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The paper [39] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. The results of [39] include regularity assumptions on the perturbation and involve excluding parameters. The paper [39] also constructs response solutions to a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing of coefficients, alternative methods for nonlinear elliptic equations) to extend the results of [39]. We weaken the regularity assumptions on the perturbation and we describe the phenomena that happens for all parameters. In the ill-posed problem, we use a recently developed time-dependent center manifold theorem which allows to reduce the problem to a finite-dimensional ODE with quasi-periodic dependence on time. The bounded and sufficiently small solutions of these ODE give solutions of the ill-posed PDE. [ABSTRACT FROM AUTHOR]

Abstract The used methods in our papers [1–13] named new modified and extended auxiliary equation methods. The calculations of our article were correct. All solutions satisfied the partial differential equations in our papers. Highlights • The hydrodynamic mathematical methods. • We present nonlinear Schrodinger equation. • Solitary wave to the generalized nonlinear higher order of dynamical equation. • Fractional generalized reaction duffing equations. [ABSTRACT FROM AUTHOR]

Abstract This comment concerns the doubtful new auxiliary equation method (Khater method) included in the above paper. We prove by simple calculation that this method is wrong. The exact solutions of the proposed nonlinear equation obtained in the above paper are all wrong. Highlights • The new auxiliary equation method is called Khater method. • We prove by simple calculation that Khater method is wrong. • We show that the exact solutions obtained in the above paper are all wrong. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals. [ABSTRACT FROM AUTHOR]

This paper studies the zero-viscosity limit for the Navier-Stokes equations with rotation, an additive white noise and nonslip boundary condition in dimension three. By establishing the well-posedness of the limit problems and constructing approximate solutions, we obtain the convergence in the mean square for the Oseen type equation with nonlinear source term, and the convergence in probability of the martingale solution of the stochastic Navier-Stokes equations with rotation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in R × R 2 with the nonlinearity η | u | 2 where η ∈ C ∖ { 0 } and low regularity initial data. If s < − 1 / 4 , the ill-posedness result in the Sobolev space H s (R 2) is known. We will prove the well-posedness in H s (R 2) for − 1 / 2 < s < − 1 / 4 by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces. [ABSTRACT FROM AUTHOR]

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the long time dynamical behavior of the solutions for d -dimensional nonlinear Schödinger equation with general nonlinearity by using Birkhoff normal form technique and the so-called tame property in Gevrey space and modified Sobolev space. [ABSTRACT FROM AUTHOR]

In this paper we deal with the following boundary value problem { − Δ p u + g (u) | ∇ u | p = h (u) f in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, in a domain Ω ⊂ R N (N ≥ 2) , where 1 ≤ p < N , g is a positive and continuous function on [ 0 , ∞) , and h is a continuous function on [ 0 , ∞) (possibly blowing up at the origin). We show how the presence of regularizing terms h and g allows to prove existence of finite energy solutions for nonnegative data f only belonging to L 1 (Ω). [ABSTRACT FROM AUTHOR]

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation ∂ t u + (− Δ) 2 u = ∇ ⋅ F (∇ u) , where F satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case F (ξ) = ± | ξ | 2 ξ , we also examine the large time behaviour and stability of global solutions for arbitrary and small initial data in VMO, respectively. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ⁡ ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]

In this paper, we use a natural iteration technique to prove the existence of solutions to nonlinear Dirichlet problems. Among the examples included is the prescribed mean curvature equation. The nature of the technique allows applications to unbounded domains, as we show with domains of the form R n − 1 × (− d / 2 , d / 2). [ABSTRACT FROM AUTHOR]

In this paper we consider an abstract Cauchy problem for a Maxwell system modeling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study a perturbation of the Cahn–Hilliard equation with nonlinear terms of logarithmic type. This new model is based on an unconstrained theory recently proposed in [5]. We prove the existence, regularity and uniqueness of solutions, as well as (strong) separation properties of the solutions from the pure states, also in three space dimensions. We finally prove the convergence to the Cahn–Hilliard equation, on finite time intervals. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fully nonlinear wave equations on the product space R 3 × T with quadratic nonlinearities and on R 2 × T with cubic nonlinearities, respectively. It is shown that for the small initial data satisfying some space-decay rates at infinity, these nonlinear equations admit global smooth solutions when the corresponding partial null conditions hold and while have almost global smooth solutions when the partial null conditions are violated. Our proof relies on the Fourier mode decomposition of the solutions with respect to the periodic direction, the efficient combinations of time-decay estimates for the solutions to the linear wave equations and the linear Klein-Gordon equations, and the global weighted energy estimates. In addition, an interesting auxiliary energy is introduced. As a byproduct, our results can be applied to the 4D irrotational compressible Euler equations of polytropic gases or Chaplygin gases on R 3 × T , the 3D relativistic membrane equation and the 3D nonlinear membrane equation on R 2 × T. [ABSTRACT FROM AUTHOR]

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established. [ABSTRACT FROM AUTHOR]

• The effect of nonlinearity on the performance of Basis Pursuit (BP) is examined. • It is shown the performance impairment from nonlinearity can be notably mitigated by using the BP a set selector. • It is shown that normalization of the Jacobian to equal column norm leads to notable improvement when the likelihood of damage is uniform in the parameter space. The Basis Pursuit (BP) is an optimization statement used to search for the sparsest solution of underconstrained problems of the form Φ x = b using ℓ 1 minimization. This paper shows that approximating nonlinear problems as linear can result in considerable compromise in the capacity of the BP to attain the minimum cardinality solution and that this impairment can be mitigated by treating the strategy as a selector of a set of unknowns that renders the problem fully constrained, subsequently solving with due consideration for the nonlinearity. It is shown that gains realized by the selector approach derive primarily from the fact that the part of the nonlinearity that projects onto the right-hand side, b , has no effect on the relative values of the solution, x , and that the orthogonal residual seldom modifies the largest entries. The paper considers damage characterization as the application domain and shows that normalization of the Jacobian to equal column norm, a step that is standard when BP is applied in most fields, but not implemented so far in structural engineering, leads to large improvements when the probability of damage across the parameter space is essentially uniform. [ABSTRACT FROM AUTHOR]

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions. • Three/four-stage SDIRK methods with controllable numerical dissipation. • Algebraically stable for time-dependent nonlinear simulations. • Embedded formulation with accurate error estimation. • Applications to both first- and second-order time-dependent problems. [ABSTRACT FROM AUTHOR]

Let P be the space of probability measures on R d. We associate a coupled nonlinear Fokker-Planck equation on R d , i.e. with solution paths in P , to a linear Fokker-Planck equation for probability measures on the product space R d × P , i.e. with solution paths in P (R d × P). We explicitly determine the corresponding linear Kolmogorov operator L ˜ t using the natural tangent bundle over P with corresponding gradient operator ∇ P. Then it is proved that the diffusion process generated by L ˜ t on R d × P is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by L ˜ t in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurability conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of "Nemytskii-type". As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schrödinger type PDEs on R d × P 2 , through the Feynman-Kac formula for the corresponding diffusion processes. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to carry out a comprehensive investigation into the question of existence and non-existence of large solutions to semilinear elliptic equations with non-linear gradient terms in bounded domains of the Euclidean space. Based on new Keller-Osserman type integrability criteria, we provide necessary and sufficient conditions for the existence of large solutions. The novelty in our study rests on the fact that the Keller-Osserman type conditions we employ take both the growth rates of the gradient term as well as those of the first order non-linearity terms into account. When the gradient term does not appear, our criteria reduces to the well-known Keller-Osserman condition. [ABSTRACT FROM AUTHOR]

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonrelativistic limit of the following nonlinear Dirac equations − i c α ⋅ ∇ u + m c 2 β u − ω u + V (x) u = W (x) | u | p − 2 u , where c denotes the speed of light, m > 0 is the mass of the electron. Then we show nonexistence of special solutions by using the nonrelativistic limit of the corresponding nonlinear Dirac equations. Here, nonexistence means no nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the global strong solutions to a micro-macro model for compressible polymeric fluids with small initial data and their long-time decay rates of all order spatial derivatives. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first present that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for d ≥ 2. Moreover, for d ≥ 3 , we show a new critical Fourier estimation that allows us to derive the long-time decay rates of the L 2 norm for all order spatial derivatives. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and multiplicity of solutions for the following class of nonlinear Dirac equations [Display omitted] where V : R 3 → R and f : [ 0 , + ∞) → R are continuous functions. It is proved that the number of solutions is at least the number of global minimum points of V when ε is small enough. [ABSTRACT FROM AUTHOR]

Negative stiffness vibration isolation devices have attracted significant attention and research interest for their ability to enhance the low-frequency vibration isolation performance of passive vibration isolators. However, their narrow linear negative stiffness region has hindered their broader application. To address this limitation, a novel design method for vibration isolators is proposed, integrating the permanent magnetic negative stiffness device (NSD) with the displacement amplification mechanism (DAM). The paper begins by presenting a specific vibration isolator design method and analyzing the kinematic relationship. Next, a single-degree-of-freedom (SDOF) nonlinear equivalent model of the vibration isolator is established using the pseudo-rigid-body model (PRBM) approach. Numerical simulations are then conducted to obtain the transmissibility of the isolator. Furthermore, two isolators—one with the displacement amplification effect and another without—are manufactured, and experimental investigations are carried out. The experimental results confirm that the incorporation of the displacement amplification mechanism effectively expands the linear negative stiffness region of the negative stiffness magnet pair, enabling wide-frequency and large-amplitude vibration isolation. Notably, these experimental findings align well with the simulation results. In summary, this paper proposes a creative design method for wide frequency and large amplitude vibration isolators, and proves it through theoretical analysis and experimental validation. • A create negative stiffness isolator based on displacement amplification is proposed. • The linear region of the negative stiffness is significantly expanded by three times. • Employ the harmonic balance method for nonlinear equation solving and simulating. • Inspire the design of large displacement amplitude negative stiffness isolators. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The dynamics of the reaction-diffusion-advection population models with linear boundary condition has been widely studied. This paper is devoted to the dynamics of a reaction-diffusion-advection population model with nonlinear boundary condition. Firstly, the stability of the trivial steady state is investigated by studying the corresponding eigenvalue problem. Secondly, the existence and stability of nontrivial steady states are proved by applying the Crandall-Rabinowitz bifurcation Theorem, the Lyapunov-Schmidt reduction method and perturbation method, in which bifurcation from simple eigenvalue and that from degenerate simple eigenvalue are both possible. The general results are applied to a parabolic equation with monostable nonlinear boundary condition, and to a parabolic equation with sublinear growth and superlinear boundary condition. Our theoretical results show that the nonlinear boundary condition can lead to the occurrence of various steady state bifurcations. Meanwhile, compared with the linear boundary condition, the nonlinear boundary condition can induce the multiplicity and growing-up property of positive steady-state solutions for the model with logistic interior growth. Finally, the numerical results show that the advection can change the bifurcation direction of some bifurcation, and affect the density distribution of the species. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear elliptic equation arising in astrophysics: (0.1) − Δ u + V (x) u = | u | 2 ⁎ − 2 u | x ′ | + Q (x) | u | q − 2 u , x : = (x ′ , x ″) ∈ R m × R n − m , where n ≥ 3 , 2 ≤ m < n , 2 ⁎ = 2 (n − 1) n − 2 , 2 < q < 2 n n − 2 and V (x) , Q (x) ∈ C (R n). The partly singular term | u | 2 ⁎ − 2 u | x ′ | in (0.1) forces us to establish a refined Sobolev inequality involving partly weighted Morrey norm, which generalizes similar inequality obtained by G. Palatucci and A. Pisante (2014) [25]. Benefiting from the new inequality, we obtain a global compactness result and some existence result for (0.1) , which extend the results obtained by Y. B. Deng et al. (2012) [12]. Our strategy turns out to be more concise because we avoid the use of Levy concentration functions. [ABSTRACT FROM AUTHOR]

The present paper is concerned with a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of a homogenization theorem (i.e., convergence of solutions as the period of oscillation goes to zero) as well as a characterization of homogenized equations. In particular, homogenized matrices are described in terms of solutions to cell-problems, which have different forms depending on the log-ratio of the spatial and temporal periods of the coefficients. At a critical ratio, the cell problem turns out to be a parabolic equation in microscopic variables (as in linear diffusion) and also involves the limit of solutions, which is a function of macroscopic variables. The latter feature stems from the nonlinearity of the equation, and moreover, some strong interplay between microscopic and macroscopic structures can explicitly be seen for the nonlinear diffusion. As for the other ratios, the cell problems are always elliptic (in micro-variable only) and do not involve any macroscopic variables, and hence, micro- and macrostructures are weakly interacting each other. Proofs of the main results are based on the two-scale convergence theory (for space-time homogenization). Furthermore, finer asymptotics of gradients, diffusion fluxes and time-derivatives with certain corrector terms are provided, and a qualitative analysis on homogenized matrices is also performed. [ABSTRACT FROM AUTHOR]

For N ≥ 3 , by the seminal paper of Brezis and Véron (1980/81), no positive solutions of − Δ u + u q = 0 in R N ∖ { 0 } exist if q ≥ N / (N − 2). In turn, for 1 < q < N / (N − 2) the existence and profiles near zero of all positive C 1 (R N ∖ { 0 }) solutions are given by Friedman and Véron (1986). In this paper, for every q > 1 and θ ∈ R , we prove that the elliptic problem (⋆) − Δ u − λ | x | − 2 u + | x | θ u q = 0 in R N ∖ { 0 } with u > 0 has a C 1 (R N ∖ { 0 }) solution if and only if λ > λ ⁎ , where λ ⁎ = Θ (N − 2 − Θ) with Θ = (θ + 2) / (q − 1). We show that (a) if λ > (N − 2) 2 / 4 , then U 0 (x) = (λ − λ ⁎) 1 / (q − 1) | x | − Θ is the only solution of (⋆) and (b) if λ ⁎ < λ ≤ (N − 2) 2 / 4 , then all solutions of (⋆) are radially symmetric and their total set is U 0 ∪ { U γ , q , λ : γ ∈ (0 , ∞) }. We give the precise behavior of U γ , q , λ near zero and at infinity, distinguishing between 1 < q < q N , θ and q > max ⁡ { q N , θ , 1 } , where q N , θ = (N + 2 θ + 2) / (N − 2). In addition, we answer open questions arising from the works of Cîrstea (2014) and Wei–Du (2017) for θ ≤ − 2 by settling the existence and sharp profiles near zero of all positive solutions of (⋆) in Ω ∖ { 0 } , subject to u | ∂ Ω = 0 , where Ω is a smooth bounded domain containing zero. [ABSTRACT FROM AUTHOR]

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations. • Exponential Parareal notably reduces time-to-solution for non-diffusive equations. • Linear analysis accurately predicts Parareal performance on nonlinear problems. • Repartitioning is essential for stabilizing exponential integrators within Parareal. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of heat equation in R + N with flux ∂ ∂ ν u = exp ⁡ (p u) on ∂ R + N for any constant p > 0. The initial trace μ (x) is a Radon measure in R N with supp μ ⊂ R + N ‾ or a nonnegative measurable function in R + N. We at first show the necessary conditions on the initial trace for the existence of solutions for every positive p , and then we divide into three cases to give sufficient conditions on the solvability of solutions, where all of the descriptions on the initial trace are determined quantitatively. For some cases, we identify the strongest singularity on the initial trace to guarantee the solvability of this nonlinear parabolic problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate pointwise space-time behavior of the compressible Oldroyd-B model in R 3 based on Green's function method. We first give the representations of two Green's matrices in the Fourier space: one is a 7 × 7 matrix for the compressible part and the other is a 6 × 6 matrix for the incompressible part. Then by using high-medium-low frequency analysis together with real analysis and Fourier analysis we get the pointwise estimates for each entry in these Green's matrices. Finally, we deal with the nonlinear problem and obtain the pointwise space-time description of the solution. It shows that the fluid density ρ , momentum m and the symmetric tensor of constrains T obey the generalized Huygens' principle as the compressible Navier-Stokes equations. As a byproduct, we extend L 2 -estimate in [47] to L p -estimate with p > 1. Moreover, the decay of T is faster than those of (ρ , m) , which also refine the result in [47]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we study global existence and pointwise decay estimates for the nonlinear Dirac equation with quadratic nonlinearity. We consider four cases depending on the spatial dimension n , the mass parameter m , and the initial data ψ 0 : i) (n , m) = (2 , 0) and ψ 0 is compactly supported; i i) (n , m) = (3 , 0) and ψ 0 is compactly supported; i i i) (n , m) = (3 , 0) and ψ 0 is not necessarily compactly supported; i v) n = 3 , m ≥ 0 and ψ 0 is compactly supported. In each of the cases i)- i i i) , we prove a small data global existence result, a sharp pointwise decay estimate and a scattering result for the global solution. In the case i v) we prove a uniform (in the mass parameter m) global existence result, a unified pointwise decay estimate, and a scattering result. [ABSTRACT FROM AUTHOR]

This paper deals with a multi-objective control problem for the Kuramoto-Sivashinsky equation by following a Stackelberg-Nash strategy. We have a distributed control called Leader, and two boundary controls called Followers, each of them has to act over the equation to influence the behavior of the state in a particular way, by reaching or approaching to many targets at once. To be more precise, the Leader wants to drive the solution to a prescribed target at a final time, and the followers have to minimize some given cost functionals, adapting themselves to what the Leader wants. The main difficulty here is that, since the Followers are in the boundary, the problem turns to be equivalent to prove a partial null controllability result for a system of nonlinear fourth-order equations with boundary coupling terms. [ABSTRACT FROM AUTHOR]

In this paper, we study the forward and the inverse problem for the fractional magnetic Schrödinger equation with a nonlinear electric potential. We first obtain the maximum principle for the linearized equation and apply it to show that the problem is well-posed under suitable assumptions for the exterior condition. Then we obtain the uniqueness in recovering both the magnetic and the nonlinear electric potentials, assumed to be analytic in terms of the solution, from the exterior data of the solution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous Neumann problem of a nonlinear diffusion equation driven by the p (x) -Laplace operator. Complete classification of global existence and blow-up in finite time of solutions is given when the initial data satisfies different conditions. We first obtain a threshold result for the solution to exist globally or to blow up in finite time when the initial energy is subcritical or critical. We also establish a decay rate of the L 2 norm for global solutions. Furthermore, we provide some sufficient conditions for the existence of global and blow-up solutions for supercritical initial energy. We finally give two-sided estimates of asymptotic behavior when the diffusion term dominates the source. This is a continuation of our previous work [17]. [ABSTRACT FROM AUTHOR]

In a previous paper, [1] , a new formulation for isothermal gas flow in a single pipeline including gravity and friction effects has been introduced and solved by finite element methods. A performance analysis has been done by using some tests both manufactured and experimental. The goal of the present work is to extend the ideas to model gas flow in a transportation network. As a first step, in this paper the network only includes pipes connected at nodes which can be emission or consumption points, or simply structural connections between pipes. In addition to the mass and momentum conservation equations inside the pipes that are partial differential equations, we need to write the mass conservation equations at the nodes. The latter appear to be a set of constraints (one per node) whose respective Lagrange multipliers are the (continuous) pressures at the nodes. Thus, the whole model is a Partial Differential-Algebraic Equation (PDAE). For numerical solution, an implicit time discretization is proposed first and then a weak formulation of the resulting semi-discrete problem is introduced. After doing a finite element discretization we are led to solve a nonlinear system of numerical equations per time step what is done by Newton's method. In order to reduce the computing time, the algorithm can be parallelized in such a way that, at each time step, every pipe can be solved separately in one processor (or core). Moreover, we show that the proposed scheme exactly conserves the mass at the nodes and in the whole network, and a new simple and compact way of computing the network line-pack over the time is given. Finally, the introduced methodology is validated by applying the computer program to a triangular network, which is a standard test in references on the subject, and to a real network for which we have got measurements that enable us to create the boundary conditions for the model and also to compare the numerical results with experimental data. • Numerical solution of Euler-like models with friction and variable topography. • An equivalent formulation with one nonlinear PDE of second order in time and space. • An implicit time discretization combined with a finite element method is used. • Accuracy has been tested with experimental data and numerical results from papers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dual fractional parabolic problem { ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) , in R + n × R , u (x , t) = 0 , in (R n ﹨ R + n) × R , where R + n : = { x ∈ R n | x 1 > 0 } is the right half space. We prove that the positive solutions are strictly increasing in x 1 direction without assuming the solutions be bounded. So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative ∂ t α and the fractional Laplacian (− Δ) s. To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator ∂ t α + (− Δ) s along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or boundedness assumption on the solutions, we develop a new novel approach lies in establishing the averaging effects for such nonlocal operator and apply these averaging effects twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions. We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems. [ABSTRACT FROM AUTHOR]

Recently, there has been a series of works on the positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations, where the notion of mass conservation follows from a straightforward analogy of that of time-dependent problems, i.e. conserving the mass = preserving cell averages during limiting. Based on such a notion, the implementations and theoretical proofs of positivity-preserving limited methods for stationary equations are unnecessarily complicated and constrained. As will be shown in this paper, in some extreme cases, their convergence could even be problematic. In this work, we clarify a more appropriate definition of mass conservation for limiters applied to stationary hyperbolic equations and establish the genuinely conservative high-order positivity-preserving limited discontinuous Galerkin methods based on this definition. The new methods are able to preserve the positivity of solutions of scalar linear equations and scalar nonlinear equations with invariant wind direction, with much simpler implementations and easier proofs for accuracy and the Lax-Wendroff theorem, compared with the existing methods. Two types of positivity-preserving limiters preserving the local mass of stationary equations are developed to accommodate for the new definition of conservation and their accuracy are investigated. We would like to emphasize that a major advantage of the original DG scheme presented in [24] is a sweeping procedure, which allows for the computation of conservative steady-state solutions explicitly, cell by cell, without iterations, even for nonlinear equations as long as the wind direction is fixed. The main contribution of this paper is to introduce a limiting procedure to enforce positivity without changing the conservative property of this original DG scheme. The good performance of the algorithms for stationary hyperbolic equations and their applications in time-dependent problems are demonstrated by ample numerical tests. • A new definition of local conservation is given for stationary hyperbolic systems. • This allows the design of positivity-preserving discontinuous Galerkin (DG) schemes in more general cases than before. • Such high order positivity-preserving DG schemes are more general than before. [ABSTRACT FROM AUTHOR]

This paper studies the local bifurcation direction, stability properties and global structure for a nonlinear pseudodifferential equation, which describes the periodic travelling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. We first obtain the precise formula of the second derivative of bifurcation parameters at the bifurcation points. In particular, their signs can be strictly judged when constant vorticity vanishes. Furthermore, we present the stability analysis for the travelling water waves that have small vorticity and amplitude. We also show that the global bifurcation curves can't form a loop. Moreover, if the total head is bounded, the existence of waves of all amplitudes from zero up to that of Stokes' highest wave has been established. [ABSTRACT FROM AUTHOR]

Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results. [ABSTRACT FROM AUTHOR]

Let (M , g , σ) be an m -dimensional closed spin manifold, with a fixed Riemannian metric g and a fixed spin structure σ ; let S (M) be the spinor bundle over M. The spinorial Yamabe-type problems address the solvability of the following equation D g ψ = f (x) | ψ | g 2 m − 1 ψ , ψ : M → S (M) , x ∈ M where D g is the associated Dirac operator and f : M → R is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when m = 2 , a solution corresponds to a conformal isometric immersion of the universal covering M ˜ into R 3 with prescribed mean curvature f ; meanwhile, for general dimensions and f ≡ c o n s t a n t ≠ 0 , a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold (S m , g) where the solution set of the spinorial Yamabe-type problem is not compact: 1). the geometric potential f is constant (say f ≡ 1) with the background metric g being a C k perturbation of the canonical round metric g S m , which is not conformally flat somewhere on S m ; 2). f is a perturbation from constant and is of class C 2 , while the background metric g ≡ g S m . [ABSTRACT FROM AUTHOR]

In this paper, we study the persistence of spatial analyticity for the solutions to the Klein-Gordon-Schrödinger system, which describes a physical system of a nucleon field interacting with a neutral meson field, with analytic initial data. Unlike the case of a single nonlinear dispersive equation, not much is known about nonlinear dispersive systems as it is harder to show the spatial analyticity of coupled equations simultaneously. The only results known so far are rather recent ones for the Dirac-Klein-Gordon system which governs the physical system when the nucleon is described by Dirac spinor fields in the case of relativistic fields. In contrast, we aim here to study the Klein-Gordon-Schrödinger system that works in the non-relativistic regime. It is shown that the radius of spatial analyticity of the solutions at later times obeys an algebraic lower bound as time goes to infinity. [ABSTRACT FROM AUTHOR]

Abstract With the development of vision and optimization techniques, visual-inertial odometry (VIO) has shown the capability of motion estimating in the GNSS-denied condition. The VIO can provide absolute pitch and roll angles estimating value, but no the absolute azimuth. In the paper, we proposed a VIO aided by compass, which can obtain the azimuth with respect to the north direction in the geographic frame. Moreover, aided by compass, the yaw angle estimating error was reduced to a greater degree, due to the measurement of azimuth. Furthermore, the consistency of the VIO backend estimator is improved as well, while the accuracy of the estimated pose states was also wholly improved. The aiding approach is a tightly-couple information fusion system of camera, IMU and magnetoresistive sensors. The optimization method is based on the pre-integration and bundle adjustment. In the paper, we derived the compass residual model based on the pre-integration model, and then its Jacobian and covariance formation were deduced to solve the nonlinear equations. The compass aided VIO software was implemented based on the Nvidia Jetson Tx2. The system was fully tested based on hardware-in-the-loop simulation and vehicle test in the real physical environment. The pose errors of VIOs with and without compass aiding were compared in the above tests. The simulation results showed that the position was and yaw errors were improved obviously; the compass aided VIO was still consistent, but the pure VIO was consistent not. The consistency character is evaluated by average NEES by Monte-Carlo in simulation. The vehicle test showed that the position error was reduced by 23%; the yaw error was reduced by 21%. As a result, the compass aided VIO not only improved the pose estimated accuracy, especially position and yaw, but also improved the consistency of VIO system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the problem { u t t − Δ u + P ( x , u t ) = f ( x , u ) in ( 0 , ∞ ) × Ω , u = 0 on ( 0 , ∞ ) × Γ 0 , u t t + ∂ ν u − Δ Γ u + Q ( x , u t ) = g ( x , u ) on ( 0 , ∞ ) × Γ 1 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) in Ω ‾ , where Ω is a bounded open subset of R N with C 1 boundary ( N ≥ 2 ), Γ = ∂ Ω , Γ 1 is relatively open on Γ, Δ Γ denotes the Laplace–Beltrami operator on Γ, ν is the outward normal to Ω, and the terms P and Q represent nonlinear damping terms, while f and g are nonlinear perturbations. In the paper we establish local and global existence, uniqueness and Hadamard well-posedness results when source terms can be supercritical or super-supercritical. [ABSTRACT FROM AUTHOR]

In this paper, we study the viscous Boussinesq equation in the whole space R n , which describes the propagation of small amplitude and long waves on the surface of water with viscous effects. Concerning the linear Cauchy problem, some qualitative properties of solutions including L m − L q estimates with 1 ⩽ m ⩽ q ⩽ ∞ , inviscid limits and asymptotic profiles of solution with respect to the small viscosity are investigated by means of the Fourier analysis and the WKB method. For another, by applying some fractional order interpolations in the harmonic analysis, we derive the L q well-posedness and estimates for small data solutions to the nonlinear viscous Boussinesq equation under some conditions for the parameter of nonlinearity. [ABSTRACT FROM AUTHOR]

We characterize lower growth estimates for subsolutions in halfspaces of fully nonlinear partial differential equations on the form F (x , u , D u , D 2 u) = 0 in terms of solutions to ordinary differential equations built upon assumptions on F. Using this characterization we derive several sharp Phragmen–Lindelöf-type theorems for certain classes of well known PDEs. The equation need not be uniformly elliptic nor homogeneous and we obtain results both in case the subsolution is bounded or unbounded. Among our results we retrieve classical estimates in the halfspace for p -subharmonic functions and extend those to more general equations; we prove sharp growth estimates, in terms of k and the asymptotic behavior of ∫ 0 R C (s) d s , for subsolutions of equations allowing for sublinear growth in the gradient of the form C (| x |) | D u | k with k ≥ 1 ; we establish a Phragmen–Lindelöf theorem for weak subsolutions of the variable exponent p -Laplace equation in halfspaces, 1 < p (x) < ∞ , p (x) ∈ C 1 , of which we conclude sharpness by finding the "slowest growing" p (x) -harmonic function together with its corresponding family of p (x) -exponents. The paper ends with a discussion of our results from the point of view of a spatially dependent diffusion problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for Hartree type equation i u t + u x x = [ K ⁎ | u | 2 ] u , with Cauchy data in modulation spaces M p , q (R). We establish global well-posedness results in M p , p ′ (R) when K (x) = λ | x | γ , (λ ∈ R , 0 < γ < 1) with no smallness condition on initial data, where p ′ is the Hölder conjugate of p. Our proof uses a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Grünrock and Chaichenets et al. to the modulation space setting and exploits polynomial growth of the Schrödinger propagator on modulation spaces. [ABSTRACT FROM AUTHOR]