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We deal with the following nonlinear Schrodinger equation with magnetic field and critical growth: { ( e i ∇ − A ( x ) ) 2 u + V ( x ) u = f ( | u | 2 ) u + | u | 2 ⁎ − 2 u in R N , u ∈ H 1 ( R N , C ) , where e > 0 is a small parameter, N ≥ 3 , 2 ⁎ = 2 N N − 2 is the critical Sobolev exponent, A ∈ C 1 ( R N , R N ) is a magnetic vector potential, V : R N → R is a continuous positive potential having a local minimum and f : R → R is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we investigate the existence and concentration of nontrivial solutions for e > 0 small enough.

In this paper we study classical solutions to the zero–flux attraction–repulsion chemotaxis–system ( ◇ ) { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) in Ω × ( 0 , t ⁎ ) , 0 = Δ v + α u − β v in Ω × ( 0 , t ⁎ ) , 0 = Δ w + γ u − δ w in Ω × ( 0 , t ⁎ ) , where Ω is a smooth and bounded domain of R 2 , t ⁎ is the blow–up time and α , β , γ , δ , χ , ξ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u ( x , 0 ) = u 0 ∈ C 0 ( Ω ¯ ) , system (◇) has a unique classical solution which becomes unbounded as t ↗ t ⁎ . The main result of this investigation is to provide an explicit lower bound for t ⁎ estimated in terms of ∫ Ω u 0 2 d x and attained by means of well–established techniques based on ordinary differential inequalities.

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system { u t = ∇ ⋅ [ ( u + α ) m 1 − 1 ∇ u − χ u ( u + α ) m 2 − 2 ∇ v ] in Ω × ( 0 , T ) , v t = Δ v − v + u in Ω × ( 0 , T ) under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0 , χ > 0 , m 1 , m 2 ∈ R and T > 0 . Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 ≠ 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω.

This paper considers the initial-boundary value problem for the nonlocal Cahn–Hilliard equation ∂ t φ + ( − Δ + 1 ) ( a ( ⋅ ) φ − J ⁎ φ + G ′ ( φ ) ) = 0 in ( 0 , T ) × Ω in an unbounded domain Ω ⊂ R N with smooth bounded boundary, where N ∈ N , T > 0 , and a ( ⋅ ) , J , G are given functions. In the case that Ω is a bounded domain and − Δ + 1 is replaced with −Δ, this problem has been studied by using a Faedo–Galerkin approximation scheme considering the compactness of the Neumann operator − Δ + 1 (cf. [8] , [16] ). However, the compactness of the Neumann operator − Δ + 1 breaks down when Ω is an unbounded domain. The present work establishes existence and energy estimates of weak solutions for the above problem on an unbounded domain.

Transmission boundary value problems for the time-harmonic Maxwell system, the Helmholtz operator, and the perturbed Dirac operator are formulated, using Clifford algebras, on bounded Lipschitz domains of R m with m ≥ 3 . It is shown how the Dirac problem decouples into several Maxwell and Helmholtz problems. Necessary and sufficient conditions are provided for well-posedness in each case, and for the Dirac problem to be equivalent to one or several independent Maxwell problems.

Recently, by using the argument of Lei & Lin (2011) [11] , Liu & Gao (2017) [13] establish the global well-posedness of mild solutions to the three-dimensional Boussinesq equations in the space χ − 1 defined by χ − 1 = { u ∈ D ′ ( R 3 ) : ∫ R 3 | ξ | − 1 | u ˆ ( ξ ) | d ξ ∞ } . However, it seems that their proof is incorrect, and has some obvious and essential mistakes. Compared with the Navier-Stokes equations, it is difficulty to obtain a global well-posedness of mild solutions to the Boussinesq system in the space χ − 1 . In this paper, we will point out the mistakes of Liu & Gao. And, furthermore, in order to understand the difficulty of the Boussinesq system better, we study an illuminating system as follows: { ∂ t u + ( u ⋅ ∇ ) u − μ ( 1 + t ) α Δ u + ∇ p = θ e 3 , in R 3 × ( 0 , ∞ ) , ∂ t θ + ( u ⋅ ∇ ) θ − k ( 1 + t ) α Δ θ = 0 , in R 3 × ( 0 , ∞ ) , ∇ ⋅ u = 0 , in R 3 × ( 0 , ∞ ) , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , in R 3 , where μ > 0 , k > 0 and α > 1 are real constant parameters. By using the time-weighted estimate, we can show that the above system has a global mild solution.

In this paper, we establish some Liouville type theorems for positive classical supersolutions in the whole space R N of the elliptic equation − Δ u + a ⋅ ∇ u = u p and of the elliptic system { − Δ u + a ⋅ ∇ u = v p − Δ v + a ⋅ ∇ v = u q . Here, p , q ∈ R and a is a smooth vector field satisfying some growth condition at infinity.

We consider the Cauchy problem for the wave equation with overdamping and semilinear source terms: ( P ) { u t t − Δ u + b ( t ) u t = N ( u ) , ( t , x ) ∈ R + × R N ( u , u t ) ( 0 , x ) = ( u 0 , u 1 ) ( x ) , x ∈ R N , with b ( t ) = b 0 ( t + 1 ) − β , b 0 > 0 , β − 1 and N ( u ) = | u | p − 1 u , p > 1 . Ikeda and Wakasugi [8] have recently showed that, when | N ( u ) | ≤ C | u | p for any p > 1 , there is a global-in-time solution to (P) for suitable small data, and that, when N ( u ) = ± | u | p , the local-in-time solution blows up within a finite time for suitable large data. To show the blow-up result, their method seems to be not applicable to our semilinear term. Our aim is to show the blow-up of solutions for suitable large data, by the method much different from theirs.

The Smith's approximation formula for the gamma function is given by Γ ( x + 1 2 ) = 2 π ( x e ) x ( 2 x tanh ⁡ 1 2 x ) x / 2 ( 1 + O ( 1 x 5 ) ) , as x → ∞ . In this paper, by means of a little known power series, we develop this formula to an asymptotic expansions: Γ ( x + 1 / 2 ) 2 π ( x / e ) x ∼ ( 2 x tanh ⁡ 1 2 x ) x / 2 exp ⁡ ( ∑ n = 3 ∞ [ ( 2 n ) ! − ( 2 n − 1 ) 2 2 n − 1 ] ( 1 − 2 1 − 2 n ) 2 n ( 2 n − 1 ) ( 2 n ) ! B 2 n x 2 n − 1 ) , as x → ∞ , and give an estimate of remainder in the above asymptotic series, where B 2 n is the Bernoulli number. Moreover, as relevant results, Ramanujan type asymptotic expansions for the gamma function Γ ( x + 1 / 2 ) are obtained; an asymptotic expansion for Wallis ratio and an estimate of the remainder are established.

In this paper we study the Strichartz estimates for the Schrodinger propagator in the context of Wiener amalgam spaces which, unlike the Lebesgue spaces, control the local regularity of a function and its decay at infinity separately. This separability makes it possible to perform a finer analysis of the local and global behavior of the propagator. Our results improve some of the classical ones in the case of large time.

In this paper, we derive L 1 − L 1 long time estimates for the strongly damped plate equation u t t + Δ 2 u + Δ 2 u t = 0 x ∈ R n , t ∈ R + , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) . In particular, we prove that ‖ u ( t , ⋅ ) ‖ L 1 ≤ C ( 1 + t ) n 4 ( ‖ u 0 ‖ L 1 + ( 1 + t ) 1 2 ‖ u 1 ‖ L 1 ) , for any t ≥ 0 , in space dimension n ≥ 5 .

We prove the global well-posedness of weak solutions for nonlinear wave equations with supercritical source and damping terms on a three-dimensional torus T 3 of the prototype u t t − Δ u + | u t | m − 1 u t = | u | p − 1 u , ( x , t ) ∈ T 3 × R + ; u ( 0 ) = u 0 ∈ H 1 ( T 3 ) ∩ L m + 1 ( T 3 ) , u t ( 0 ) = u 1 ∈ L 2 ( T 3 ) , where 1 ≤ p ≤ min ⁡ { 2 3 m + 5 3 , m } . Notably, p is allowed to be larger than 6.

In this paper we use variational methods to establish a Berestycki-Lions type result for two classes of quasilinear Schrodinger equations in R N in the zero-mass situation.

In this paper we study the quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source: u t = ∇ ⋅ ( D ( u ) ∇ u − S ( u ) ∇ v ) + f ( u ) , v t = Δ v − a 1 v + b 1 w , w t = Δ w − a 2 w + b 2 u , under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n ( n ≥ 1 ), where a i , b i > 0 ( i = 1 , 2 ), D , S ∈ C 2 ( [ 0 , ∞ ) ) and f : R → R is a smooth function generalizing the logistic source f ( s ) = b − μ s r for all s ≥ 0 with b ≥ 0 , μ > 0 and r ≥ 1 . We obtain the global boundedness of solutions in four cases: (i) the self-diffusion dominates the cross-diffusion; (ii) the logistic source suppresses the cross-diffusion; (iii) the logistic dampening balances the cross-diffusion with μ > 0 suitably large; (iv) the self-diffusion and the logistic source both balance the cross-diffusion to some extent with μ > 0 arbitrary. As corollaries, we also consider the global boundedness of solutions for the quasilinear attraction-repulsion chemotaxis model with logistic source: u ˜ t = ∇ ⋅ ( D ( u ˜ ) ∇ u ˜ ) − χ ∇ ⋅ ( u ˜ ∇ z ) + ξ ∇ ⋅ ( u ˜ ∇ w ˜ ) + f ( u ˜ ) , z t = Δ z − ρ z + η u ˜ , w ˜ t = Δ w ˜ − δ w ˜ + γ u ˜ , where χ , η , ξ , γ , ρ , δ > 0 .

In this paper we use variational methods to establish existence of solitary waves for a class of generalized Kadomtsev-Petviashvili ( G K P ) equation in R N . The positive and zero mass cases are considered. The main argument is to find a Palais Smale sequence satisfying a property related to Pohozaev identity, as in [21] , which was used for the first time by [23] .

In this paper, we investigate the existence of sign-changing solution for a class nonlocal Kirchhoff-Schrodinger-type equation − ( a + b ∫ R 3 | ∇ u | 2 d x ) △ u + V ( x ) u = f ( x , u ) , x ∈ R 3 , where a and b are positive constants. With the help of the constraint variational method and via a direct approach, we show the existence of sign-changing solution and prove that the sign-changing solution has precisely two nodal domains. This work can be regarded as the complement for some results of the literature.

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models: u t t + ( − Δ ) σ u + μ ( − Δ ) δ u t = f ( u , u t ) , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) with σ ≥ 1 , μ > 0 and δ ∈ ( 0 , σ 2 ) . Here the function f ( u , u t ) stands for the power nonlinearities | u | p and | u t | p with a given number p > 1 . We are interested in investigating L 1 estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faa di Bruno's formula. By assuming additional L m regularity on the initial data, we use ( L m ∩ L q ) − L q and L q − L q estimates with q ∈ ( 1 , ∞ ) and m ∈ [ 1 , q ) , to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on L q spaces.

In this paper, we study the asymptotic stability of the following density-suppressed motility system ( ⁎ ) { u t = Δ ( γ ( v ) u ) + μ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v + u − v , x ∈ Ω , t > 0 , in a bounded domain with smooth boundary, where the motility function γ ( v ) and μ are positive. Under the conditions that the motility function γ ( v ) has the lower-upper bounded and that μ is large enough, we derive that system (⁎) possesses a unique global solution in three-dimensional space. Moreover, we show that if the global classical solution exists of system (⁎) in any dimensional space, then the solution will converge to the equilibrium ( 1 , 1 ) exponentially as t → + ∞ when μ > K 0 16 with K 0 = max 0 ≤ v ≤ ∞ ⁡ | γ ′ ( v ) | 2 γ ( v ) .

We study the Schrodinger systems with linear and nonlinear coupling terms (doubly coupled nonlinear Schrodinger system for short) which arise naturally in nonlinear optics, and in the Hartree–Fock theory for Bose–Einstein condensates, among other physical problems. First, for small linear coupling constant, we get existence of a nontrivial bound state solution to the system via perturbation method, furthermore, we prove each component of the bound state solution is nonnegative by energy estimate. Second, we establish a version of Pohozaev–Nehari identity and prove a nonexistence result for the more general system when the spatial dimension N ≥ 4 .

We consider the Cauchy problem for the relativistic Vlasov-Poisson-Fokker-Planck system in the whole space. For perturbative initial data with suitable regularity, we obtain the global classical solutions and prove the exponential time decay rate to the equilibrium around a global relativistic Maxwellian.

In the paper, it is proved that a pair of analytic functions is simultaneously approximated by shifts ( ζ ( s + i γ k h ) , ζ ( s + i γ k h , α ) ) of the Riemann and Hurwitz zeta-functions with transcendental parameter α, where 0 γ 1 γ 2 ⋯ ⩽ γ n ⩽ ⋯ is a sequence of imaginary parts of non-trivial zeros of ζ ( s ) . For this, the weak Montgomery conjecture on pair correlation of zeros of ζ ( s ) is used.

In this paper, we study the following quasilinear Schrodinger equation of Choquard type − △ u + V ( x ) u − △ ( u 2 ) u = ( I α ⁎ | u | p ) | u | p − 2 u , x ∈ R N , where N ≥ 3 , 0 α N , 2 ( N + α ) N ≤ p 2 ( N + α ) N − 2 and I α is a Riesz potential. Under appropriate assumptions on V ( x ) , we establish the existence of positive solutions.

In this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in R 2 , − Δ u + V ( x ) u = ( 1 | x | μ ⁎ F ( u ) | x | β ) f ( u ) | x | β + e h ( x ) in R 2 , where V is a positive continuous potential, 0 μ 2 , β ≥ 0 , 2 β + μ ≤ 2 , e is a small parameter and F ( s ) is the primitive function of f ( s ) . Suppose that the nonlinearity f ( s ) is of critical exponential growth in the sense of Trudinger-Moser inequality, we prove the existence and multiplicity of solutions by variational methods.

We consider the question of exponential decay to equilibrium of solutions of an abstract class of degenerate evolution equations on a Hilbert space modeling the steady Boltzmann and other kinetic equations. Specifically, we provide conditions suitable for construction of a stable manifold in a particular “reverse L ∞ ” norm and examine when these do and do not hold.

In this paper, we show the global well-posedness and blow-up result of the solutions with the energy below the threshold for the combined nonlinear Klein–Gordon equation u t t − Δ u + u = | u | p − 1 u − | u | q − 1 u , d ≥ 3 , in the energy space H 1 ( R d ) × L 2 ( R d ) , where 1 + 4 d q p ≤ 1 + 4 d − 2 . We give a threshold of blow up and global well-posedness using a modified variational approach, in the spirit of Ibrahim et al. (2011) [7] .

In this paper we consider the following Schrodinger–Poisson system in the whole R 3 , { − Δ u + u + λ ϕ u = f ( u ) in R 3 , − Δ ϕ = u 2 in R 3 , where λ > 0 and the nonlinearity f is “asymptotically cubic” at infinity. This implies that the nonlocal term ϕu and the nonlinear term f ( u ) are, in some sense, in a strict competition. We show that the system admits a least energy sign-changing and radial solution obtained by minimizing the energy functional on the so-called nodal Nehari set.

This work is concerned with a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production: u t = ∇ ⋅ ( ( 1 + u ) − α ∇ u ) − ∇ ⋅ ( u ( 1 + u ) β − 1 ∇ v ) + f ( u ) , v t = Δ v − v + u γ , with nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain, where α , β ∈ R and γ > 0 . The logistic type source term f ( u ) satisfies that either f ( u ) ≡ 0 or f ( u ) = r u − μ u k with r ∈ R , μ > 0 and k > 1 . The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameters conditions, which improves the known results.

The aim of this paper is to give an existence and uniqueness of solutions for the Cauchy problem of the Navier–Stokes type equations associated to the sublaplacian provided by the left invariant vector fields on the Heisenberg group. To avoid the difficulty of the non-commutative which is intrinsic in 2-step stratified Lie groups, we construct the solenoidal space by using the right invariant vector fields on the Heisenberg group.

In this paper we study the Kirchhoff equation with Hartree-type nonlinearities − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = ( I α ⁎ | u | p ) | u | p − 2 u in R 3 , where a , b > 0 , V : R 3 → R is a potential function and I α is a Riesz potential of order α ∈ ( 0 , 3 ) . Under certain assumptions on V ( x ) , we prove the equation has a ground state for all ( 3 + α ) / 3 p 3 + α by variational methods.

We study the gradient blow-up phenomena for a Hamilton–Jacobi equation with degenerate diffusion u t = ( | u x | p − 2 u x ) x + | u x | q in an interval for q > p ≥ 2 . In the main theorem, we construct a gradient blow-up solution with a small L 1 initial datum.

We provide a complete description of anisotropic scaling limits of stationary linear random field (RF) on Z 3 with long-range dependence and moving average coefficients decaying as O ( | t i | − q i ) in the ith direction, i = 1 , 2 , 3 . The scaling limits are taken over rectangles in Z 3 whose sides increase as O ( λ γ i ) , i = 1 , 2 , 3 when λ → ∞ , for any fixed γ i > 0 , i = 1 , 2 , 3 . We prove that all these limits are Gaussian RFs whose covariance structure is determined by the fulfillment or violation of the balance conditions γ i q i = γ j q j , 1 ≤ i j ≤ 3 . The paper extends recent results in Puplinskaitė and Surgailis (2015, 2016) [27] , [28] , Pilipauskaitė and Surgailis (2016, 2017) [31] , [29] on anisotropic scaling of long-range dependent RFs from dimension 2 to dimension 3.

In this paper, we study a quasilinear differential equation ( | u ′ ( x ) | p − 2 u ′ ( x ) ) ′ + w ( x ) | u ( x ) | q − 2 u ( x ) = 0 in ( 0 , b ) , where q > p > 1 . A theorem related to some oscillation properties depending on the initial parameters is developed. On the basis of this result, we also show that the identical zero set of solutions will determine the function w ( x ) uniquely under some minor assumption. Essentially, the main methods using in this work are the scaling argument and Prufer-type substitutions.

This paper deals with convergence of a solution for the parabolic–parabolic Keller–Segel system { ( u λ ) t = Δ u λ − χ ∇ ⋅ ( u λ ∇ v λ ) in Ω × ( 0 , ∞ ) , λ ( v λ ) t = Δ v λ − v λ + u λ in Ω × ( 0 , ∞ ) , where Ω is a bounded domain in R n ( n ≥ 2 ) with smooth boundary, χ , λ > 0 are constants, to that for the parabolic–elliptic Keller–Segel system { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) in Ω × ( 0 , ∞ ) , 0 = Δ v − v + u in Ω × ( 0 , ∞ ) as λ ↘ 0 .

The paper focuses on a class of supercritical quasilinear Schrodinger elliptic equations − Δ u + V ( x ) u + Δ ( u 2 ) u = λ f ( u ) , x ∈ R N , where N ≥ 3 . We suppose that the nonlinearity f ( t ) : R → R is continuous and just superlinear in a neighborhood of t = 0 . To discuss the existence of positive solutions, the potential V ( x ) are considered in three cases, that are, compact potential, well potential and periodic potential. Some techniques are also employed here, such as Jeanjean's monotonicity trick, blow-up techniques and the Moser iteration.

We consider the existence of homoclinic orbits at the origin of a Hamiltonian system q ¨ + V ′ ( q ) = 0 in R N ( N ≥ 3 ) where V has a strict global maximum at q = 0 and a singularity at a point e ≠ 0 , namely V ( q ) → − ∞ as q → e . We establish via variational methods the existence of a generalized homoclinic orbit q ˜ that may enter the singularity e without assuming the strong force condition of Gordon. Moreover when V ∼ − 1 / | q − e | α ( 0 α 2 ) near e, we give a bound for the number of collisions of q ˜ based on the Morse index of approximated solutions. As a consequence we obtain that q ˜ is classical (non-collision) orbit for α ∈ ] 1 , 2 [ and enters the singularity e at most one time in R if α ∈ ] 0 , 1 ] .

We study, for given T > 0 , the Abel-like equation θ ′ = f 0 + ∑ j = 1 m f j θ j + ∑ k = 1 n a k ( θ ) J τ k ( θ ) where each f j : [ 0 , T ] → R ( j ∈ { 0 , . . . , m } ) is a T-periodic function of bounded variation and each term a k ( θ ) J τ k ( θ ) ( k ∈ { 1 , . . . , n } ) is a jump in θ ′ of state dependent amplitude a k ( θ ) at state dependent instant τ k ( θ ) ∈ [ 0 , T ) . Under appropriate conditions, the equation is shown to admit an absolutely continuous T-periodic solution θ with derivative θ ′ of bounded variation having preassigned jumps a k ( θ ) J τ k ( θ ) . By the contraction principle, a bound is obtained for the rate of point-wise convergence (to the solution) of a sequence of Picard iterates. An algorithm to calculate iterated convolutions is developed and applied to a Riccati equation and to an Abel equation from cosmology.

In this paper, the large time decay of the micropolar fluid equations on H ˙ s ( R n ) (where n = 2 , 3 and s ≥ 0 ) are studied. We show, for each s ≥ 0 , that t s / 2 ‖ ( u , w ) ( ⋅ , t ) ‖ H ˙ s ( R n ) → 0 as t → ∞ for Leray global solutions with arbitrary initial data in L 2 ( R n ) . When the vortex viscosity is present, we obtain a (faster) decay for the micro-rotational field: ‖ w ( ⋅ , t ) ‖ H ˙ s ( R n ) = o ( t s + 1 2 ) . Some related results are also included.

In this paper, we consider the following elliptic problem: (0.1) − Δ u + V ( x ) u + λ ( κ α ⁎ | u | p ) | u | p − 2 u = ( κ β ⁎ | u | q ) | u | q − 2 u , x ∈ R 3 , where λ > 0 , 0 α 3 , 0 β 3 , 1 + α 3 p 3 + α , q > 1 and κ α ( x ) = | x | α − 3 , κ β ( x ) = | x | β − 3 . If V = 1 , we show the existence of a ground state solution and multiple solutions, as well as the nonexistence result for (0.1) . Furthermore, we investigate the limit behavior of ground state solutions of (0.1) as λ → 0 + . Ground state solutions are also obtained if the potential V is not a constant.

In this article, we establish the existence of type I nontopological bubbling solutions for the Einstein–Maxwell–Higgs equation in R 2 . Our solutions bubble at some string points and the local profile near those points comes from the radially symmetric solutions. We also prove that such bubbling solutions cannot exist on compact surfaces.

In this paper, we are interested in weak solutions of the following Schrodinger–Kirchhoff equation involving double critical nonlinearities: { − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u = u 2 ⁎ ( s 1 ) − 1 | x | s 1 + u 2 ⁎ ( s 2 ) − 1 | x | s 2 , in R 3 ∖ { 0 } , u ∈ D 1 , 2 ( R 3 ) , u ≥ 0 in R 3 , where a , b are two positive constants, 2 ⁎ ( s ) = 2 ( 3 − s ) and 0 ≤ s 1 s 2 2 . A natural strategy is to search the weak solutions of the equation as critical points of a suitable functional via variational methods. But by dilation invariance we see that the functional doesn't satisfy ( P S ) condition. We will make use of several techniques to recover compactness. Our results show that the higher order term dominates lower order term and there exist multi-scale characteristics in this nonlinear problem.

This paper is devoted to the following fully parabolic chemotaxis system with Lotka–Volterra competitive kinetics { u t = Δ u − χ 1 ∇ ⋅ ( u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v − χ 2 ∇ ⋅ ( v ∇ w ) + μ 2 v ( 1 − v − a 2 u ) , x ∈ Ω , t > 0 , w t = Δ w − λ w + b 1 u + b 2 v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions, where Ω ⊂ R n is a bounded domain with smooth boundary. We mainly consider the global existence and boundedness of classical solutions in the three dimensional case, which extends and partially improves the results of Bai–Winkler (2016) [1] , Xiang (2018) [25] , as well as Lin–Mu–Wang (2015) [10] , etc.

This paper deals with the oscillation problem for the nonlinear differential equation ( | x ′ | p ( t ) − 2 x ′ ) ′ + λ t p ( t ) | x | p ( t ) − 2 x = 0 , where λ is a positive parameter and p ( t ) > 1 is a nondecreasing and smooth function. Some (non)oscillation criteria are given for this equation. The obtained results are the best possible in a certain sense. We use the Riccati technique and the function sequence technique. Moreover, we propose a conjecture concerning the oscillation problem for this equation in the case when p ( t ) tends to infinity as t → ∞ .

A non-degeneracy theorem for positive solutions of the standard Kirchhoff model is proved in all cases except for n ≥ 5 and b ∫ R n | ∇ Q | 2 d x = 2 ( n − 4 ) n − 4 2 / ( n − 2 ) n − 2 2 . In particular, for dimensions n ≥ 5 and b ∫ R n | ∇ Q | 2 d x 2 ( n − 4 ) n − 4 2 / ( n − 2 ) n − 2 2 , we show that there exist two non-degenerate positive solutions of the Kirchhoff equations, which seem to be completely different from the result of the standard Schrodinger equation. The effects of the non-local term on the positive solution set are also studied. We show that the non-local term has no effect on the structure of the positive solution set for dimensions 1 ≤ n ≤ 3 and the effects eventually appear for dimensions n ≥ 4 . With the non-degeneracy property of positive solutions of the limit problem, we construct concentrated solutions of a singularly perturbed Kirchhoff problem via the well-known Lyapunov–Schmidt reduction method. For dimensions n ≥ 5 , the existence result of concentrated solutions is completely new and is not implied by the main result of G.M. Figueiredo et al. in [19] , where a generalized singularly perturbed Kirchhoff problem was considered.

This paper is concerned with the asymptotic behavior of the solution for the generalized Hamilton–Jacobi equation u t + f ( D u ) = Δ u . It is known that one dimensional equation v t + C 0 v x 2 = v x x has a self-similar solution v ( x 1 + t ) if v + ≠ v − where v + = v ( t , + ∞ ) , v − = v ( t , − ∞ ) . This kind of solution is called planar diffusion wave in multi-dimension (M-D). In this paper, it is shown that under some smallness conditions, the solutions to the generalized Hamilton–Jacobi equation converge to the above diffusion wave v ( x 1 + t ) with C 0 = 1 2 ∂ 2 f ( ξ 1 , ξ 2 , ⋯ , ξ n ) ∂ ξ 1 2 | ξ 1 = 0 , as time tends to infinity. The convergence rate in time is also obtained.

For α > 0 and 0 p ∞ , the maximum principle in Fock spaces F α p states that for entire functions f and g in F α p , there exists a radius R > 0 such that | f ( z ) | ⩽ | g ( z ) | for all | z | ⩾ R , then ‖ f ‖ p , α ⩽ ‖ g ‖ p , α . The answer to the problem for p ⩾ 1 can be found in [15] . In this article, motivated by the work of Božin and Karapetrovic [1] , we prove that the maximum principle in Fock space F α p when 0 p 1 is not true.

We consider meromorphic solutions to general differential-difference equation of type L ( z , f ) + ∑ j = 1 q c j ( z ) f n j ( z + ζ j ) = ∑ j = 1 r γ j e λ j z , where L ( z , f ) ≢ 0 is a linear differential-difference polynomial of f with small function coefficients, and 1 n 1 ⋯ n q , γ j ( 1 ≤ j ≤ r ) are nonvanishing constants, c j ( z ) ( 1 ≤ j ≤ q ) are nonvanishing small functions. If the equation admits a transcendental meromorphic solution f of finite order satisfying that λ ( f ) σ ( f ) , λ ( 1 / f ) σ ( f ) , then representation of f and relations between coefficients in ψ ( z , f ) and γ i , λ j can be obtained.

A foundational result in the theory of Lyndon words (words that are strictly earlier in lexicographic order than their cyclic permutations) is the Chen-Fox-Lyndon theorem which states that every word has a unique non-increasing decomposition into Lyndon words. This article extends this factorization theorem, obtaining the proportion of these decompositions that are strictly decreasing. This result is then used to count primitive pseudo orbits (sets of primitive periodic orbits) on q-nary graphs. As an application we obtain a diagonal approximation to the variance of the characteristic polynomial coefficients q-nary quantum graphs., Comment: 10 pages

We consider the static vacuum Einstein spacetime when the spatial factor is conformal to a n-dimensional pseudo-Euclidean space. The most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described. We obtain the entire set of solutions of the reduced system, where the classical Schwarzschild solution arises as a particular solution. In addition, we show that the Riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature.

The paper investigates longtime dynamics of the Kirchhoff wave equation with strong damping and critical nonlinearities: u t t − ( 1 + ϵ ‖ ∇ u ‖ 2 ) Δ u − Δ u t + h ( u t ) + g ( u ) = f ( x ) , with ϵ ∈ [ 0 , 1 ] . The well-posedness and the existence of global and exponential attractors are established, and the stability of the attractors on the perturbation parameter ϵ is proved for the IBVP of the equation provided that both nonlinearities h ( s ) and g ( s ) are of critical growth.

We study the following time fractional complex nonlinear Ginzburg–Landau equation: { e − i ω 0 C D t α u − △ u = e i γ | u | p − 1 u , x ∈ R N , t > 0 , u ( 0 , x ) = u 0 ( x ) , x ∈ R N , where 0 α 1 , γ ∈ R , − π + π α 2 ω π − π α 2 , p > 1 , u 0 ∈ L q ( R N ) ( q ≥ q c = N ( p − 1 ) 2 and q ≥ 1 ) is a complex-valued function, and D t α 0 C u = ∂ ∂ t 0 I t 1 − α ( u ( t , x ) − u ( 0 , x ) ) , where I t 1 − α 0 denotes a left Riemann–Liouville fractional integral of order 1 − α . By defining two operators and establishing some estimates of them, we prove the well-posedness of the mild solution for this problem in C ( [ 0 , T ] , L q ( R N ) ) and L 2 r q α N ( r − q ) ( ( 0 , T ) , L r ( R N ) ) , where r satisfies 1 / q − 1 / r 2 / N . Moreover, we also obtain the existence of global solutions when ‖ u 0 ‖ L q c ( R N ) is sufficiently small.