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In this paper, we consider the asymptotic behavior of solutions to the Cauchy problem for the three-dimensional model of compressible reactive fluid, which can be described by a compressible Navier–Stokes type system with potential external force. First, the existence of the stationary solution is shown in the case that the external force is small enough. Next, making use of the energy method, we prove that the stationary solution is time-asymptotically stable provided that the external force and the initial perturbation are sufficiently small. Finally, we obtain the time-decay rate of the solution toward the stationary solution by combining the $��L^p�-�L^q�$ estimates for the corresponding linear problem and the energy estimates for the nonlinear system.

We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators of mixed order.

The setting that we introduce is very general and comprises, for instance, the sum of two fractional Laplacians, or of a fractional Laplacian and a Laplacian, as particular cases (the situation in which there are infinitely many operators, and even a continuous distribution of operators, can be considered as well).

New bits of functional analysis are introduced to deal with this problem. An eigenvalue analysis divides the existence theory into two streams, one related to a mountain pass method, the other to a linking technique.

The uniformization theory of Gromov hyperbolic spaces investigated by Bonk, Heinonen, and Koskela, generalizes the case where a classical Poincaré ball type model is used as the starting point. In this paper, we develop this approach in the case where the underlying domain is unbounded, corresponding to the classical Poincaré half-space model. More precisely, we study conformal densities via Busemann functions on Gromov hyperbolic spaces and prove that the deformed spaces are unbounded uniform spaces. Furthermore, we show that there is a one-to-one correspondence between the bilipschitz classes of proper geodesic Gromov hyperbolic spaces that are roughly starlike with respect to a point on the Gromov boundary and the quasisimilarity classes of unbounded locally compact uniform spaces. Our result can be understood as an unbounded counterpart of the main result of Bonk, Heinonen, and Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque. 270 (2001).

We construct a unique local strong solution to the finitely extensible nonlinear elastic (FENE) dumbbell model of Warner-type for an incompressible polymer fluid (described by the Navier–Stokes–Fokker–Planck equations) interacting with a flexible elastic shell. The latter occupies the flexible boundary of the polymer fluid domain and is modeled by a beam equation coupled through kinematic boundary conditions and the balance of forces. In the 2D case for the co-rotational Fokker–Planck model we obtain global-in-time strong solutions.

A main step in our approach is the proof of local well-posedness for just the solvent–structure system in higher-order topologies which is of independent interest. Different from most of the previous results in the literature, the reference spatial domain is an arbitrary smooth subset of $�R^3$, rather than a flat one. That is, we cover viscoelastic shells rather than elastic plates. Our results also supplement the existing literature on the Navier–Stokes–Fokker–Planck equations posed on a fixed bounded domain.

We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realize the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals—the positive and negative Dirac propagators—global in space and in time, with distinguished complex-valued geometric phase functions. As applications, we relate the Cauchy evolution operators with the Feynman propagator and construct Cauchy surfaces covariances of quasifree Hadamard states.

Of interest in this work is the following equation:

We present a characterization of $��(�\mu �,�\nu �)�$-dichotomies in terms of the admissibility of certain pairs of weighted spaces for nonautonomous discrete time dynamics acting on Banach spaces. Our general framework enables us to treat various settings in which no similar result has been previously obtained as well as to recover and refine several known results. We emphasize that our results hold without any bounded growth assumption and the statements make no use of Lyapunov norms. Moreover, as a consequence of our characterization, we study the robustness of $��(�\mu �,�\nu �)�$-dichotomies, that is, we show that this notion persists under small but very general linear perturbations.

We establish the pullback asymptotic compact of the family probability measures with respect to probability distributions of the solutions of the 2-$�D$ nonautonomous stochastic Navier–Stokes equations, and prove the existence and uniqueness of a pullback measure attractor. The structures of pullback measure attractors is characterized by complete solutions, which is an extension of the notation of evolution systems of measures introduced and developed by Da Prato and Röckner in [Rendiconti Lincei-Matematica e Applicazioni 17 (2006), no. 4, 397–403] and [Seminar on Stochastic Analysis, Random Fields and Applications, 115–122, Springer, 2007]. Moreover, for stochastic systems containing periodic deterministic forcing terms, we show the pullback measure attractors are also periodic under certain conditions.

We study the following Kirchhoff-type equation

Via a new variational principle established by Moameni (C. R. Math. Acad. Sci. Paris. 355 (2017) 1236–1241), we shall show that, for each $��p�>�2�$, there exists $��{\lambda }^{\ast }�>�0�$ such that for each $��\lambda �\in �(�0�,�{\lambda }^{\ast }�)�$ Equation (0.1) has a positive solution with negative energy. Furthermore, by using the improved Clark theorem, we can obtain a sequence of solutions with negative energy converging to zero in $��L^{\infty }��\left(�\Omega �\right)��$ without the restriction of $�\lambda$.

We study the Carleson measures on nontangentially accessible (NTA) and admissible for the Dirichlet problem (ADP) domains in the Heisenberg groups $�H^n$ and provide two characterizations of such measures: (1) in terms of the level sets of subelliptic harmonic functions and (2) via the 1-quasiconformal family of mappings on the Korányi–Reimann unit ball. Moreover, we establish the $�L^2$-bounds for the square function $�S_{\alpha }$ of a subelliptic harmonic function and the Carleson measure estimates for the BMO boundary data, both on NTA domains in $�H^n$. Finally, we prove a Fatou-type theorem on $��(�\epsilon �,�\delta �)�$-domains in $�H^n$. Our work generalizes results by Capogna–Garofalo and Jerison–Kenig.