid \gamma \mid }{2&#x0002B;2\alpha }},\mid \gamma \mid &#x0003D;0,1,\dots, m-1,t\ge 1, $$and C3(1+t)−1+|γ|2+2α≤||∇γψ(·,t)||L2≤C4(1+t)−|γ|2+2α,|γ|=m,t≥1.$$ {C}_3{\left(1&#x0002B;t\right)}&#x0005E;{-\frac{1&#x0002B;\mid \gamma \mid }{2&#x0002B;2\alpha }}\le {\left\Vert {\nabla}&#x0005E;{\gamma}\psi \Big(\cdotp, t\Big)\right\Vert}_{L&#x0005E;2}\le {C}_4{\left(1&#x0002B;t\right)}&#x0005E;{-\frac{\mid \gamma \mid }{2&#x0002B;2\alpha }},\mid \gamma \mid &#x0003D;m,t\ge 1. $$ [ABSTRACT FROM AUTHOR]

The (3 + 1)‐dimensional Vakhnenko–Parkes mathematical model has a wide range of applications in science and engineering. In this paper, the model studied is investigated and analyzed by using two effective schemes such as sine‐Gordon expansion method and its newly developed version, rational SGEM. Moreover, many novel properties of model studied are extracted in detail. Furthermore, their wave distributions properties and graphs are also plotted under the strain conditions. Interactions of the gravitational potential and high‐frequency wave perturbation properties are also reported in a detailed manner. [ABSTRACT FROM AUTHOR]

In this paper, a semi‐discrete rotational velocity‐correction projection method is proposed to solve the Kelvin–Voigt viscoelastic fluid equations. In this method, the rotational velocity‐correction projection method can preserve the divergence free of the velocity u$$ u $$, and the nonlinear equation was linearized. Then, the unconditional stability and optimal convergence order will be provided by the numerical analysis. Finally, our analysis are confirmed by some numerical results, and the algorithm is effective. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem for the time‐fractional two‐dimensional Cahn–Hilliard equation. The ill‐posedness and a conditional stability of the inverse problem are proved. We present a new iterative variational regularization method to solve it. The convergence rates of the regularized solutions under the a priori regularization parameter choice rule and a posteriori regularization parameter choice rule are obtained. Numerical examples illustrate the effectiveness and stability of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we systematically present some main results of the fractional calculus, almost periodic functions, fuzzy functions, fuzzy fractional calculus, and almost periodic generalized fuzzy dynamic equations on timescales. Moreover, the potential future research of almost periodic fractional fuzzy dynamic equations on timescales is discussed. The results presented in this survey can be applied to study the qualitative theory of almost periodic fractional fuzzy dynamic equations and fuzzy fractional calculus on timescales. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fractional Schrödinger equation ε2s(−Δ)su+V(x)u=P(x)f(u)+Q(x)|u|2s∗−2uinℝN,$$ {\varepsilon}&#x0005E;{2s}{\left(-\Delta \right)}&#x0005E;su&#x0002B;V(x)u&#x0003D;P(x)f(u)&#x0002B;Q(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{2_s&#x0005E;{\ast }-2}u\kern0.30em \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;N, $$where ε>0$$ \varepsilon >0 $$ is a parameter, s∈(0,1),2s∗=2NN−2s,N>2s,(−Δ)s$$ s\in \left(0,1\right),{2}_s&#x0005E;{\ast }&#x0003D;\frac{2N}{N-2s},N>2s,{\left(-\Delta \right)}&#x0005E;s $$ is the fractional Laplacian and f$$ f $$ is a superlinear and subcritical nonlinearity. Under a local condition imposed on the potential function, combining the penalization method and the concentration‐compactness principle, we prove the existence of a positive solution for the above equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to solve an inverse source problem of time fractional diffusion‐wave equation in a cylinder. The ill‐posedness, existence, and uniqueness of the inverse source problem are proved. We propose an iterative generalized quasi‐boundary value regularization method to deal with the inverse source problem. The convergence rates of the regularized solution are obtained under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule. Numerical examples are presented to show the validity and stability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we provide sufficient conditions on the incompressible viscous resistive MHD equations and the viscous non‐resistive MHD equations to ensure the energy equality. Part of our results also holds for the ideal MHD equations. [ABSTRACT FROM AUTHOR]

In this paper, the improved Fan subequation method is employed to construct exact traveling wave solutions for a generalized fifth‐order dispersive equation. By making use of bifurcation theory of dynamical systems, we have succeeded to obtain the phase portraits of the subequations involving all possible parameter conditions, and many different types of traveling wave solutions as well, which include more general soliton solutions, kink solutions, triangular function solutions, rational solutions, and Jacobian elliptic function solutions with double periods and so on. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors and could be applicable to explain diversity in qualitative features of wave phenomena. [ABSTRACT FROM AUTHOR]

The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control f$$ f $$ acts on the right end of it. As a first step, we prove the existence of a solution for the homogeneous problem, then we prove some estimates on its energy. Thanks to them, we prove an observability inequality, and using the notion of solution by transposition, we prove that the initial problem is null controllable. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem for a pseudo‐parabolic Kirchhoff equation with logarithmic nonlinearity. Using the potential well method, we obtain a threshold result of global existence and finite‐time blow‐up for the weak solutions with initial energy J(u0)≤d$$ J\left({u}_0\right)\le d $$. When the initial energy J(u0)>d$$ J\left({u}_0\right)&gt;d $$, we find another criterion for the vanishing solution and blow‐up solution. We also establish the decay rate of the global solution and estimate the life span of the blow‐up solution. Meanwhile, we study the existence of the ground state solution to the corresponding stationary problem. [ABSTRACT FROM AUTHOR]

In this paper, we concern the Cauchy problem of two‐dimensional (2D) compressible nematic liquid crystal flows with vacuum as far‐field density. Under a geometric condition for the initial orientation field, we establish a blowup criterion in terms of the integrability of the density for strong solutions to the compressible nematic liquid crystal flows. This criterion generalizes previous results of compressible nematic liquid crystal flows with vacuum, which concludes the initial boundary problem and Cauchy problem. [ABSTRACT FROM AUTHOR]