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The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

In this paper, we obtain the existence of positive solution for a system of quasilinear Schrodinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrodinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrodinger equations.

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.

In this paper, we are concerned with variable coefficients plate system subjected to three partially distributed feedbacks: time-varying delay, frictional and viscoelastic dissipations. This work is devoted to, without any prior quantification of both decay rate of relaxation function and growth rate of frictional dissipation near the origin, establish a general decay result which corresponds to a certainly stable ODE. Our result extends the decay result obtained for some kind of problems with finite history to problem with infinite history. Moreover, this paper allows a wider class of kernels of infinite history, and the usual exponential and polynomial decay rates are only special cases. The proof is based on the multiplier method and some techniques about convex functionals.

The present paper concerns a class of abstract mixed variational problems governed by a strongly monotone Lipschitz continuous operator. With the existence and uniqueness results in the literature for the problem under consideration, we prove a general convergence result, which shows the continuous dependence of the solution with respect to the data by using arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Then we consider an associated optimal control problem for which we prove the existence of optimal pairs. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to mixed variational problems. To provide an example, we illustrate our results in the study of a mathematical model which describes the equilibrium of an elastic body in frictional contact with a foundation.

In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed.

This paper deals with a class of thin-film equation, which was considered in Li et al. (Nonlinear Anal Theory Methods Appl 147:96–109, 2016), where the case of lower initial energy ( $$J(u_0)\le d$$ and d is a positive constant) was discussed, and the conditions on global existence or blow-up are given. We extend the results of this paper on two aspects: Firstly, we consider the upper and lower bounds of blow-up time and asymptotic behavior when $$J(u_0)d$$ .

In this paper, we consider a modified three-dimensional incompressible Boussinesq model. The model considered in this paper has viscosity in the velocity equations, but no diffusivity in the temperature equation. To bypass the difficulty caused by the absence of thermal diffusion, we make use of the maximal $$L_t^{p}L_x^{q}$$ regularity for the heat kernel to establish the global regularity result.

In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$30$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .

In the paper, an equilibrium problem for an elastic body with a thin elastic and a volume rigid inclusion is analyzed. It is assumed that the thin inclusion conjugates with the rigid inclusion at a given point. Moreover, a delamination of the thin inclusion is assumed. Inequality type boundary conditions are considered at the crack faces to prevent a mutual penetration between the faces. A passage to the limit is justified as the rigidity parameter of the thin inclusion goes to infinity. The main goal of the paper is to analyze an optimal control problem with a cost functional characterizing a deviation of the displacement field from a given function. A rigidity parameter of the thin inclusion serves as a control function. An existence theorem to this problem is proved.

In this article, we have studied the transmission problem of a system of hyperbolic equations consisting of a free wave equation and a wave equation with dissipation on the boundary, each one acting on a part of its one-dimensional domain. This paper proves the equivalence between the exponential stability previously proven by Liu and Williams (Bull Aust Math Soc 97:305–327, 1998) and the inequality observability on the boundary as a result of this paper. First of all, we have built an auxiliary problem on where we extracted some slogans to be used later. Then we have introduced a number $${\mathcal {X}}>0$$ representing the difference between the speed of wave propagation in each part of the domain, and we proved one observability inequality on the boundary. Finally, we proved the equivalence between the two properties.

This paper is concerned with the long-time behaviour of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows introduced in Lin and Liu (Commun Pure Appl Math, 48:501–537, 1995) with a non-autonomous forcing bulk term and order parameter field boundary conditions. In this paper, we prove the existence of pullback attractors and estimate the upper bound of its fractal dimension under some suitable assumptions.

Umeda et al. (Jpn J Appl Math 1:435–457, 1984) considered a rather general class of symmetric hyperbolic–parabolic systems: $$A^{0}z_{t}+\sum_{j=1}^{n}A^{j}z_{x_{j}}+Lz=\sum_{j,k=1}^{n}B^{jk}z_{x_{j}x_{k}}$$ and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices $${L}$$ and $${B^{jk}(j,k=1,\ldots,n)}$$ are both assumed to be real symmetric. So far there are no general results in case that $${L}$$ and $${B^{jk}}$$ are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of $${L^{1}({\mathbb{R}}^3)}$$ - $${L^2({\mathbb{R}}^3)}$$ type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of $${L^p({\mathbb{R}}^n)}$$ - $${L^{q}({\mathbb{R}}^n)}$$ - $${L^{r}({\mathbb{R}}^n)}$$ estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.

This paper is concerned with the asymptotic behavior on $${\partial\Omega}$$ of boundary blow-up solutions to semilinear elliptic equations $$\left\{\begin{array}{ll} \Delta u=b(x)f(u),~~ &x\in \Omega, \\ u(x)=\infty, ~~ &x\in\partial\Omega,\end{array} \right.$$ where b(x) is a nonnegative function on $${\Omega}$$ and may vanish on $${\partial\Omega}$$ at a very degenerate rate; f is nonnegative function on [0,∞) and normalized regularly varying or rapidly varying at infinity. The main feature of this paper is to establish a unified and explicit asymptotic formula when the function f is normalized regularly varying or grows faster than any power function at infinity. The effect of the mean curvature of the nearest point on the boundary in the second-order approximation of the boundary blow-up solution is also discussed. Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.

This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).

In this paper we study the approximation of random diffusion by nonlocal diffusion with properly rescaled kernels in age-structured models. First we show that solutions of age-structured models with nonlocal diffusion under Dirichlet and Neumann boundary conditions converge to solutions of the corresponding age-structured models with random diffusion under Dirichlet and Neumann boundary conditions, respectively. Then we prove that the principal eigenvalues of the nonlocal operators in age-structured models under Dirichlet and Neumann boundary conditions converge to the principal eigenvalues of the corresponding Laplace operators in age-structured models under Dirichlet and Neumann boundary conditions, respectively.

Previously, the incompressible limit of the equations of Hookean elastodynamics has been established in the whole space. The present paper is devoted to the study of the incompressible limit in a bounded domain. Here, the main difficulty lies in the interactions between the large operator and the boundary. By designing some delicate semi-norms and energy estimates, we obtain the uniform estimates of the solutions. As a result, the incompressible limit is established by classical compactness argument.

This paper is devoted to classifying the dynamical behavior of solutions to a free boundary problem of a predator–prey model with a nonlocal reaction term. We investigate the spreading-vanishing dichotomy, that is, the asymptotic behavior of predator and prey species. Moreover, some criteria are also established for spreading and vanishing. At last, when spreading occurs, we obtain an upper and a lower bound of the asymptotic spreading speed.

In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay and a very general class of memory kernel functions. Under some suitable conditions and by introducing a suitable Lyapunov functional, we state and prove explicit and general stability results of the energy. We give also some applications to illustrate our results. This work generalizes the previous results without time delay term to those with time-varying delay.

In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Benard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity $$u_2$$ or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as $$\Delta u_2$$ , then we can obtain the global well-posedness, inviscid limit and some exponential decay estimates. Our key observation is the nice symmetrical structure hidden in the corresponding system, which plays an extremely important role in the global well-posedness studied here.

In this paper, we consider the signal-dependent diffusion and sensitivity in a chemotaxis–competition population system with two different signals in a two-dimensional bounded domain. We consider more general signal production functions and assume that the signal-dependent diffusion is a decreasing function which may be degenerate with respect to the density of the corresponding signal. We first obtain the global existence and uniform-in-time bound of classical solutions and show that the blow-up effect can be precluded for signal-dependent diffusion and sensitivity with certain properties. Then, by constructing Lyapunov functionals, we study the global attractivity of nonzero (boundary/positive) homogeneous steady states under three different strengths of competition. In particular, we obtain that the nonzero boundary constant steady states are globally asymptotically stable when they are globally attractive, which means no pattern formation occurs, while for interior constant steady state, its global attractivity can imply the global stability for some special signal production functions. Finally, numerical simulations show that for large signal sensitivity, different signal production functions can lead to various complex spatial–temporal patterns around the positive homogeneous steady state. In particular, for a given signal production mechanism, various patterns are observed for different population growth rates.

In this paper, we propose a new viscosity for a coupled compressible Navier–Stokes/Allen–Cahn system because it describes the motion of a gas in a flowing liquid. The viscosity depends on two different variables (the density and the unknown function in Allen–Cahn equations). We establish blow-up criterions for strong solutions to initial-boundary value problem. We also show that the strong solutions can be improved to classical solutions. Precisely, when the viscosity only depends on the density, we prove the global existence and uniqueness of the Navier–Stokes/Allen–Cahn equations.

In this paper, we are concerned with existence, uniqueness, regularity, and continuous dependence upon the initial data for mild solutions of an equation of viscoelasticity in power type materials in the $$L^q$$ -setting. We also analyze the continuation of this solution up a maximal time of existence and a blow-up alternative. Finally, we obtain the global well-posedness for the problem.

In this paper, we are concerned with an SIRS epidemic reaction–diffusion system with standard incidence infection mechanism in a spatially heterogeneous environment. We first establish the uniform bounds of solutions and then derive the threshold dynamics in terms of the basic reproduction number $$\mathcal {R}_0$$ . Our main focus is on the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible, infected or recovered population is small, and our results show that the disease always exists in the entire habitat in each case of small diffusion rate. This suggests that restricting the diffusion (migration) rate of population is not an effective strategy of disease eradication.

The paper is devoted to a reaction-diffusion problem describing diffusion and consumption of nutrients in a biological tissue consisting of small cells periodically arranged in an extracellular matrix. Cells consume nutrients with a rate proportional to cell area and to nutrient concentration. The dependence on the nutrient concentration can be linear or nonlinear. The cells are modeled by a potential approximating the Dirac’s delta-function. The potential has a periodically distributed support of small measure. The problem contains two small parameters: the diameter of a cell and the distance between the cells (in comparison with the characteristic macroscopic size). In the multi-dimensional formulation assuming some restriction on the relation of parameters, we prove convergence of solution of this problem to the solution of a limiting homogenized problem. We show that the problem is non-homogenizable in classical sense if this restriction fails.

The object of the present paper is to consider the Dirichlet boundary value problem of the coupled linear quasi-static theory of elasticity for porous isotropic elastic infinite strip. The general representations of a regular solutions of a system of considered equations for a homogeneous isotropic medium are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. Using the Fourier method, the Dirichlet BVP is solved effectively (in quadratures) for the infinite strip.

In this paper, we study a Lotka–Volterra cooperative system with nonlocal dispersal under worsening habitats. By constructing appropriate vector super-/subsolutions combined with the monotone iteration scheme, we obtain the existence of bounded and positive forced waves connecting zero equilibrium to the coexistence state of the limiting system corresponding to the most favorable resource with the wave speed at which the habitat is worsening. Compared with the existence result of traveling wave to the homogeneous system, we find that the wave phenomenon is more likely to occur in such shifting environment. Here, we remove the common assumption that the dispersal kernels are compactly supported and symmetric. Further, we investigate the tail behavior of the forced waves, especially for the rate of convergence to the extinction state, and derive the long-time dynamics of two species. Our result shows that weak interspecies cooperation and nonlocal diffusion pattern cannot prevent species from disappearing eventually under such a worsening habitat.

In this paper, we introduce a new Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, governed by an approximating function h. We characterize the well-posedness in terms of the metric properties of the family of approximating sets, under various assumptions on h. Then, we use the well-posedness properties in order to obtain convergence results of the solution with respect to the data. The proofs are based on arguments of monotonicity combined with the properties of the Clarke directional derivative. Our results provide mathematical tools in the study of a large number of static problems in Contact Mechanics. To provide an example, we consider a mathematical model which describes the equilibrium of a rod–spring system with unilateral constraints. We prove the unique weak solvability of the model, and then we illustrate our abstract convergence results in the study of this contact problem and provide the corresponding mechanical interpretations.

In this paper, we consider the initial value problem for the compressible fluid models of Korteweg type in $${\mathbb {R}}^n(n\ge 3)$$ and asymptotic profile of global solutions and the corresponding convergence rate are established. The structure of the nonlinear term plays a very important role in constructing asymptotic profile. The proof is based on the decay estimate of solutions operator, decay estimate and weighted decay estimate of global solutions.

In previous work (Alves et al. in Z Angew Math Phys 69:106, 2018), by using the linear semigroup theory, Alves et al. investigated the existence and exponential stability results for a Moore–Gibson–Thompson model encompassing memory of type 1, 2 or 3 in a history space framework. In this paper, we continue to consider the similar problem with type 1 and establish explicit and general decay results of energy for system in both the subcritical and critical cases, by introducing suitable energy and perturbed Lyapunov functionals and following convex functions ideas presented in Guesmia (J Math Anal Appl 382:748–760, 2011). Our results allow a much larger class of the convolution kernels which improves the earlier related results.

In this paper, we are concerned with the generalized FORQ/MCH equation, which includes the celebrated Fokas–Olver–Rosenau–Qiao equation (or also called the modified Camassa–Holm equation). Firstly, a priori estimates in the transport equation theory and 1-D Morse-type estimates were applied to derive a blow-up criterion. Then, we exploit the characteristic ordinary differential equation to construct a conservative property, which leads to the precise blow-up scenario. Finally, making use of the fine structure and conservation laws, we present a blow-up result for the strong solutions with respect to the initial data.

In this paper, we consider the inviscid Boussinesq system in the presence of magnetic field in three space dimensions. We prove that the system admits infinitely many global weak solutions with large initial data. Our main tool is based on the convex integration method developed by De Lellis and Szekelyhidi in the context of incompressible Euler system.

We consider the problem of a small body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible steady Navier–Stokes equations in an exterior domain in a half-space, with appropriate boundary conditions on the wall, the body and at infinity. In this paper, we first prove in a very general setup the existence of weak solutions for the problem with the body. Then, we show that any such solution can be truncated and then extended to provide a weak solution for a simplified problem where the body is replaced by a (small) source term with compact support. This simplified problem was already shown to possess strong solutions. We then prove a weak–strong uniqueness theorem to show the uniqueness of solutions for the simplified problem. Finally, we show that this also implies the uniqueness of solutions for the problem of the moving body which proves that the solutions of both problems have the same asymptotic behavior at infinity.

This paper deals with a drift-diffusion system being subjected to Robin boundary conditions. Under appropriate hypotheses on the data, a local existence result in time is obtained by using a fixed-point argument combined with some a priori estimates.

In this paper, we deal with the semilinear wave equations with a local internal damping and dynamic Wentzell boundary conditions with a memory term. The stabilization estimate is more difficult to obtain since the physical energy of the system not only contains the $$H^1$$ Sobolev norm of the solution but also depends on the memory term on the boundary. The exponential stabilization is attained by constructing new Lyapunov functionals and using multiplier methods. To illustrate the results, numerical simulations are given in the last part.

The global existence and regularity problem on the magnetohydrodynamic (MHD) equations with fractional dissipation is not well understood for many ranges of fractional powers. This paper examines this open problem from a different perspective. We construct a class of large solutions to the d-dimensional ( $$d=2,3$$ ) MHD equations with any fractional power. The process presented here actually assesses that an initial data near any function whose Fourier transform lives in a compact set away from the origin always leads to a unique and global solution.

The main objective of this paper is to study the temporal approximation scheme of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics. We first prove the existence of a global attractor in V by using the Sobolev compactness embedding theorem, and then we prove its global attractor and stationary statistical properties of the temporal approximation scheme convergence to those of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics as the time step goes to zero.

In this paper, we consider a nonlinear divergence form parabolic equation with time-dependent coefficient and inhomogeneous Neumann boundary condition. We establish the new sufficient conditions on nonlinear functions to guarantee that the positive solution $$u(\pmb {x},t )$$ exists globally. Under the conditions to guarantee that the positive solution blows up, by establishing the Sobolev inequality in multidimensional space and constructing the unified functionals, we obtain upper and lower bounds of the blow-up time $$t^*$$ .

This paper is concerned with the Cauchy problem of the Quasigeostrophic equation on the torus $${\mathbb {T}}^2$$ . We prove the almost sure existence and uniqueness of the global solution with the random initial data in $${\mathcal {H}}^s({\mathbb {T}}^2), s\in [-1,0)$$ .

This paper studies the existence, nonexistence and uniqueness of positive solutions for a class of quasilinear equations. We also analyze the behavior of these solutions with respect to two parameters $$\kappa $$ and $$\lambda $$ that appear in the equation. The proof of our main results relies on bifurcation techniques, the sub- and supersolution method and a construction of an appropriate large solution.

In this paper, we present a class of general solutions of the unsteady Oseen equations in terms of two scalar functions. We also give a condition for a divergence-free vector to be a possible solution of Oseen equations.

In this paper, the Cauchy problem to the 1D compressible magnetohydrodynamics equations is considered. We establish the global existence and uniqueness of strong solution when the viscosity coefficient is assumed to be constant or density dependent. The analysis is based on some new mathematical techniques and the weighed Caffarelli–Kohn–Nirenberg inequality to get the $$L^p(1\le p

In this paper, we prove global well-posedness for the incompressible Oldroyd-B model under some general initial conditions by using Littlewood–Paley theory. In particular, we can obtain global well-posedness for the 2D case with large velocity, which is very different from the previous works.

We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state., arXiv admin note: text overlap with arXiv:1711.03371, arXiv:2001.01509

In this paper, we study thermoelastic Bresse systems where the heat conductions, acting on the axial force, are modeled for both Fourier’s and Cattaneo’s laws. For the Fourier’s case, we prove exponential stability of solutions if and only if the condition of equal wave speeds is satisfied. For the Cattaneo’s case, we characterize the exponential stability by a new condition on the coefficients of the system. We also prove, in the general case, polynomial stability of solutions. These results complement previous results obtained by Liu and Rao (Z Angew Math Phys 60:54–69, 2009), Fatori and Rivera (IMA J Appl Math 75:881–904, 2010) and Dell’Oro (J Differ Equ 258:3902–3927, 2015).

The aim of this paper is to provide some results in the study of an abstract optimization problem in reflexive Banach spaces and to illustrate their use in the analysis and control of static contact problems with elastic materials. We start with a simple model problem which describes the equilibrium of an elastic body in unilateral contact with a foundation. We derive a variational formulation of the model which is in the form of minimization problem for the stress field. Then we introduce the abstract optimization problem for which we prove existence, uniqueness and convergence results. The proofs are based on arguments of lower semicontinuity, monotonicity, convexity, compactness and Mosco convergence. Finally, we use these abstract results to deduce both the unique solvability of the contact model and the existence and the convergence of the optimal pairs for an associated optimal control problem.

This paper deals with a fourth-order parabolic PDE arising in the theory of epitaxial growth of crystal. For the stationary problem, we find a ground-state solution on its corresponding Nehari manifold by Lagrange multiplier method. As far as the evolution problem is concerned, we study the dynamics for both the global solution and the blow-up solution. Especially, for the global solution, we prove that the energy functional decays exponentially and we get the concretely decay rate. For the blow-up solution, the exponential growth of the solution is obtained, and we also obtain the growth rate. The behavior of the energy functional as $$t\rightarrow T$$ is also discussed, where T is the blow-up time. Moreover, we prove that there exists blow-up solution with arbitrary high initial energy. Finally, for the low initial energy case, we give some equivalent conditions for the solutions blow up in finite time and exist globally, respectively. Our results extend the results got by Escudero et al. (Eur J Appl Math 24(3):437–453, 2013; J Differ Equ 254(6):2515–2531, 2013; Math Method Appl Sci 37(6):793–807, 2014; J Math Pures Appl 103(4):924–957, 2015).

In this paper, we study the existence and nonexistence of traveling wave solution for the nonlocal dispersal predator–prey model with spatiotemporal delay. This model incorporates the Leslie–Gower functional response into the Lotka–Volterra-type system, and both species obey the logistic growth. We explore the existence of traveling wave solution for $$c\ge c^{\star }$$ by using the upper-lower solutions and the Schauder’s fixed point theorem. Furthermore, the nonexistence of traveling wave solution for $$c