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In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u= \lambda u+\mu u\ln u^2+|u|^{2^{**}-2}u, &x\in\Omega,\\ u=\dfrac{\partial u}{\partial \nu}=0, &x\in\partial\Omega \end{cases} \end{eqnarray*} is considered, where $\Omega\subset \mathbb{R}^{N}$ is a bounded smooth domain with $N\geq5$. Some interesting phenomenon occurs due to the uncertainty of the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at lest one nontrivial weak solution under some appropriate assumptions of $\lambda$ and $\mu$. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced.

In this paper, we prove scattering asymptotics for the 2D (discrete dimension) cubic resonant system. This scattering result was used in Zhao \cite{Z1} as an assumption to obtain the scattering for cubic NLS on $\mathbb{R}^2\times \mathbb{T}^2$ in $H^1$ space. Moreover, the 1D analogue is proved in Yang-Zhao \cite{YZ}. Though the scheme is also tightly based on Dodson \cite{D}, the 2D case is more complicated which causes some new difficulties. One obstacle is the failure of `$l^2$-estimate' for the cubic resonances in 2D (we also discuss it in this paper, which may have its own interests). To fix this problem, we establish weaker estimates and exploit the symmetries of the resonant system to modify the proof of \cite{YZ}. At last, we make a few remarks on the research line of `long time dynamics for NLS on waveguides'., 29 pages, comments are welcome!

In this paper, we study a free boundary problem, which arises from an optimal trading problem of a stock that is driven by a uncertain market status process. The free boundary problem is a variational inequality system of three functions with a degenerate operator. The main contribution of this paper is that we not only prove all the four switching free boundaries are no-overlapping, monotonic and $C^{\infty}$-smooth, but also completely determine their relative localities and provide the optimal trading strategies for the stock trading problem.

In a previous paper(2021), the author studied the asymptotic behavior of coexistence steady-states to the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. As a result, he proved that the asymptotic behavior can be characterized by a limiting system that consists of a semilinear elliptic equation and an integral constraint. This paper studies the set of solutions of the limiting system. The first main result gives sufficient conditions for the existence/nonexistence of nonconstant solutions to the limiting system by a topological approach using the Leray-Schauder degree. The second main result exhibits a bifurcation diagram of nonconstant solutions to the one-dimensional limiting system by analysis of a weighted time-map and a nonlocal constraint., 32 pages, 3 figures

We are interested in the multi-dimentional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the "div-curl" structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. The new ingredient lies in the introduction of effective flux $(\theta,\mathcal{G})$, which enables us to capture the dissipation arising from \textit{combination} of density and deformation tensor. In absence of compatible conditions, the partial dissipation is found in non-Newtonian compressible fluids, which is weaker than that of usual Navier-Stokes equations., Comment: title changed and some proofs in the paper are rearranged

We consider stochastic wave map equation on real line with solutions taking values in a $d$-dimensional compact Riemannian manifold. We show first that this equation has unique, global, strong in PDE sense, solution in local Sobolev spaces. The main result of the paper is a proof of the Large Deviations Principle for solutions in the case of vanishing noise., The current paper is an expanded and corrected version of the previous submission. Major change is the addition of Lemma 5.5. Martin Ondrej\'at's name has been added as a new author. The title of the paper has also been modified to a more suitable one to our results

We are concerned with the following constrained minimization problem: $$e(a_{1},a_{2},\beta) := \inf\left\{E_{a_{1},a_{2},\beta}(u_{1},u_{2}): \|u_{1}\|_{L^{2}(\mathbb{R}^{3})} = \|u_{2}\|_{L^{2}(\mathbb{R}^{3})} = 1\right\},$$ where $E_{a_{1},a_{2},\beta}$ is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters $a_{1}, a_{2}, \beta$ and two trapping potentials $V_1(x)$ and $V_2(x)$. In this paper, we obtain the existence of minimizers of $e(a_{1},a_{2},\beta)$ for possible $a_{1}, a_{2}$ and $\beta$ under suitable conditions on the potentials, which generalizes the results of the papers [16,17,18] in different senses., Comment: 32 pages

The paper considers the system of pressureless gas dynamics in one space dimension. The question of solvability of the initial-boundary value problem is addressed. Using the method of generalized potentials and characteristic triangles, extended to the boundary value case, an explicit way of constructing measure-valued solutions is presented. The prescription of boundary data is shown to depend on the behavior of the generalized potentials at the boundary. We show that the constructed solution satisfies an entropy condition and it conserves mass, whereby mass may accumulate at the boundary. Conservation of momentum again depends on the behavior of the generalized boundary potentials. There is a large amount of literature where the initial value problem for the pressureless gas dynamics model has been studied. To our knowledge, this paper is the first one which considers the initial-boundary value problem.

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the $C^\infty[0,1]$ steady-state solution by the method of upper and lower solution, and the uniqueness in both $W^{1,2}(0,1)$ and $L^\infty(0,1)$ by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in $C([0,1]\times [0,+\infty))\cap C^{2,1}([0,1]\times (0,+\infty))$, which, for any continuous initial value, converges to the steady-state solution in $C[0,1]$ (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species., 38 pages, 5 figures

The aim of this paper is twofold. The first is to study the asymptotics of a parabolically scaled, continuous and space-time stationary in time version of the well-known Funaki-Spohn model in Statistical Physics. After a change of unknowns requiring the existence of a space-time stationary eternal solution of a stochastically perturbed heat equation, the problem transforms to the qualitative homogenization of a uniformly elliptic, space-time stationary, divergence form, nonlinear partial differential equation, the study of which is the second aim of the paper. An important step is the construction of correctors with the appropriate behavior at infinity.

The paper considers the existence of one- or two-hump solutions of a singularly perturbed nonlinear Schrodinger (NLS) equation, which is the standard NLS equation with a third order perturbation. In particular, this equation appears in the field of nonlinear optics, where it is used to describe pulses in optical fibers near a zero dispersion wavelength. It has been shown formally and numerically that the perturbed NLS equation has one- or multi-hump solutions with small oscillations at infinity, called generalized one- or multi-hump solutions. The main purpose of the paper is to provide the first rigorous proof of the existence of generalized one- or two-hump solutions of the singularly perturbed NLS equation. The several invariant properties of the equation, i.e., the translational invariance, the gauge invariance and the reversibility property, are essential to obtain enough free constants to prove the existence. The ideas and methods presented here may be applicable to show existence of generalized 2 k -hump solutions of the equation.

This paper focuses on the study of ground states and nontrivial solutions for the following Hamiltonian elliptic system: { − Δ u + V ( x ) u = f 1 ( x , v ) , x ∈ R 2 , − Δ v + V ( x ) v = f 2 ( x , u ) , x ∈ R 2 , where V ∈ C ( R 2 , ( 0 , ∞ ) ) and f 1 , f 2 : R 2 × R → R have critical exponential growth. The strongly indefinite features together with the critical exponent bring some new difficulties in our analysis. In this paper, we develop a direct approach and use an approaching argument to seek Cerami sequences for the energy functional and estimate the minimax levels of such sequences. In particular, under some general assumptions imposed on the nonlinearity f i , we obtain the existence of ground states and nontrivial solutions for the above system as well as the following system in bounded domain, { − Δ u = f 1 ( v ) , x ∈ Ω , − Δ v = f 2 ( u ) , x ∈ Ω , u = 0 , v = 0 , x ∈ ∂ Ω . Our results improve and extend the related results of de Figueiredo-do O-Ruf (2004) [20] ; (2011) [21] , of Lam-Lu (2014) [30] , and of de Figueiredo-do O-Zhang (2020) [22] .

Consider the advective Fisher-KPP equation u t = u x x − β u x + f ( u ) on the half line [ 0 , ∞ ) with Dirichlet boundary condition at x = 0 . In a recent paper [10] , the authors considered the problem without advection (i.e., β = 0 ) and constructed a new type of entire solution U ( x , t ) , which, under the additional assumption f ″ ( u ) ≤ 0 , is concave and U ( ∞ , t ) = 1 for all t ∈ R . In this paper, we consider the equation with advection and without the additional assumption f ″ ( u ) ≤ 0 . In case β = 0 , using a quite different approach from [10] we construct an entire solution U ˜ which is similar as U in the sense that U ˜ ( ∞ , t ) ≡ 1 and U ˜ ( ⋅ , t ) is asymptotically flat as t → − ∞ , but different from U in the sense that it does not have to be concave. Our result reveals that the asymptotically flat (as t → − ∞ ) property rather than the concavity is more essential for such entire solutions. In case β 0 , we construct another new entire solution U ˆ which is completely different from the previous ones in the sense that U ˆ ( ∞ , t ) increases from 0 to 1 as t increasing from −∞ to ∞.

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction: { ( − Δ ) p s u + ( − Δ ) q s u + V ( e x ) ( | u | p − 2 u + | u | q − 2 u ) = h ( u ) + | u | r − 2 u in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0 , in R N , } where e is a positive parameter, 0 s 1 , 2 ⩽ p q N / s , ( − Δ ) t s ( t ∈ { p , q } ) is the fractional t-Laplace operator, while V : R N ↦ R and h : R ↦ R are continuous functions. The analysis developed in this paper covers both critical and supercritical cases, that is, we assume that either r = q s ⁎ : = N q / ( N − s q ) or r > q s ⁎ . The main results establish the existence of multiple positive solutions as well as related concentration properties. In the first case, due to the strong influence of the critical term, the result holds true for “high perturbations” of the subcritical nonlinearity. In the second framework, the result holds true for “low perturbations” of the supercritical nonlinearity. The concentration properties are achieved by combining topological and variational methods, provided that e is small enough and in close relationship with the set where the potential V attains its minimum.

This paper concerns the existence of bounded classical solutions to the attraction-repulsion chemotaxis system with logistic source (⋆) { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + f ( u ) , x ∈ Ω , t > 0 , 0 = Δ v − β v + α u , x ∈ Ω , t > 0 0 = Δ w − δ w + γ u , x ∈ Ω , t > 0 in a smooth bounded domain Ω ⊆ R N ( N ≥ 1 ) , subject to nonnegative initial data and homogeneous Neumann boundary conditions, where f ( u ) ≤ a − b u r for all u ≥ 0 with some a ≥ 0 , b > 0 and r ≥ 1 . Here χ , α , ξ , β as well as γ and δ are positive constants. It is proved that the corresponding system ( ⋆ ) possesses a unique global bounded classical solution in the balance case χ α = ξ γ with r > 2 N − 2 N or, the attraction domination case ξ α > ξ γ with b ≥ ( N − 2 ) + N ( χ α − ξ γ ) and r = 2 , respectively. The study of this paper improves the results in Li-Xiang (2016) [12] , Xu-Zheng (2018) [39] , Wang (2016) [26] as well as Zhao et al. (2017) [42] and Tello-Winkler (2007) [23] , in which, the assumption b > ( N − 2 ) + N ( χ α − ξ γ ) (see Li-Xiang (2016) as well as Wang (2016) and Zhao et al. (2017)) or b > ( N − 2 ) + N χ (see Tello-Winkler (2007)) or r > 2 N + 2 N + 2 (see Xu-Zheng (2018)) or r > N 2 + 4 N − N 2 (see Li-Xiang (2016)) are intrinsically required.

In this paper, we investigate well-posedness and well-posedness in the generalized sense for a new class of isoperimetric-type constrained variational control problems. For this purpose, in the first part of the current paper, we introduce new versions for the concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with the considered curvilinear integral functional. Thereafter, we define the approximating solution set of the considered class of isoperimetric-type constrained variational control problems. By using these completely new elements, we formulate and prove several characterization results on well-posedness and well-posedness in the generalized sense for the problem under study. Also, in order to highlight the theoretical results and tools derived in the paper, some illustrative examples are provided.

In this paper, we give a Moser-type theorem for C l -smooth hyperbolic-type degenerate Hamiltonian system with the following Hamiltonian H = 〈 ω , y 〉 + 1 2 v 2 − u 2 d + P ( x , y , u , v ) , ( x , y , u , v ) ∈ T n × R n × R 2 , which is associated with the standard symplectic structure, with d ≥ 1 . Due to the difficulty coming from the degeneracy, our result is quite different from L. Chierchia and D. Qian's work [8] (non-degenerate case). An interesting phenomenon shown in degenerate case is the l-regularity of above Hamiltonian system not only relies on the tori's dimension n but also strongly relies on the degenerate index d. Under arbitrary small perturbation P, we prove that if l ≥ ( 5 d + 2 ) ( 8 τ + 3 ) , where τ > n − 1 , the above hyperbolic-type degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class C β for any β ≤ 8 τ + 2 . Our result can be seen a generalization of paper [42] from analytic case to C l -smooth case and can also be seen a generalization of paper [8] from non-degenerate case to degenerate case.

This paper is concerned with Friedrichs extensions for a class of Hamiltonian systems. The non-symmetric problems are usually complicated and have unexpected properties. Here, Friedrichs extensions of a class of singular Hamiltonian systems including non-symmetric cases are characterized by imposing some constraints on each element of domains D ( H ) of the maximal operators H. These characterizations are given independent of principal solutions. It is interesting that by the results in the paper the Friedrichs extension of each of a large class of non-symmetric Hamiltonian systems has similar form to that of a symmetric Hamiltonian system. In addition, Friedrichs extensions of regular Hamiltonian systems are characterized incidently, J -self-adjoint Friedrichs extensions are studied, and a result is given for elements of D ( H ) , which makes the expression of the Friedrichs extension domain simpler. All results for Hamiltonian systems are finally applied to Sturm-Liouville operators with matrix-valued coefficients.

In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces. One of our basic tools consists in a Hardy type inequality proved in the present paper that allows us to establish Sobolev embeddings into Lebesgue spaces with weights in anisotropic Lebesgue spaces.

We are concerned with two kinds of singular limits of the Cauchy problem to the two-layer rotating shallow water equations as the Rossby number and the Froude number tend to zero. First we construct the uniform estimates for the strong solutions to the system under the condition that the Froude number is small enough. Different from the previously studied cases, the large operator of this model is not skew-symmetric. One of the key new ideas in this paper is to obtain the uniform estimates using the special structure of the system rather than the antisymmetry of the large operator. After that the convergence of the equations with ill-prepared data to a two-layer incompressible Navier-Stokes system is proved with the help of Strichartz estimates constructed in this paper.

In this paper, motivated by recent papers on the stabilization of evolution problems with nonlocal degenerate damping terms, we address an extensible beam model with degenerate nonlocal damping of Balakrishnan-Taylor type. We discuss initially on the well-posedness with respect to weak and regular solutions. Then we show for the first time how hard is to guarantee the stability of the energy solution (related to regular solutions) in the scenarios of constant and non-constant coefficient of extensibility. The degeneracy (in time) of the single nonlocal damping coefficient and the methodology employed in the stability approach are the main novelty for this kind of beam models with degenerate damping.

In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p > 2 ) { n t + u ⋅ ∇ n = ∇ ⋅ ( | ∇ n | p − 2 ∇ n ) − χ ∇ ⋅ ( n ∇ c ) , c t + u ⋅ ∇ c − Δ c = − c n , u t + ∇ π = Δ u + n ∇ φ , div u = 0 in a bounded domain Ω of R 3 with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem to find an optimal p-value ( p ≥ 2 ) which ensures that the solution is global bounded. In particular, the closer the value of p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist whenever p > p ⁎ ( ≈ 2.012 ) . It improved the result of [21] , [22] , in which, the authors established the global bounded solutions for p > 23 11 . Moreover, we also consider the large time behavior of solutions, and show that the weak solutions will converge to the spatially homogeneous steady state ( n ‾ 0 , 0 , 0 ) . Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of n is proved in the sense of L ∞ -norm, not only in L p -norm or weak-* topology.

In this paper, we consider the Schnakenberg model with heterogeneity on the interval ( − 1 , 1 ) . We first construct stationary solutions which concentrate at a suitable point by using the Liapunov-Schmidt reduction method. Moreover, by investigating the associated linearized eigenvalue problem, we establish the linear stability of the solutions above. Iron, Wei, and Winter (2004) established the existence and stability of multi-peak symmetric stationary solutions in non-heterogeneity case. In their work, the one-peak solution is always stable. For the symmetric heterogeneity case, Ishii and Kurata (2019) gave the analysis of one-peak symmetric solutions in details and revealed a destabilization effect of the heterogeneity. In this paper, we reveal that the mechanism which the location of the concentration point and the stability with respect to eigenvalues of order o ( 1 ) are determined by the interaction of the heterogeneity with the associated Green's function. In particular, we not suppose that the heterogeneity is symmetric. Also, by a typical example, we performed several numerical simulations to illustrate our main results.

In this paper, we consider an eigenvalue problem defined on a two-patch domain. Our first aim is to show that the two-patch eigenvalue problem is equivalent to the eigenvalue problem of a measure differential equation defined on an one-patch domain. Our second aim is to study the existence of principal eigenvalue of the measure differential equation, and we will prove the principal eigenvalue is continuously depending on the weight measure in the weak⁎ topology of the measure space. Our third aim is to solve a minimization problem on principal eigenvalues. Some main results of this paper have interesting relations with population dynamics. We will interpret these results in terms of survival chances and optimal distribution of resources.

In the present paper, we study the following planar Choquard equation: { − Δ u + V ( x ) u = ( I α ⁎ F ( u ) ) f ( u ) , x ∈ R 2 , u ∈ H 1 ( R 2 ) , where V ( x ) is an 1-periodic function, I α : R 2 → R is the Riesz potential and f ( t ) behaves like ± e β t 2 as t → ± ∞ . A direct approach is developed in this paper to deal with the problems with both critical exponential growth and strongly indefinite features when 0 lies in a gap of the spectrum of the operator − △ + V . In particular, we find nontrivial solutions for the above equation with critical exponential growth, and establish the existence of ground states and geometrically distinct solutions for the equation when the nonlinearity has subcritical exponential growth. Our results complement and generalize the known ones in the literature concerning the positive potential V to the general sign-changing case, such as, the results of de Figueiredo-Miyagaki-Ruf (1995) [16] , of Alves-Cassani-Tarsi-Yang (2016) [4] , of Ackermann (2004) [1] , and of Alves-Germano (2018) [5] .

We correct an error in our paper “Inviscid limit of the compressible Navier–Stokes equations for asymptotically isothermal gases”, which was made at four distinct points in the document. The main result therein (Theorem 1.4) is correct as stated, with the only exception that the requirement α > 1 / 2 must be imposed in point (3) of Definition 1.1 of the paper.

Motivated by the traditional Lotka-Volterra competitive models, this paper proposes and analyzes a class of stochastic reaction-diffusion partial differential equations. In contrast to the models in the literature, the new formulation enables spatial dependence of the species. In addition, the noise process is allowed to be space-time white noise. In this work, well-posedness, regularity of solutions, existence of density, and existence of an invariant measure for stochastic reaction-diffusion systems with non-Lipschitz and non-linear growth coefficients and multiplicative noise are considered. By combining the random field approach and infinite integration theory approach in SPDEs for mild solutions, analysis is carried out. Then this paper develops a Lotka-Volterra competitive system under general setting; longtime properties are studied with the help of newly developed tools in stochastic calculus.

This paper derives an explicit formula for the effective diffusion tensor by using the solutions to some effective cell problems after homogenizing Road effective boundary conditions (EBCs). The concept of Road EBCs was proposed recently by H. Li and X. Wang, and in this paper, we extend the effective conditions on closed curves to those on patterns, especially on the included nodes. We also prove that homogenization process commutes with the derivation of Road EBCs. By analyzing the effective diffusion tensor, we obtain several rules for maximizing its trace with given Road-effective-diffusivity/scale and length/scale in each cell and define a notion of balanced patterns. Moreover, we give an estimate of the trace of the effective diffusion tensor of patterns satisfying some connectedness as Road-effective-diffusivity/scale goes to infinity.

In this paper we study the well-posedness for the inhomogeneous nonlinear Schrodinger equation i ∂ t u + Δ u = λ | x | − α | u | β u in Sobolev spaces H s , s ≥ 0 . The well-posedness theory for this model has been intensively studied in recent years, but much less is understood compared to the classical NLS model where α = 0 . The conventional approach does not work particularly for the critical case β = 4 − 2 α d − 2 s . It is still an open problem. The main contribution of this paper is to develop the well-posedness theory in this critical case (as well as non-critical cases). To this end, we approach to the matter in a new way based on a weighted L p setting which seems to be more suitable to perform a finer analysis for this model. This is because it makes it possible to handle the spatially decaying factor | x | − α in the nonlinearity more efficiently. This observation is a core of our approach that covers the critical case successfully.

In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its principal eigenvalue and the DMP. Meanwhile a positive subsolution provides the coercivity constant. The numerical simulations show that the nonlinear finite element approximate solutions do not oscillate if the DMP is fulfilled. The characterization of the DMP and the mesh sizes guaranteeing the existence of positive sub- and supersolutions of the nonlinear finite element approximate problem, in the case of variable coefficients and all types of boundary conditions are some of the novelties of this paper. The excellent performance of the method is tested in two examples with boundary layers caused by very small diffusion.

This paper solves the stability problem on a partially dissipated system of magnetohydrodynamic equations near a background magnetic field. Large-time behavior of the corresponding linearized system is also obtained. These results presented in this paper rigorously confirm a nonlinear phenomenon observed in physical experiments that the magnetic field actually stabilizes electrically conducting fluids.

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C 1 / 2 -Holder continuous. The coefficient of the time-dependent damping is given by μ ( 1 + t ) λ , ( 0 λ , 0 μ ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that 0 λ 1 , 0 μ or λ = 1 , 2 μ . The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.

In the article, we prove the large data scattering for two models, i.e. the defocusing quintic nonlinear Schrodinger equation on R 2 × T and the defocusing cubic nonlinear Schrodinger equation on R 3 × T . Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper is the second project of our series work (two papers, together with [31] ) on large data scattering for the defocusing critical NLS with integer index nonlinearity on low dimensional waveguides. At this point, this category of problems are almost solved except for two remaining resonant system conjectures and the quintic NLS problem on R × T .

We consider the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions on an interval. Our previous papers [2] , [3] obtained the global bifurcation diagram of stationary solutions, which includes the secondary bifurcation from the odd symmetric solution due to the symmetric breaking effect. This paper derives the stability/instability of all symmetric solutions and instability of a part of asymmetric solutions. To do so, we use the exact representation of symmetric solutions and show the distribution of eigenvalues of the linearized eigenvalue problem around these solutions. And we show the instability of asymmetric solutions by the SLEP method. Finally, our results with respect to stability are supported by some numerical simulations.

This paper is concerned with the stationary problem for a diffusive Lotka-Volterra predator-prey-mutualist model with a protection zone under homogeneous Neumann boundary conditions. Compared with the case where the mutualist is absent in [12] , this paper aims to reveal the effects of mutualism coefficients α and β on the existence, number and stability of steady states. It turns out that when α is large, the model has at most one steady state and it is stable (if it exists); however existence of multiple steady states is examined for large β, moreover asymptotic profiles of steady states are established as β tends to infinity.

Let A ( s , t ) be a continuous function with a positive lower bound m, and Ω be a bounded domain in R N . In this short note, we propose an iterative procedure for finding nonnegative solutions of the following Kirchhoff-Carrier type equations { − A ( ‖ u ‖ p , ‖ ∇ u ‖ 2 ) Δ u = g ( x , u ) x ∈ Ω , u = 0 x ∈ ∂ Ω . The main advantage of our procedure is that the convergent proof of the iterative sequence depends only on comparison principle of the Laplace operator instead of comparison principle of Kirchhoff-Carrier type operator itself. Therefore, we almost need no restrictions on A ( s , t ) except for continuous and a positive lower bound. This removes away the monotonicity assumption of A ( s , t ) used in most papers based on sub-supersolution method. As applications of the abstract result obtained by our iterative method, some concrete examples are also studied in Section 2 of this paper.

We consider the parabolic type equation in R n : (0.1) ( ∂ t + H ) y ( t , x ) = 0 , ( t , x ) ∈ ( 0 , ∞ ) × R n ; y ( 0 , x ) ∈ L 2 ( R n ) , where H can be one of the following operators: (i) a shifted fractional Laplacian; ( i i ) a shifted Hermite operator; ( i i i ) the Schrodinger operator with some general potentials. We call a subset E ⊂ R n as a stabilizable set for (0.1) , if there is a linear bounded operator K on L 2 ( R n ) so that the semigroup { e − t ( H − χ E K ) } t ≥ 0 is exponentially stable. (Here, χ E denotes the characteristic function of E, which is treated as a linear operator on L 2 ( R n ) .) This paper presents different geometric characterizations of the stabilizable sets for (0.1) with different H. In particular, when H is a shifted fractional Laplacian, E ⊂ R n is a stabilizable set for (0.1) if and only if E ⊂ R n is a thick set, while when H is a shifted Hermite operator, E ⊂ R n is a stabilizable set for (0.1) if and only if E ⊂ R n is a set of positive measure. Our results, together with the results on the observable sets for (0.1) obtained in [1] , [19] , [25] , [33] , reveal such phenomena: for some H, the class of stabilizable sets contains strictly the class of observable sets, while for some other H, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for (0.1) where H is the Schrodinger operator with some general potentials.

In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition: { ( − Δ ) p s u ( x ) = f ( u ( x ) ) , x ∈ Ω , u ( x ) > 0 , x ∈ Ω , u ( x ) = 0 , x ∈ R n ∖ Ω , where ( − Δ ) p s is the fractional p-Laplacian defined as (0.1) ( − Δ ) p s u ( x ) = C n , s , p P . V . ∫ R n | u ( x ) − u ( y ) | p − 2 [ u ( x ) − u ( y ) ] | x − y | n + s p d y with 0 s 1 and p ≥ 2 . We first establish a maximum principle in unbounded domains involving the fractional p-Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p-Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39] , which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p-Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f ( ⋅ ) and on the domain Ω. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators.

In this paper, we investigate steady incompressible Euler flows with nonvanishing vorticity in a planar bounded domain. Let q be a harmonic function that corresponds to an irrotational flow. This paper proves that if q has k isolated local extremum points on the boundary, then there exist two kinds of steady Euler flows with small vorticity supported near these k points. For the first kind, near each maximum point the vorticity is positive and near each minimum point the vorticity is negative. For the second kind, near each minimum point the vorticity is positive and near each maximum point the vorticity is negative. Moreover, near these k points, the flow is characterized by a semilinear elliptic equation with a given profile function in terms of the stream function. The results are achieved by solving a certain variational problem for the vorticity and studying the limiting behavior of the extremizers.

In this paper, we study a free boundary problem modeling solid tumor growth with vasculature which supplies nutrients to the tumor; this is characterized in the Robin boundary condition. It was recently established [Discrete Cont. Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a threshold value μ ⁎ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for 0 μ μ ⁎ and linearly unstable for μ > μ ⁎ . In this paper we further study the nonlinear stability of the radially symmetric stationary solution, which introduces a significant mathematical difficulty: the center of the limiting sphere is not known in advance owing to the perturbation of mode 1 terms. We prove a new fixed point theorem to solve this problem, and finally obtain that the radially symmetric stationary solution is nonlinearly stable for 0 μ μ ⁎ when neglecting translations.

In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter–Saxton equation (1.1) , and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.

The initial boundary value problems for compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma are considered on exterior domains in this paper. With the radial symmetry assumption, the global existence of solutions to compressible Navier-Stokes-Poisson equations with the large initial data on a domain exterior to a ball in R n ( n ≥ 1 ) is proved. Moreover, without any symmetry assumption, the global existence of smooth solutions near a given constant steady state for both compressible Navier-Stokes-Poisson and Euler-Poisson equations on an exterior domain in R 3 with physical boundary conditions is also established with the exponential stability. A key issue addressed in this paper is on the global-in-time regularity of solutions near physical boundaries. This is in particular so for the 3-D compressible Navier-Stokes-Poisson equations to which global smooth solutions of initial boundary value problems are seldom found in literature to the best of knowledge.

In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half plane and in the half space ($\mathbb{R}_+^2$ or $\mathbb{R}_+^3$), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary conditions, vorticity becomes unbounded near the boundary. The novelty of this paper is to introduce boundary layer norms that capture the unbounded vorticity and to derive sharp estimates on this vorticity that are uniform in the inviscid limit., Comment: this greatly revised and shortened the previous version

In this paper we discuss existence of pointwise solutions for dynamical models of viscoplasticity. Among other things, this work answers the question about necessity of safe-load conditions in case of viscoplasticity, which arise in the paper of K. Chelminski (2001) [11] . We proved that solutions can be obtained without assuming any kind of safe-load conditions. Moreover, in the manuscript we consider much more general model than in the above mentioned paper. Namely, we consider the model with mixed boundary conditions and we allow a possible disturbance of the inelastic constitutive function by a globally Lipschitz function. Presented approach shows that via the same methods one can prove existence of pointwise solutions for: coercive models, self-controlling models, models with polynomial growth (not necessary of single valued) and monotone-gradient type models of viscoplasticity.

In this paper, we use the Gromov-Hausdorff distances between two global attractors (which belong to disjoint phase spaces) and two dynamical systems to consider the continuous dependence of the global attractors and the stability of the dynamical systems on global attractors induced by the reaction diffusion equation under perturbations of the domain. The novelty of the paper is to compare any two systems in different phase spaces without the process of “pull-backing” the perturbed systems to the original domain.

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Ito formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Ito formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random “noise”, under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process.

The paper investigates the well-posedness and longtime dynamics of the Kirchhoff wave model with structural nonlinear damping: u t t − ϕ ( ‖ ∇ u ‖ 2 ) Δ u + σ ( ‖ ∇ u ‖ 2 ) ( − Δ ) θ u t + f ( u ) = g ( x ) , with θ ∈ [ 1 / 2 , 1 ) . We find a new critical exponent p ⁎ ≡ N + 2 N − 2 ( > p θ ≡ N + 2 θ N − 2 , N ≥ 3 ) and show that when the growth exponent p of the nonlinearity f ( u ) is up to the range: 1 ≤ p p ⁎ : (i) the IBVP of the equation is well-posed and its solution is of additionally global regularity when t > 0 ; (ii) for each θ ∈ [ 1 / 2 , 1 ) , the related solution semigroup has in natural energy space H an optimal global attractor A θ whose compactness and attractiveness are in the regularized space H 1 + θ where A θ lies, and an optimal exponential attractor E θ ⁎ whose compactness, boundedness of the fractional dimension and the exponential attractiveness are in H 1 + θ where E θ ⁎ lies, respectively; (iii) the family of global attractors { A θ } θ ∈ [ 1 / 2 , 1 ) is upper semi-continuous at each point θ 0 ∈ [ 1 / 2 , 1 ) . The paper breaks though the longstanding existed growth restriction: 1 ≤ p ≤ p θ for p θ had been considered a uniqueness index, deepens and extends the results in literature [6] , [25] , [27] .

In this paper, we investigate multiple existence of homoclinic solutions for a periodically perturbed N-dimensional autonomous differential equation with a degenerate homoclinic solution of degeneracy degree d. Known results were obtained with a functional perturbation, which is regarded as an infinite-dimensional parameter. In this paper we consider a single parameter perturbation, a special form of former's functional perturbation, and prove that the single parameter is enough to unfold all possibilities of linearly independent homoclinic solutions bifurcated from the unperturbed degenerate homoclinic one, which actually improves the known results. Furthermore, we prove that those homoclinic solutions are all transversal, showing co-existence of multiple chaotic motions.

The main purpose of this paper is to investigate mathematically gas discharge. Townsend discovered α- and γ-mechanisms which are essential for ionization of gas, and then derived a threshold of voltage at which gas discharge can happen. In this derivation, he used some simplification such as discretization of time. Therefore, it is an interesting problem to analyze the threshold by using the Degond–Lucquin-Desreux–Morrow model and also to compare the results of analysis with Townsend's theory. Note that gas discharge never happens in Townsend's theory if γ-mechanism is not taken into account. In this paper, we study an initial–boundary value problem to the model with α-mechanism but no γ-mechanism. This problem has a trivial stationary solution of which the electron and ion densities are zero. It is shown that there exists a threshold of voltage at which the trivial solution becomes unstable from stable. Then we conclude that gas discharge can happen for a voltage greater than this threshold even if γ-mechanism is not taken into account. It is also of interest to know the asymptotic behavior of solutions to this initial–boundary value problem for the case that the trivial solution is unstable. To this end, we establish bifurcation of non-trivial stationary solutions by applying Crandall and Rabinowitz's Theorem, and show the linear stability and instability of those non-trivial solutions.