This article investigates the pullback dynamics and statistical solutions for the semi-dissipative nonlinear Schrödinger equations describing the model of Bose–Einstein condensate with an impurity. The principal difficulty comes from the semi-dissipative property described by the fact that the first equation is dissipative and the second one is conservative. First, we prove that the generated evolution process of the solution mappings possesses a pullback attractor, which pullback attracts any bounded subset of the phase space with respect to a certain mixed topology. Second, we establish that the evolution process has a family of invariant Borel probability measures, and these measures satisfy the Liouville type equation. Finally, we verify the equivalence between the statistical solutions and invariant measures. The obtained results reveal that the Boson wave function plays the dominant role while the impurity wave function plays the subordinate role in the evolution dynamics of the addressed model.
The current study incorporates the seasonality and time delayed effects of the pathogen’s incubation period into the classic S–I epidemic model. We focus on the existence of time periodic traveling wave solutions to simulate disease invasion in complicated backgrounds, and some key difficulties thus arise spontaneously to invalidate the utilization of classical Laplace transform methods. For R0>1 ${\mathcal{R}}_{0}{ >}1$ , we characterize the critical wave speed c*, construct upper and lower solutions, and apply two fixed point theorems to prove the existence of traveling waves for c ∈ [c*, ∞), while the nonexistence is obtained both for the cases R0<1 ${\mathcal{R}}_{0}{< }1$ and R0>1 ${\mathcal{R}}_{0}{ >}1$ and c ∈ (0, c*). Additionally, by employing the uniform persistence theory, entire solution technique, and parabolic comparison principle, we still establish the persistence of traveling wave solutions. For c > c*, the persistence at the invaded end is proved by a contradiction–normalization argument, which reduces the loss of persistence to a nontrivial periodic solution of the linearized delayed equation. The critical case c = c* is obtained through a leftmost half-level normalization and compactness. For c ∈ (0, c*), the Laplace-transform argument is replaced by a left-tail normalization and a comparison construction for an auxiliary problem without the spatial shift. Finally, a novel numerical method is also developed to investigate the effects of seasonality and delay kernels on the minimum wave speed c*.
This paper investigates the local well-posedness and blow-up criteria for a new Alice–Bob b-family peakon system within the framework of Besov spaces, under suitable conditions on the correction term R(u, v). Furthermore, we establish the global existence of solutions for the Alice–Bob Camassa–Holm peakon system, provided that the correction term R(u, v) is at most linearly increasing in v.
In this paper, we investigate the asymptotic behavior of the strong solution to the Cauchy problem of 3D compressible magnetohydrodynamics (MHD) equations with density-dependent viscosities. Based on the global existence of large strong solution and some uniform-in-time energy estimates, by applying the Fourier splitting method, we obtain the optimal decay rates of the lower and higher order derivatives of the global strong solution provided that the initial data belongs to Lp0 ${L}^{{p}_{0}}$ with p 0 ∈ [1, 2), additionally.
This paper is devoted to dealing with the global existence and blow-up property for a mixed local and nonlocal parabolic equation with logarithmic nonlinear source and different initial energy levels. For subcritical and critical initial energy levels, we obtain the existence of the global weak solution by using Galerkin approximation method and prove the finite time blow-up phenomenon of the weak solution by combining a family of modified potential wells with convexity method. Meanwhile, the decay estimate of the L 2 norm of the global weak solution, the lower and upper bounds of the maximal existence time of the finite time blow-up weak solution are deduced in this case. For the case of supercritical initial energy level, we give the sufficient conditions on the existence of global and nonglobal weak solutions by using ω-limit set method. In particular, we study the large time behavior of the global weak solution and show that the global weak solution will converge to the corresponding stationary solution and vanish as t → +∞. Moreover, we obtain the existence of the finite time blow-up weak solution with arbitrarily high initial energy level.
