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We prove that the wave operators of the scattering theory for the fourth order Schrö­dinger operator ^2 + V(x) on R^4 are bounded in L^p(R^4) for the set of p ’s of (1,ınfty) depending on the kind of spectral singularities of H at zero which can be described by the space of bounded solutions of (^2 + V(x))u(x)=0 .

The chemotaxis-Navier-Stokes system (⋆){nt+u⋅∇namp;= Δn−∇⋅(nχ(c)∇c),ct+u⋅∇camp;= Δc−nf(c),ut+(u⋅∇)uamp;= Δu+∇P+n∇Φ,∇⋅uamp;= 0\\begin{equation*} (\\star )\\qquad \\qquad \\qquad \\quad \\begin {cases} n_t + u\\cdot \\nabla n & =\\ \\ \\Delta n - \\nabla \\cdot (n\\chi (c)\\nabla c),\\\[1mm] c_t + u\\cdot \\nabla c & =\\ \\ \\Delta c-nf(c), \\\[1mm] u_t + (u\\cdot \\nabla )u & =\\ \\ \\Delta u + \\nabla P + n \\nabla \\Phi , \\\[1mm] \\nabla \\cdot u & =\\ \\ 0 \\end{cases} \\qquad \\qquad \\qquad \\quad \\end{equation*} is considered under boundary conditions of homogeneous Neumann type for nn and cc, and Dirichlet type for uu, in a bounded convex domain Ω⊂R3\\Omega \\subset \\mathbb {R}^3 with smooth boundary, where Φ∈W1,∞(Ω)\\Phi \\in W^{1,\\infty }(\\Omega ) and χ\\chi and ff are sufficiently smooth given functions generalizing the prototypes χ≡const.\\chi \\equiv const. and f(s)=sf(s)=s for s≥0s\\ge 0. It is known that for all suitably regular initial data n0,c0n_0, c_0 and u0u_0 satisfying 0≢n0≥00\\not \\equiv n_0\\ge 0, c0≥0c_0\\ge 0 and ∇⋅u0=0\\nabla \\cdot u_0=0, a corresponding initial-boundary value problem admits at least one global weak solution which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. The present paper shows that after some relaxation time, this solution enjoys further regularity properties and thereby complies with the concept of eventual energy solutions, which is newly introduced here and which inter alia requires that two quasi-dissipative inequalities are ultimately satisfied. Moreover, it is shown that actually for any such eventual energy solution (n,c,u)(n,c,u) there exists a waiting time T0∈(0,∞)T_0\\in (0,\\infty ) with the property that (n,c,u)(n,c,u) is smooth in Ω¯×[T0,∞)\\bar \\Omega \\times [T_0,\\infty ) and that n(x,t)→n0¯,c(x,t)→0andu(x,t)→0\\begin{eqnarray*} n(x,t)\\to \\overline {n_0}, \\qquad c(x,t)\\to 0 \\qquad \\mbox {and} \\qquad u(x,t)\\to 0 \\end{eqnarray*} hold as t→∞t\\to \\infty, uniformly with respect to x∈Ωx\\in \\Omega. This resembles a classical result on the three-dimensional Navier-Stokes system, asserting eventual smoothness of arbitrary weak solutions thereof which additionally fulfill the associated natural energy inequality. In consequence, our results inter alia indicate that under the considered boundary conditions, the possibly destabilizing action of chemotactic cross-diffusion in (⋆\\star) does not substantially affect the regularity properties of the fluid flow at least on large time scales.

This paper deals with the parabolic-parabolic-elliptic forager-exploiter model under homogeneous Neumann boundary conditions. It is shown that if the taxis effects of exploiters are suitably weak, its classical solution is globally bounded in arbitrary dimensions. Moreover, the foragers and exploiters will approach spatially homogeneous distributions in the large time limit.

This paper is framed in a series of studies on attraction–repulsion chemotaxis models combining different effects: nonlinear diffusion and sensitivities and logistic sources, for the dynamics of the cell density, and consumption and/or production impacts, for those of the chemicals. In particular, herein we focus on the situation where the signal responsible of gathering tendencies for the particles’ distribution is produced, while the opposite counterpart is consumed. In such a sense, this research complements the results in Frassu et al. (Math Methods Appl Sci 45:11067–11078, 2022) and Chiyo et al. (Commun Pure Appl Anal, 2023, doi: 10.3934/cpaa.2023047), where the chemicals evolve according to different laws.

Metric spaces are generalized by many scholars. Recently, Khatami and Mirzavaziri use a mapping called t-definer to popularize the triangle inequality and give a generalization of the notion of a metric, which is called a ★-metric. In this paper, we prove that every ★-metric space is metrizable. Also, we study the total boundedness and completeness of ★-metric spaces.

We derive a fractional Cahn-Hilliard equation (FCHE) by considering a gradient flow in the negative order Sobolev space H-α, α ϵ [0,1], where the choice α = 1 corresponds to the classical Cahn-Hilliard equation while the choice α = 0 recovers the Allen-Cahn equation. The existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fractional order α and that it indeed reduces the free energy. We then turn to the delicate question of the L∞ boundedness of the solution and establish an L∞ bound for the FCHE in the case where the nonlinearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier-Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semidiscrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order α. It is observed that the nature of the solution of the FCHE with a general α > 0 is qualitatively (and quantitatively) closer to the behavior of the classical Cahn-Hilliard equation than to the Allen-Cahn equation, regardless of how close to zero the value of α is. An examination of the coarsening rates of the FCHE reveals that the asymptotic rate is rather insensitive to the value of α and, as a consequence, is close to the well-established rate observed for the classical Cahn-Hilliard equation.

This paper investigates the resilient annular finite‐time synchronization and boundedness problems for master‐slave systems under dynamic event‐triggered scheme (ETS), actuator faults, and scaling attacks. A comprehensive model for characterizing scaling attacks is presented. The switched dynamic ETS is developed to reduce the bandwidth pressure. Two different internal variables are designed, each of which changes based on the switching manifold. Simultaneously, the resilient problem of switched master‐slave systems with respect to stochastic scaling attacks is considered, where there are limited malicious signals introduced by the adversary. Based on the Lyapunov‐like function involving the different internal dynamic variables, a set of criteria is established in the form of linear matrix inequalities (LMIs), which guarantee the annular finite‐time boundedness. Finally, two examples are presented to verify the theoretical findings. This work has addressed the resilient annular finite‐time synchronization control problem for master‐slave systems under dynamic ETS, actuator faults, and scaling attacks. A scaling attack is used in the attack strategy, associated with a Bernoulli‐distributed random variable. An adaptive mechanism is used to mitigate any potential effects of actuator faults that may occur within the system.

This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as $$\\eqalign{& u_t = \\Delta (\\gamma (v)u) + \\rho u-\\mu u^l,\\quad x\\in \\Omega ,\\;t > 0, \\cr & v_t = \\Delta v-v + u,\\quad x\\in \\Omega ,\\;t > 0.} $$It is shown that whenever ρ ∈ ℝ, μ > 0 and $$l > \\max \\left\\{ {\\displaystyle{{n + 2} \\over 2},2} \\right\\},$$then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium $$\\left( {{\\left( {\\displaystyle{{\\rho _ + } \\over \\mu }} \\right)}^{1/(l-1)},{\\left( {\\displaystyle{{\\rho _ + } \\over \\mu }} \\right)}^{1/(l-1)}} \\right)$$as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds $Y\\rightarrow X$ with a rational section, provided that $\\dim (Y)\\leq 5$ and $Y$ is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs $(X, \\Delta )$ with $K_X+\\Delta$ numerically trivial and not of product type, in dimension at most four.

In this paper, we study Hausdorff operator H μ on weighted mixed norm Fock spaces F ϕ p , q for 1 ≤ p , q ≤ ∞ . The boundedness and compactness of H μ on F ϕ p , q are characterized, and we give when H μ on F ϕ p , q is power bounded or uniformly mean ergodic.