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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.

The stabilization properties of dissipative Timoshenko systems have been attracted the attention and efforts of researchers over the years. In the past 20 years, the studies in this scenario distinguished primarily by the nature of the coupling and the type or strength of damping. Particularly, under the premise that the Timoshenko beam model is a two-by-two system of hyperbolic equations, a large number of papers have been devoted to the study of the so-called partially damped Timoshenko systems by assuming damping effects acting only on the angle rotation or vertical displacement (Almeida Junior et al. in Math Methods Appl Sci 36:1965–1976, 2013; in Z Angew Math Phys 65:1233–1249, 2014; Alves et al. in SIAM J Math Anal 51(6):4520–4543, 2019; Ammar-Khodja et al. in J Differ Equ 194:82–115, 2003; Munoz Rivera and Racke in Discrete Contin Dyn Syst Ser B 9:1625–1639, 2003; J Math Anal Appl 341:1068–1083, 2008; Santos et al. in J Differ Equ 253(9):2715–2733, 2012). In these cases, the desired exponential decay property of the energy solutions is achieved when the non-physical equal wave speed assumption plays the role to stabilization according since the pioneering Soufyane’s paper (C R Acad Sci Paris 328(8):731–734, 1999). Recent results due to Almeida Junior et al. (Z Angew Math Phys 68(145):1–31, 2017; Z Angew Math Mech 98(8):1320–1333, 2018; IMA J Appl Math 84(4):763–796, 2019; Acta Mech 231:3565–3581, 2020) show that the second vibration mode or simply second spectrum of frequency and it’s damaging consequences appears as a lost element in analysis of stabilization and now it’s more clear that the damping importance into stabilization scenario of Timoshenko type systems. This paper considers a one-dimensional viscoelastic Timoshenko type system in the light of the second spectrum of frequency where the equal wave speed assumption is not needed for getting the exponential decay property. Precisely, we consider the so-called truncated version for the Timoshenko system according studies due to Elishakoff (Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010; ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11, 2015; Int J Solids Struct 109:143–151, 2017; J Sound Vib 435:409–430, 2017; Int J Eng Sci 116:58–73, 2017; Acta Mech 229:1649–1686, 2018; Z Angew Math Mech 98(8):1334–1368, 2018; Math Mech Solids, 2019) and we added a viscoelastic damping acting on shear force. We firstly prove the global well-posedness of the system by Faedo–Galerkin approximation. By assuming minimal conditions on the relaxation function, we establish an optimal explicit and energy decay rate for which exponential and polynomial rates are special cases. This result is new and substantially improves earlier results in the literature where the equal wave speeds plays the role for getting the stability properties. It is likely to open more research areas to Timoshenko system and probably others.

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 investigate the initial value problem for the 3D magneto-micropolar fluid equations with mixed partial viscosity. The main purpose of this paper is to establish global well-posedness of classical small solutions. More precisely, we prove that the global stability of perturbations near the steady solution is given by a background magnetic field. The proof is mainly based on the energy estimate and bootstrapping argument.

This paper is concerned with the following memory-type Timoshenko system $$\begin{aligned} {\left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K(\varphi _x+\psi )_x =0,\\ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+\displaystyle \int \limits _0^{+\infty } g(s)\psi _{xx}(t-s){\mathrm{d}}s=0,\\ \end{array}\right. } \end{aligned}$$ with Dirichlet boundary conditions, where g is a positive nonincreasing function satisfying, for some nonnegative functions $$\xi $$ and G, $$\begin{aligned}g'(t)\le -\xi (t)G(g(t)),\qquad \forall t\ge 0.\end{aligned}$$ Under appropriate conditions on $$\xi $$ and G, we establish some new decay results that generalize and improve many earlier results in the literature such as Mustafa (Math Methods Appl Sci 41(1): 192–204, 2018), Messaoudi et al. (J Integral Equ Appl 30(1): 117–145, 2018) and Guesmia (Math Model Anal 25(3): 351–373, 2020). We consider the equal speeds of propagation case, as well as the nonequal-speed case. Moreover, we delete some assumptions on the boundedness of initial data used in many earlier papers in the literature.

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.

The current paper is devoted to the study of the stability of space–time periodic traveling wave solutions and positive space–time periodic entire solutions of nonlocal dispersal cooperative systems in space–time periodic habitats. We first show the existence, uniqueness and stability of positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ for such nonlocal dispersal cooperative system. The existence of space–time periodic traveling wave solution connecting $${\mathbf {0}}$$ and positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ has been established by Bao, Shen and Shen (Commun. Pure Appl. Anal. 18: 361–396, 2019). In this paper, by using comparison principle and a weight function, we further show that the space–time periodic traveling wave solution for nonlocal dispersal cooperative system is asymptotically stable, as long as the initial value is uniformly bounded in a weighted space.

In this paper, a chemotaxis model with bounded chemotactic sensitivity and signal absorption is considered under homogeneous Neumann boundary conditions in the ball $$\Omega =B_R(0)\subset {\mathbb {R}}^n$$, where $$R>0$$ and $$n\ge 2$$. Here, S is a scalar function with $$S(s,t)\in C^2([0,\infty )\times [0,\infty ))$$. Moreover, for some positive constant K, $$|S(s,t)|\le K$$ for all $$s,t\in [0,\infty )$$. For all appropriately regular and radially symmetric initial data ($$u_0,v_0$$) fulfilling $$u_0\ge 0$$ and $$v_0>0$$, the present paper shows that there is a globally defined pair (u, v) of radially symmetric functions which are continuous in $$({{\overline{\Omega }}} \backslash \{0\}) \times [0, \infty )$$ and smooth in $$({{\overline{\Omega }}} \backslash \{0\}) \times (0, \infty )$$, and which solve the corresponding initial-boundary value problem for ($$\star $$) with $$(u(\cdot , 0), v(\cdot , 0))=\left( u_{0}, v_{0}\right) $$ in an appropriate generalized sense. Moreover, in the two-dimensional setting, it is shown that these solutions are global mass-preserving in the flavor of the identity $$\begin{aligned} \int \limits _\Omega u(x,t)=\int \limits _\Omega u_0(x)\quad \text {for all }t>0 \end{aligned}$$and any nontrivial of these globally defined solutions eventually becomes smooth and satisfies $$\begin{aligned} u(\cdot , t) \rightarrow \frac{1}{|\Omega |}\int \limits _\Omega u_{0},\quad \text { and } \quad v(\cdot , t) \rightarrow 0 \quad \text { as } t \rightarrow \infty \end{aligned}$$uniformly with respect to $$x\in \Omega $$.

In this paper, we study the long-time behavior toward rarefaction waves for the Cauchy problem to a one-dimensional Navier–Stokes equations for a reacting mixture. It is shown that under the condition adiabatic exponent $$\gamma $$ is close to 1, the global stability is established. In this paper, the initial perturbation can be large.

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 are concerned with the equation $$\begin{aligned} u_t= \sum _{i=1}^N\frac{\partial }{\partial x_i} \left( a_i(x)\left| u_{x_i} \right| ^{p_i(x) - 2} u_{x_i} \right) +\sum _{i=1}^N\frac{\partial b_i(u)}{\partial x_i},\ \ (x,t) \in \Omega \times (0,T), \end{aligned}$$ where $$a_i(x)|_{x\in \partial \Omega }=0$$ and $$a_i(x)|_{x\in \Omega }>0$$ . By the theory of anisotropic variable exponent Sobolev spaces, we study the well-posedness of weak solutions of this equation. Since $$a_i(x)$$ is degenerate on the boundary, the stability of weak solutions may be established without any boundary value condition. The main feature which distinguishes this paper from other related works lies in the fact that we propose a novel analytical method to deal with the stability of weak solutions.

This paper is concerned with the zero-flux logistic chemotaxis system with nonlinear signal production: $$u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u)$$ , $$v_t=\Delta v-v+g(u)$$ in a bounded and smooth domain $$\Omega \subset {\mathbb {R}}^n$$ ( $$n\ge 1$$ ), where $$\chi >0$$ is a constant, and $$f,g\in C^1({\mathbb {R}})$$ generalize the prototypes $$f(s)=s-\mu s^{\alpha }$$ and $$g(s)=s(s+1)^{\beta -1}$$ with $$\alpha >1$$ and $$\beta ,\mu >0$$ . The existing studies, for the case that the self-diffusion or the logistic-type source prevails over the cross-diffusion singly, have asserted the global boundedness of solutions under the assumption that $$\beta 0$$ suitably large. In the present paper, we prove that if $$\alpha -1=\beta =\frac{2}{n}$$ , then due to the self-diffusion and the logistic kinetics working together, the solutions are globally bounded regardless of the size of $$\mu >0$$ . Note that $$\alpha -1=\beta =\frac{2}{n}$$ reduces to $$\alpha =2$$ and $$\beta =1$$ if $$n=2$$ . Hence, our result covers that of the two-dimensional classical chemotaxis system with logistic source: $$u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+u-\mu u^2$$ , $$v_t=\Delta v-v+u$$ (Osaki et al. in Nonlinear Anal 51:119–144, 2002).

In this paper, we study the incompressible inviscid limit of the viscous two-fluid model in the whole space $${\mathbb {R}}^3$$ with general initial data in the framework of weak solutions. By applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of the densities and the velocity, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Euler equations in the time interval provided that the latter exists. Moreover, thanks to the Strichartz’s estimates of linear wave equations, we also obtain the convergence rates. The main ingredient of this paper is that our wave equations include the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolution of the solutions.

This paper studies the lower bound of the lifespan $T^{*}$ for the heat equation $u_t=\Delta u$ in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a nonlinear radiation condition on partial boundary: the normal derivative $\partial u/\partial n=u^{q}$ on $\Gamma_1\subseteq \partial\Omega$ for some $q>1$, while $\partial u/\partial n=0$ on the other part of the boundary. Previously, under the convexity assumption of $\Omega$, the asymptotic behaviors of $T^{*}$ on the maximum $M_{0}$ of $u_{0}$ and the surface area $|\Gamma_{1}|$ of $\Gamma_{1}$ were explored. In this paper, without the convexity requirement of $\Omega$, we will show that as $M_{0}\rightarrow 0^{+}$, $T^{*}$ is at least of order $M_{0}^{-(q-1)}$ which is optimal. Meanwhile, we will also prove that as $|\Gamma_{1}|\rightarrow 0^{+}$, $T^{*}$ is at least of order $|\Gamma_{1}|^{-\frac{1}{n-1}}$ for $n\geq 3$ and $|\Gamma_{1}|^{-1}\big/\ln\big(|\Gamma_{1}|^{-1}\big)$ for $n=2$. The order on $|\Gamma_{1}|$ when $n=2$ is almost optimal. The proofs are carried out by analyzing the representation formula of $u$ in terms of the Neumann heat kernel., Comment: 31 pages, 7 figures

In a preceding paper, we have studied the radiative heating of a semi-transparent body $$\varOmega $$ (e.g., glass) by a black radiative source S surrounding it, black source at absolute uniform temperature u(t) at time t between time 0 and time $$t_\mathrm{f}$$ , the final time of the radiative heating. This problem has been modeled by an appropriate coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. In the present paper, u being considered as the control variable, we want to adjust the absolute temperature distribution $$(x,t) \mapsto T(x,t)$$ inside the semi-transparent body $$\varOmega $$ near a desired temperature distribution $$T_\mathrm{d}(\cdot ,\cdot )$$ during the time interval of radiative heating $$]0,t_\mathrm{f}[$$ by acting on u, the purpose being to deform $$\varOmega $$ to manufacture a new object. In this respect, we introduce the appropriate cost functional and the set of admissible controls $$U_\mathrm{ad}$$ , for which we prove the existence of optimal controls. Introducing the state space and the state equation, a first-order necessary condition for a control $$u:t \mapsto u(t)$$ to be optimal is then derived in the form of a Variational Inequality by using the implicit function theorem and the adjoint problem. We close this paper by some numerical considerations.

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 is concerned with a time-periodic and delayed nonlocal reaction–diffusion population model with monostable nonlinearity. Under quasi-monotone or non-quasi-monotone assumptions, it is known that there exists a critical wave speed $$c_*>0$$ such that a periodic traveling wave exists if and only if the wave speed is above $$c_*$$ . In this paper, we first prove the uniqueness of non-critical periodic traveling waves regardless of whether the model is quasi-monotone or not. Further, in the quasi-monotone case, we establish the exponential stability of non-critical periodic traveling fronts. Finally, we illustrate the main results by discussing two types of death and birth functions arising from population biology.

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.

In dell’Isola et al. (Zeitschrift für Angewandte Math und Physik 66(6):3473–3498, 2015, Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016), the concept of pantographic sheet is proposed. The aim is to design a metamaterial showing: (i) a large range of elastic response; (ii) an extreme toughness in extensional deformation; (iii) a convenient ratio between toughness and weight. However, these required properties must coexist with non-detrimental mechanical characteristics in the presence of other kinds of imposed displacements. The aim of this paper is to prove via numerical simulations that pantographic sheets may effectively resist to coupled bending and extensional deformations. The four-parameter model introduced shows its versatility as it is able to encompass all the considered types of (large) deformations. The numerical integration scheme which we use is based on the same concepts exploited in Turco et al. (Zeitschrift für Angewandte Math und Physik 67(4):1–28, 2016): They prove that the Hencky-type discretization is very efficient also in nonlinear large deformations and large displacements regimes. In Part II of this paper, we will show that the used models are very effective to describe experimental evidence.

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 a Timoshenko system only with thermodiffusion effects. We establish exponential energy decay of the system with two kinds of boundary conditions under the assumption of the equal wave speeds. Our result extends the recent result by Aouadi et al. in Z. Angew. Math. Phys., 70, 117 (2019).

In this paper, we concerned with a haptotaxis system proposed as a model for fusogenic oncolytic virotherapy and syncytia, accounting for interaction between uninfected cancer cells, infected cancer cells, syncytia cancer cells, extracellular matrix and oncolytic virus. We present the global classical solution in two-dimensional bounded domains under appropriate regularity assumptions on the initial data.

In this paper, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin-Voigt type acting on the axial force, under fully Dirichlet boundary conditions. First, using a general criteria of Arendt-Batty, we prove the strong stability of our system. Finally, using a frequency domain approach combined with the multiplier method, we prove that the energy of our system decays polynomially with different rates., arXiv admin note: text overlap with arXiv:2008.11596, arXiv:2007.08316

The paper concerns an equilibrium problem for an elastic plate with a thin rigid inclusion. Both vertical and horizontal displacements of the plate are considered in the frame of the model. It is assumed that the inclusion crosses the external boundary of the plate. Moreover, the inclusion is delaminated from the plate which leads to appearance of a crack between the inclusion and the plate. To provide a mutual nonpenetration between crack faces, we consider nonlinear boundary conditions with unknown set of a contact. We prove a solution existence of the equilibrium problem and analyze an inverse problem with unknown elasticity tensor. To formulate the inverse problem, additional data are imposed to be determined from a boundary measurement. An existence of a solution to the inverse problem is proved, and stability properties are established.

In this paper, we consider the following pseudo-relativistic Choquard equations: $$\begin{aligned} (-\Delta +m^{2})^{s} u+wu=R_{N,t}\left( \frac{1}{|x-y|^{N-2t}}*u^{p}\right) u^{q}, \quad \mathrm{in} \;\;\mathbb {R}^{N}, \end{aligned}$$ where $$s,t\in (0,1)$$ , mass $$m>0$$ , $$w>-m^{2s}$$ , $$2

In this paper, we consider the standard linear solid model in $$\mathbb {R}^N$$ coupled with the Fourier law of heat conduction. First, we give the appropriate functional setting to prove the well-posedness of this model under certain assumptions on the parameters (that is, $$00$$ (that is, when the only dissipation comes through the heat conduction), we still have asymptotic stability, but with a slower decay rate. We also prove the optimality of the previous decay rate for the solution itself by using the eigenvalues expansion method. Finally, we complete the results in Pellicer and Said-Houari (Appl Math Optim 80: 447–478, 2019) by showing how the condition $$0

In this paper, we consider the Fornberg–Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in the proof. Besides, we also prove that the family of solitary solutions is orbitally stable and decays exponentially when wave speed c is bigger than 1.

This paper aims to study the existence and concentration of solutions for the stationary Dirac equation in $$\mathbb {R}^3$$ with critical nonlinearities: $$\begin{aligned} i\varepsilon \sum \limits _{k=1}^3\alpha _k\partial _k u-a\beta u+V(x)u=P(x)f(|u|)u+Q(x)g(|u|)u, \end{aligned}$$ where $$\varepsilon >0$$ is a small parameter and $$a>0$$ is a constant. We also show the semiclassical solutions $$\omega _\varepsilon $$ with maximum points $$x_\varepsilon $$ concentrating at a special set $${\mathcal {H}}_{P}$$ characterized by V(x), P(x) and Q(x), and for any sequence, $$x_\varepsilon \rightarrow x_0\in {\mathcal {H}}_{P}, v_\varepsilon (x):=\omega _\varepsilon (\varepsilon x+x_\varepsilon )$$ converges in $$H^1(\mathbb {R}^3,\mathbb {C}^4)$$ to a least energy solution u of $$\begin{aligned} i\sum \limits _{k=1}^3\alpha _k\partial _k u-a\beta u+V(x_0)u=P(x_0)f(|u|)u+Q(x_0)g(|u|)u. \end{aligned}$$

The problem of the small oscillations of a spherical layer of an inviscid fluid about a rigid spherical body in the gravitational field has been studied by Laplace in the case of a fluid layer of small depth. His results have been rediscovered by R. Wavre by using his method of the uniform process. The second author and his collaborators have studied the case of a layer of viscous fluid by means of the methods of the functional analysis. In this paper, we consider the case of a layer of viscoelastic fluid that obeys to the simpler Oldroyd’s law. Using the classical methods for the calculation of the potential and the methods of the functional analysis, we obtain from the variational form of the equations of the motion, an operatorial equation in a suitable Hilbert space. We reduce the problem of the small oscillations to the study of an operator pencil and so, we can precise the location of the spectrum and prove the existence of three sets of real eigenvalues. We give a theorem of existence and unicity of the solution of the associated evolution problem by means of the semi - groups theory.

In this paper, we study the nonlinear viscoelastic plate equation modeling the deformation of a suspension bridge, given by $$\begin{aligned} u_{tt}(x,y,t)+\Delta ^2 u(x,y,t) -\displaystyle \int \limits _0^{t} g(t-\tau )\Delta ^2 u(\tau )\;\hbox {d}\tau +a(x, y) \vert u_t\vert ^m u_t+\vert u\vert ^{\rho }u=0, \end{aligned}$$ in the bounded domain $$\Omega = (0, \pi )\times (-d, d)$$ , which is assumed to be hinged on its vertical edges and free on its remaining horizontal edges. We prove, with general conditions on the relaxation function, that the decay rate of energy is similar to that of the relaxation function regardless of the presence or the absence of the frictional damping.

We study the Cauchy problem for the higher-order nonlinear 3D Hartree-type equation $$\begin{aligned} \left\{ \begin{array} [c]{ll} i\partial _{t}u+\frac{1}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\left( \left| x\right| ^{-1}*\left| u\right| ^{2}\right) u,&{}{}\quad t>0, x\in {\mathbb {R}}^{3},\\ u\left( 0,x\right) =u_{0}\left( x\right) ,&{}{}\quad x\in {\mathbb {R}}^{3}, \end{array} \right. \end{aligned}$$ where $$*$$ denotes the convolution in $${\mathbb {R}}^{3}$$ . Cubic Hartree-type nonlinearity usually behaves critically for large time in three-space-dimensional case. In the present paper, we develop the factorization technique for the case of the higher-order Hartree-type equation (1.1) to show that the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Hartree-type equation has a modified character.

In this paper, we consider a three-dimensional chemotaxis-Stokes system 0.1 $$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),&\quad&x\in \Omega , t>0,\\&c_t+u\cdot \nabla c=\Delta c-nf(c),&\quad&x\in \Omega ,t>0,\\&u_t+\nabla P=\Delta u+n\nabla \phi ,&\quad&x\in \Omega ,t>0,\\&\nabla \cdot u=0,&\quad&x\in \Omega ,t>0,\\&(\nabla n^m- n S(x,n,c)\cdot \nabla c)\cdot \nu =\frac{\partial c}{\partial \nu }=0,u=0,&\quad&x\in \partial \Omega ,t>0,\\&n(x,0)=n_0(x),c(x,0)=c_0(x),u(x,0)=u_0(x),&\quad&x\in \Omega \end{aligned} \right. \end{aligned}$$ in a bounded domain $$\Omega \subset {\mathbb {R}}^3$$ with smooth boundary, where $$\phi $$ , f and S are given functions with values in $$\Omega $$ , $$[0,\infty )$$ and $${\mathbb {R}}^{3\times 3}$$ , respectively, under the no-flux boundary condition for n, c and Dirichlet boundary condition for u. It is also required that $$f\in C^1([0,\infty ))$$ is locally bounded in $$[0,\infty )$$ , that S satisfies $$|S(x,n,c)|\le n^{l-2}S_0(c)$$ with some nondecreasing $$S_0:[0,\infty )\rightarrow [0,\infty )$$ , and that m fulfills 0.2 $$\begin{aligned} m>\frac{5}{3}l-\frac{20}{9}\quad \text {and}\quad m>-\frac{3}{4}l+\frac{21}{8}\quad \text {with}\quad \frac{25}{12}\ge l>2. \end{aligned}$$ Then, with any reasonably regular initial data, the corresponding initial-boundary problem for (0.1) processes a global in time and bounded solution. This result extends the previous global boundedness result with $$m>m_*(l)$$ , where $$\begin{aligned} m_*(l)=\left\{ \begin{aligned}&l-\frac{5}{6},&\quad&\text {if }\,\frac{31}{12}\ge l>2,\\&\frac{7}{5}l-\frac{28}{15},&\quad&\text {if }\,l>\frac{31}{12}, \end{aligned} \right. \end{aligned}$$ ([15]) since both $$\begin{aligned} l-\frac{5}{6}\ge \frac{5}{3}l-\frac{20}{9} \end{aligned}$$ and $$\begin{aligned} l-\frac{5}{6}\ge -\frac{3}{4}l+\frac{21}{8} \end{aligned}$$ hold on $$\frac{25}{12}\ge l>2$$ .

This paper deals with a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption $$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot \left( \frac{u}{v}\nabla v\right) +f(u),&(x,t)\in \Omega \times (0,\infty ), \\&v_t=\Delta v+h(v,w),&(x,t)\in \Omega \times (0,\infty ), \\&w_t=\Delta w- w+u,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth convex bounded domain $$\Omega \subset {\mathbb {R}}^{2}$$ , where $$\chi >0$$ . When $$f(u)=0$$ , $$h(v,w)=-v+w$$ , we prove that the solution (u, v, w) of this model is globally uniformly bounded in time and exponentially converges to the steady state $$(\overline{u_{0}},\overline{u_{0}},\overline{u_{0}})$$ in the norm of $$L^{\infty }(\Omega )$$ provided that $$\chi 0$$ , where $$\overline{u_{0}}:=\frac{1}{|\Omega |}\int _{\Omega }u_{0}(x)\mathrm{d}x$$ . Moreover, in the case of $$f(u)=ru-\mu u^{2}$$ , $$h(v,w)=-v+w$$ , where $$r\in {\mathbb {R}}$$ , and $$\mu >0$$ , we obtain the global existence of solutions for this system. Furthermore, under the conditions of $$f(u)=ru-\mu u^{2}$$ , $$h(v,w)=-vw$$ and $$\mu >0$$ , $$r>0$$ , the solution (u, v, w) is also global in time.

In this paper, we are concerned with the large-time behavior of the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers (electrons or holes) in semiconductor device. For the Cauchy problem in $${\mathbb {R}}^3$$ , the global existence and uniqueness of smooth solutions, when the initial data are small perturbations of an equilibrium state, are obtained. Also, the solutions tend to the corresponding equilibrium state exponentially fast as the time tends to infinity. The analysis is based on the elementary $$L^2$$ -energy method, but various techniques are introduced to establish a priori estimates.

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.

For the non-isentropic compressible micropolar fluids in three dimensions, we show that the space–time behaviors of the fluid density, momentum and energy contain both generalized Huygens’ wave and diffusion wave as the non-isentropic compressible Navier–Stokes equation, while it only contains the diffusion wave for the micro-rational velocity. All of the previous estimates containing Huygens’ waves for wave behaviors of compressible flow models rely heavily on the conservative structure, for instance, the isentropic and non-isentropic Navier–Stokes equations. In this paper, after using the decomposition of fluid and electromagnetic parts to study three smaller Green’s matrices, we find that one of them is like a non-isentropic Navier–Stokes equation, but its energy is not conservative anymore due to the presence of the micro-rational velocity. We overcome this difficulty by providing a refined estimate for this Green’s matrix. Another difficulty is in proving the pointwise estimate of the micro-rational velocity only contains the diffusion wave, although its nonlinear terms contain both the Huygens’ wave and the diffusion wave. It is solved by using the relation of these two waves and developing new nonlinear estimates. Our pointwise estimate directly yields $$L^p$$ -estimate with $$p>1$$ , which is a generalization of the usual $$L^2$$ -estimate.

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.

This paper is concerned with the long-time behavior of Moore-Gibson-Thompson equation with degenerate memory effect, given by $$\begin{aligned} \tau u_{ttt}+\alpha (x)u_{tt}-b\Delta u_t-c^2\Delta u+\int \limits _0^tg(t-s)div\big (\beta (x)\nabla u(s)\big )ds=0, \quad ~in ~\Omega \times (0,\infty ), \end{aligned}$$ where structural damping and molecular relaxation are both locally distributed, satisfying $$\begin{aligned} \alpha (x)+\beta (x)\ge \delta ,\quad for~a.e. ~x\in \Omega . \end{aligned}$$ Under general conditions on memory function g(t), namely, $$g'(t)\le -\xi (t)H(g(t)),$$ and some basic restrictions imposed on $$\xi (t)$$ and $$H(\cdot )$$ , we show that the solution energy decays to zero in a general rate. Moreover, through specific examples, our result allows to deal with a much larger class of g(t), including but are not limited to polynomial or exponential decay. This will improve some related results in the literature.

In this paper, we consider the following Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases: 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda Q(x)u+\omega _{1}|u|^{2}u+\omega _{2}(K\star |u|^{2})u=0, &{} x\in {\mathbb {R}}^{3},\\ \displaystyle u>0,\quad u\in H^{1}({\mathbb {R}}^{3}),\\ \end{array}\right. } \end{aligned}$$ where $$\lambda >0$$ , $$Q\in C({\mathbb {R}}^{3},{\mathbb {R}})$$ is a potential well, $$\star $$ denotes the convolution, $$K(x)=\frac{1-3\cos ^{2}\theta }{|x|^{3}}$$ and $$\theta =\theta (x)$$ is the angle between the dipole axis determined by the vector x and the vector (0, 0, 1). When $$(\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}$$ lies in the defined unstable regime, under some suitable conditions on Q, the existence and concentration of nontrivial solutions to problem (0.1) are proved by using variational methods. In particular, the trapping potential is allowed to be sign-changing. Moreover, when $$(\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}$$ lies in the stable regime, we show that problem (0.1) with small bounded sign-changing potential has only trivial solutions.