ABSTRACT The presence of a delay in a thermoelastic system destroys the well-posedness and the stabilizing effect of the heat conduction [1]. To avoid this problem, we add to the system, at the delayed equation, a Kelvin–Voigt damping. At first, we prove the well-posedness of the system by the semigroup theory. Next, under appropriate assumptions, we prove the exponential stability of the system by

ABSTRACT The main aim of this paper is to prove the Ulam–Hyers stability of solutions for a new form of nonlinear fractional Langevin differential equations involving two fractional orders in the ψ-Caputo sense. Prior to proceeding to the main results, the proposed system is converted into an equivalent integral form by the help of fractional calculus. Next, we proceed to investigate the existence

This paper is devoted to study the dynamical behavior of a Belousov–Zhabotinsky reaction–diffusion system with nonlocal effect and find the essential difference between it and classical equations. First, we prove the existence of the solution by using the comparison principle and constructing monotonic sequences. Furthermore, the uniqueness is given by using fundamental solution and Gronwall's inequality

ABSTRACT In this paper, we establish blow-up rates for a higher-order semilinear parabolic equation with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild

ABSTRACT In this paper, an exact method named Riccati–Bernoulli sub-ODE method and a numerical method named Subdomain finite element method are proposed for solving the nonlinear generalized regularized long wave (GRLW) equation. For this purpose, Bäcklund transformation of the Riccati–Bernoulli equation and sextic B-spline functions are used for the exact and numerical solutions, respectively. The

In this article, we make several comparisons of Riemann solutions of compressible Euler equations in the modified Chaplygin gas state ( p = A ρ − ρ − 1 ) and classical Chaplygin gas state ( p = − ρ − 1 ). Firstly, through the analysis of characteristic method, we construct the Riemann solutions for the modified Chaplygin gas state and compare the Riemann solutions with different coefficients A. Then

In this paper, we introduce a single projection method with the Bregman distance technique for solving pseudomonotone variational inequalities in a real reflexive Banach space. The algorithm is designed such that its step size is determined by a self-adaptive process and there is only one computation of projection per iteration during implementation. This improves the convergence of the method and

The wave equation is considered, for all times t ∈ R , in a bounded plane domain G ϵ with an internal crack. The distance from one of the crack tips to the external boundary of G ϵ is proportional to a small parameter ϵ > 0 . Dirichlet or Neumann condition is given on the whole boundary of G ϵ . Near the crack tip, the first derivatives of solutions have square-root ( ∼ r − 1 / 2 ) singularities. The

ABSTRACT In this paper, we consider a viscoelastic equation with nonlinear feedback localized on a part of the boundary and in the presence of infinite-memory term. With imposing a more general condition on the relaxation function, we establish a more general stability result that generalizes and improves many earlier results in the literature. Our results are obtained without imposing any restrictive

The main goal of this work is to study a semilinear pseudo-parabolic equation with nonstandard growth conditions. For initial energy J ( u 0 ) ≤ d , by using the generalized potential well method, we establish a sharp threshold result on global existence or blow-up weak solution. On the other hand, we also show the asymptotic behavior of solutions and the blow-up time estimates. The optimal estimations:

ABSTRACT In this paper, we investigate a reaction–diffusion equation u t − d u x x = u ( a + b u p − 1 ∫ g ( t ) h ( t ) u q  d x ) with double free boundaries. We study blowup phenomena and asymptotic behavior of time-global solutions. For u 0 ( x ) = σ ϕ ( x ) , when a ≥ λ 1 , σ > 0 , if h 0 ≥ π 2 d a , then u will blow up in finite time. Meanwhile, we also prove when T ∗ < + ∞ , the solution must

In this paper we discuss the solvability of Langevin equations with two Hadamard fractional derivatives. The method of this discussion is to study the solutions of the equivalent Volterra integral equation in terms of Mittag–Leffler functions. The existence and uniqueness results are established by using Schauder's fixed point theorem and Banach's fixed point theorem, respectively. An example is given

In this paper, we consider Carleman estimates for the damped fourth-order plate operators ∂ t t − ρ ∂ t Δ + Δ 2 in a bounded smooth domain with Robin boundary conditions. Because the appearance of such kind of structural damping, the speed of propagation for solutions to the plate equation is infinite and the corresponding properties of the solution similar to heat equations, and is significantly different

This paper is devoted to study the existence of renormalized solutions of the parabolic Dirichlet equations of prototype: ∂ b ( x , u ) ∂ t = L u + f ( x , t ) in   Ω × ( 0 , T ) , L u = d i v ( | ∇ u | p ( x ) − 2 ∇ u + c ( x , t ) | u | γ ( x ) + F ) + l ( x , t ) | u | δ ( x ) b ( x , u ) | t = 0 = b ( x , u 0 ( x ) ) in   Ω , u = 0 on   ∂ Ω × ( 0 , T ) where b ( x , s ) ,   c ( x , t ) ,   l (

ABSTRACT In this paper, we study the regularity properties of solutions of dynamic evolution problems in perfect plasticity. We prove that for any space dimension and for any closed convex set of constraints containing zero as an interior point, the solutions are regular in space during a short time interval if the data are smooth and compactly supported. The result is based on the hyperbolic structure

ABSTRACT In this paper, we consider an abstract second-order viscoelastic equation with infinite memory, time-varying delay and a nonlinear source term. To the best of our knowledge, there is no blow-up result of solutions for the nonlinear viscoelastic equation with time-varying delay. Moreover, our result extends the blow-up result obtained for problems without memory term to those with infinite

ABSTRACT In this article, we study the L p solution of the Fredholm integral equation with parameters. On the one hand, we use Hilbert-type inequality to study Chandrasekhar-type integral operators, generalize the reachability results for the norm of the Chandrasekhar integral operator, and establish the existence and uniqueness results of the solutions of Chandrasekhar-type integral equations with

ABSTRACT In this paper, we study the nonexistence of solutions to the quasilinear Schrödinger equations: (1) − Δ p u + V ( x ) | u | p − 2 u − Δ p ( | u | 2 α ) | u | 2 α − 2 u = g ( x , u ) , x ∈ R N , (1) where p-Laplacian operator Δ p u = d i v ( | ∇ u | p − 2 ∇ u ) , 1 < p < N and α > 1 / 2 is a parameter. Under suitable conditions on g ( x , u ) , the nonexistence of solutions to Equation (1)

ABSTRACT This paper is motivated by the study of the following quasilinear Schrödinger equation − Δ u + V ( x ) u − [ Δ ( 1 + u 2 ) 1 2 ] u 2 ( 1 + u 2 ) 1 2 = λ h ( u ) , x ∈ R N , where N ≥ 3 , λ > 0 is a parameter and V ( x ) is a given positive potential. As an example, the nonlinearity includes the pure power type of h ( u ) = | u | p − 2 u for the well-studied case 12 − 4 6 < p < 2 ∗ , and the

ABSTRACT In this paper, we investigate the initial value problem of the generalized Degasperis–Procesi equation. We establish some new local-in-space blow-up criterion. Our results extend the corresponding results in the previous literature.

ABSTRACT In this paper, we focus on the existence of multiplicity of solutions for the singular quasilnear Schrödinger–Kirchhoff type problem: − a + b ∫ R 3 1 + α 2 2 | u | 2 ( α − 1 ) | ∇ u | 2 d x Δ u + α 2 Δ ( | u | α ) | u | α − 2 u = λ V ( x ) | u | p − 2 u + G ( x ) | u | 4 u , where x ∈ R 3 , a>0, b ≥ 0 , 0 < α < 1 , 1

We consider resolvents ( A ϵ + 1 ) − 1 of elliptic second-order differential operators A ϵ = − div   a ( x / ϵ ) ∇ in R d with ε-periodic measurable matrix a ( x / ϵ ) and study the asymptotic behaviour of ( A ϵ + 1 ) − 1 , as the period ε goes to zero. We provide a construction for the leading terms of the ‘operator asymptotics’ of ( A ϵ + 1 ) − 1 in the sense of L 2 -operator-norm convergence and

ABSTRACT We investigate the Schrödinger operators H ϵ = − Δ + W + V ϵ in R 2 with the short-range potentials V ϵ which are localized around a smooth closed curve γ. The operators H ϵ can be viewed as an approximation of the heuristic Hamiltonian H = − Δ + W + a ∂ ν δ γ + b δ γ , where δ γ is Dirac's δ-function supported on γ and ∂ ν δ γ is its normal derivative on γ. Assuming that the operator − Δ

A reaction–diffusion SIS epidemic model with saturated incidence rate and logistic source for the susceptible individuals is considered. We establish the uniform bounds of parabolic system and investigate the extinction and persistence of infectious diseases in terms of the basic reproduction number. We further analyze the asymptotic profiles of the endemic equilibrium for small and large movement

ABSTRACT In this paper, we investigate a porous thermoelastic system where the heat conduction is given by Coleman–Gurtin's law. We show that the system is exponentially or polynomially stable depending on a relationship between the coefficients of wave propagation speed.

ABSTRACT The aim of the present paper is to prove the existence result for a class of differential inclusions governed by the difference of two subdifferentials of nonconvex functions in Hilbert spaces. More precisely, by using the Moreau-Yosida regularization, the existence of local solutions for the following differential inclusion u ˙ ( t ) + ∂ Φ 1 ( u ( t ) ) − ∂ Φ 2 ( u ( t ) ) ∋ f ( t )  for

We revisit the scattering result below the ground state of Y. Gao and H. Wu [Scattering for the focusing H ˙ 1 2 -critical Hartree equation in energy space. Nonlinear Anal. 2010;73:1043–1056] on the radial focusing energy-subcritical Hartree equation in d = 5, using the method in Dodson and Murphy [A new proof of scattering below the ground state for the 3D radial focusing cubic NLS. preprint, arXiv:1712

ABSTRACT We analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modeled by a periodic geometry with periodicity ϵ > 0 . A resonator volume of thickness ε is connected with thin channels (opening ϵ 3 ) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit ϵ → 0 and find that

In this paper, we analytically and geometrically investigate the complexity of Maxwell–Bloch system by giving new insight. In the first place, the existence of homoclinic orbits is rigorously proved by means of the generalized Melnikov method. More precisely, for 6a−2b>c and d>0, it is certified analytically that Maxwell–Bloch system has two nontransverse homoclinic orbits. Secondly, Jacobi stability

ABSTRACT Solutions ( u , v ) to the chemotaxis system u t = ∇ ⋅ ( ( u + 1 ) m − 1 ∇ u − u ( u + 1 ) q − 1 ∇ v ) , τ v t = Δ v − v + u in a ball Ω ⊂ R n , n ≥ 2 , wherein m , q ∈ R and τ ∈ { 0 , 1 } are given parameters with m−q>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in L p ( Ω ) for some p > p 0 := n 2 ( 1 − ( m − q ) ) . For radially symmetric solutions, we show

ABSTRACT This paper investigates the limit cycles for planar differential system x ˙ = λ x − y + P n ( x , y ) ,   y ˙ = x + λ y + Q n ( x , y ) , where P n ( x , y ) , Q n ( x , y ) are homogeneous polynomials of degree n. Using the method of upper and lower solutions we show that this system has a unique limit cycle under some suitable conditions. Moreover, by employing the monotone iterative technique

ABSTRACT We study the homogenization of a double porosity diffusion problem in a thin highly heterogeneous composite medium formed by two materials separated by an imperfect interface, where the solution and its flux exhibit jumps. By applying homogenization techniques specific to the thin periodic domain under study, one derives the limit problem.

In this paper, we study the proximal incremental aggregated gradient (PIAG) method for minimizing the sum of smooth component functions and a nonsmooth function, both of which are allowed to be nonconvex. We show the linear convergence rate result under the metric subregularity and proper separation condition. Our method is developed based on a special auxiliary Lyapunov function sequence, and we also

ABSTRACT In this paper, sufficient initial conditions for finite-time blowup of smooth solutions of the irrotational compressible Euler equations with time-dependent damping are established. Our blowup conditions reveal that for sufficiently large initial velocity, fixed background density and with no largeness assumption on the initial density, the velocity of the fluid must collapse in finite time

ABSTRACT In this paper, we study global existence and blow-up for a periodic modified Camassa–Holm equation in nonhomogeneous Sobolev spaces. Also, we provide a key blow-up criteria to investigate norm inflation and ill-posedness problem for the equation in the critical Sobolev space.

ABSTRACT In this paper, we consider the pseudo-relativistic type Schrödinger equations with general nonlinearities. By studying the related constrained minimization problems, we obtain the existence of ground states via applying the concentration-compactness principle. Then some properties of the ground states have been discussed, including regularity, symmetry and etc. Furthermore, we prove that the

ABSTRACT This paper deals with the critical curves for a fast diffusive p-Laplacian equation with nonlocal inner source under nonlinear boundary flux in half-line. We obtain new critical global existence curve and critical Fujita curve by constructing various self-similar super- and sub-solutions.

ABSTRACT In this paper we consider uniform stabilization of the wave equation with variable coefficients in a bounded domain. The nonlinear damping is put partly on the interior of the domain and partly on the acoustic boundary. Under some checkable conditions on the coefficients, the energy decay results are established by Riemannian geometry method.

ABSTRACT In this paper, we generalize the minimal p-mean width of convex bodies (including the classic minimal mean width) to the Orlicz Brunn–Minkowski–Firey theory. The concept of Orlicz mean width of a convex body K in R n , w ϕ ( K ) , is introduced. Then we study the minimization problems of the form min { w ϕ ( T K ) : T ∈ S L ( n ) } and show that bodies which appear as solutions of such problems

ABSTRACT We prove the existence and uniqueness of a bounded random solution for a nonlinear stochastic integrodifferential equation on R + . We also show that the sequence of successive approximations converges to the solution x ( t , ω ) almost surely and in mean square at each t ∈ R + . An application of our result to a stochastic differential system is also given in Section 4.

ABSTRACT In this paper, we investigate a class of Kirchhoff models with integro-differential damping given by a possibly vanishing memory term in a past history framework and a nonlinear nonlocal strong dissipation u t t + α μ △ 2 u − △ p u − ∫ − ∞ t μ ( t − s ) △ 2 u ( s ) d s − N ∫ Ω | ∇ u ( t ) | 2 d x △ u t + f ( u ) = h , defined in a bounded Ω of R N . Our main goal is to show the well-posedness

ABSTRACT For electrodynamic equations related to non-conducting dispersible medium we consider the inverse problem of recovering two variable coefficients from a given phaseless information of solutions to the equations. One of these coefficients is the permittivity while the second one characterizes the time dispersion of the medium. We suppose that unknown coefficients differ from given constants

This paper is concerned with maximization and minimization problems related to a boundary value problem involving a p-Laplacian type operator. These optimization problems are formulated relative to the rearrangement of a fixed function. Firstly, by introducing a truncated function, we establish the existence and uniqueness of the solution of the boundary value problem involving a p-Laplacian type operator

ABSTRACT In this paper, first we establish the weak type B M O estimates of the commutators of the maximal operator, the fractional maximal operator and some commutators related to linear operators on the Herz type spaces with variable exponent. Subsequently the weak type Lipschitz estimates of Calderón–Zygmund singular integral commutator and fractional integral commutator from Herz-type Hardy spaces

ABSTRACT The purpose of this paper is to investigate a structural interaction model coupled with dynamic von Karman equations, without neither rotational inertia nor clamped boundary conditions. Our fundamental goal is to establish the existence and the uniqueness of the weak solution for the so-called global functional energy. The stability is also discussed. At the end, a numerical study based on

(2020). Correction. Applicable Analysis. Ahead of Print.

This work is devoted to studying a wave equation with degenerate nonlocal nonlinear damping and source terms. By potential well theory, we show the asymptotic stability of energy in the presence of a degenerate damping of polynomial type when the initial energy is small. Also, we firstly derive some sufficient conditions on initial data which lead to finite time blow-up.

In this paper, we consider a steady-state flow the thermomicropolar fluid through a thin straight channel. The flow is governed by the prescribed pressure drop between channel's ends. The heat exchange between the fluid inside the channel and the exterior medium is allowed through the upper wall, whereas the lower wall is insulated. Using the asymptotic analysis with respect to the thickness of the

ABSTRACT In this paper we obtain the interior Hölder regularity of the gradient for the elliptic p ( x ) -Laplacian equation with the logarithmic function d i v A ∇ u ⋅ ∇ u p ( x ) − 2 2 ln e + A ∇ u ⋅ ∇ u 1 / 2 A ∇ u = d i v | f | p ( x ) − 2 ln e + f f under some proper assumptions on the Hölder continuous functions p , f and A. This extends previous results obtained by Giannetti and Passarelli di

ABSTRACT In this paper, we consider the global regularity of 2D magnetic Bénard fluid equations with almost magnetic diffusion and thermal diffusivity and without kinematic viscosity. We focus on this goal in two ways. In one way, the magnetic diffusion and thermal diffusivity are separately given by D 2 and D 3 two Fourier multipliers whose symbols m 2 and m 3 are respectively given by m 2 ( ξ ) ≡

We are interested in studying the existence of solution for the generalized quasilinear Schrödinger equation: (P) − d i v ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , u ∈ H 1 ( R N ) (P) in R N , where N ≥   3 , g and h are suitable smooth functions and V is a potential that may change sign and be unbounded. By using a change of variables and variational arguments, we

ABSTRACT The A − φ − B magnetodynamic Maxwell system given in its potential and space-time formulation is a popular model considered in the engineering community. It allows to model some phenomena such as eddy current losses in multiple turn winding. Indeed, in some cases, they can significantly alter the performance of the devices, and consequently can no more be neglegted. It turns out that this

ABSTRACT This paper focus on the quantitative stability of a class of two-stage stochastic linear variational inequality problems whose second stage problems are stochastic linear complementarity problems with fixed recourse matrix. Firstly, we discuss the existence of solutions to this two-stage stochastic problems and its perturbed problems. Then, by using the corresponding residual function, we

ABSTRACT This paper is concerned with the inverse scattering and the transmission eigenvalues for anisotropic periodic layers. For the inverse scattering problem, we study the Factorization method for shape reconstruction of the periodic layers from near field scattering data. This method provides a fast numerical algorithm as well as a unique determination for the shape reconstruction of the scatterer

ABSTRACT In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog

In this paper, we study the following modified quasilinear Schrödinger equation − Δ u + V ( x ) u − κ u Δ ( u 2 ) + q h 2 ( | x | ) | x | 2 ( 1 + κ u 2 ) u + q ∫ | x | + ∞ h ( s ) s ( 2 + κ u 2 ( s ) ) u 2 ( s ) d s u = | u | p − 2 u i n   R 2 , where κ > 0 , q>0, p>6 and V ∈ C ( R 2 , R ) . We develop a more direct and simpler approach to prove the existence of ground state solutions. Our method is

The blow-up of solutions to a coupled system of semilinear wave equations in the sub-critical and critical cases is investigated. The system contains scattering dampings and combined nonlinearities. The upper bound lifespan estimates of solutions which are related to the indexes of nonlinear terms are established by applying the functional method and iteration technique.

ABSTRACT In this paper, we consider the regularity criteria of 3D incompressible Boussinesq equations. By using the Littlewood–Paley decomposition technique to establishing a regularity criterion in terms of the one-directional derivative of velocity in Besov spaces. Our result improves some previous works (Liu Q. A regularity criterion for the Navier–Stokes equations in terms of one-directional derivative

ABSTRACT In the subcritical energy case, local well-posedness is established in the radial energy space for a class of fractional inhomogeneous Choquard equations. The best constant of a Gagliardo–Nirenberg type inequality is obtained. Moreover, a sharp threshold of global existence versus blow-up dichotomy is obtained for mass super-critical and energy subcritical solutions.

ABSTRACT The stationary Stokes equations for a generalized Newtonian fluid with nonlinear unilateral, and slip and leak boundary conditions are investigated. Boundary conditions include the generalized Clarke gradient and the convex subdifferential, and the variational formulation of the problem is the variational–hemivariational inequality for the velocity field. Existence and uniqueness result for