Global existence and long time behavior of solutions to some Oldroyd type models in hybrid Besov spaces

Long time behavior of solutions to the compressible MHD system in multi-dimensions

Global existence and long time behavior of solutions to a model of ferroelectric materials

In this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for Jμ(u0)<0 and decay rate of H01 and L2-norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.

In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H‐system in a bounded smooth domain of . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions but also obtain the decay rate of the norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were.

In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].

Dynamics for a class of nonlinear 2D Kirchhoff–Boussinesq models is studied. These nonlinear plate models are characterized by the presence of a nonlinear source that alone leads to finite-time blow up of solutions. In order to counteract, restorative forces are introduced, which however are of a supercritical nature. This raises natural questions such as: (i) wellposedness of finite energy (weak) solutions, (ii) their regularity, and (iii) long time behavior of both weak and strong solutions. It is shown that finite energy solutions do exist globally, are unique and satisfy Hadamard wellposedness criterium. In addition, weak solutions corresponding to “strong” initial data (i.e., strong solutions) enjoy, likewise, the full Hadamard wellposedness. The proof is based on logarithmic control of the lack of Sobolev's embedding. In addition to wellposedness, long time behavior is analyzed. Viscous damping added to the model controls long time behaviour of solutions. It is shown that both weak and (resp. strong) solutions admit compact global attractors in the finite energy norm, (resp. strong topology of strong solutions). The proof of long time behavior is based on Ball's method [2] and on recent asymptotic quasi-stability inequalities established in [11]. These inequalities enable to prove that strong attractors are finite-dimensional and the corresponding trajectories can exhibit C ∞ smoothness.

We study the incompressible Hall-MHD system, an important model in plasma physics akin to the Navier-Stokes equations, using harmonic analysis tools. Chapter \ref{intro} consists of an introduction of the Hall-MHD system and its derivation from a two-fluid Euler-Maxwell system, along with a review of the mathematical preliminaries. Chapter \ref{w} concerns the well-posedness of the Hall-MHD system. For completeness, a proof of the global-in-time existence of the Leray-Hopf type weak solutions is included. In addition, we include a proof of the regularity criterion in \cite{D1}, which is of particular interest as it highlights the dissipation wavenumbers formulated via Littlewood-Paley theory. We then exploit the regularizing effect of diffusion and use a classical fixed point theorem to prove local-in-time existence of solutions to the generalized Hall-MHD system in certain Besov spaces as well as global-in-time existence of solutions to the hyper-dissipative electron MHD equations for small initial data in critical Besov spaces. Long time behaviour of solutions to the Hall-MHD system is studied in Chapter \ref{l}. We reproduce the proof of algebraic decay of weak solutions to the fully dissipative Hall-MHD system in \cite{CS1}; we then present our study of strong solutions to the Hall-MHD systems with mere one diffusion featuring the Fourier splitting technique. Under certain moderate assumptions, we show that the magnetic energy decays to $0$ and the kinetic energy converges to a certain constant in the resistive inviscid case, while the opposite happens in the viscous non-resistive case. Inspired by \cite{CDK}, we study the long time behaviour of solutions to the Hall-MHD system from the viewpoint of the determining Fourier modes. Via Littlewood-Paley theory, we formulate the determining wavenumbers, which bounds the low frequencies essential to the long time behaviour of the solutions. The fact that the determining wavenumbers can be estimated in a certain average sense suggests that the Hall-MHD system has finite degrees of freedom in a certain sense.

In this paper we study the long time behavior of solutions for an optimal control problem associated with the viscous incompressible electrically conducting fluid modeled by the magnetohydrodynamic (MHD) equations in a bounded two dimensional domain through the adjustment of distributed controls. We first construct a quasi-optimal solution for the MHD systems which possesses exponential decay in time. We then derive some preliminary estimates for the long-time behavior of all admissible solutions of the MHD systems. Next we prove the existence of a solution for the optimal control problem for both finite and infinite time intervals. Finally, we establish the long-time decay properties of the solutions for the optimal control problem.

In this paper, we investigate the generalized Boussinesq equation (gBq) as a model for the water wave problem with surface tension. Our study begins with the analysis of the initial value problem within Sobolev spaces, where we derive improved conditions for global existence and finite-time blow-up of solutions, extending previous results to lower Sobolev indices. Furthermore, we explore the time-decay behavior of solutions in Bessel potential and modulation spaces, establishing global well-posedness and time-decay estimates in these function spaces. Using Pohozaev-type identities, we demonstrate the non-existence of solitary waves for specific parameter regimes. A significant contribution of this work is the numerical generation of solitary wave solutions for the gBq equation using the Petviashvili iteration method. Additionally, we propose a Fourier pseudo-spectral numerical method to study the time evolution of solutions, particularly addressing the gap interval where theoretical results on global existence or blow-up are unavailable in the Sobolev spaces. Our numerical results provide new insights by confirming theoretical predictions in covered cases and filling gaps in unexplored scenarios. This comprehensive analysis not only clarifies the theoretical and numerical landscape of the gBq equation but also offers valuable tools for further investigations.

Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy

This paper deals with the long time behavior of solutions of the heat equation with a memory boundary of polynomial type. We rst establish the comparison principle and prove the local existence of classical solutions. Then we study the global existence and blowup of the solutions by using the comparison principle and Greens function. At last, we consider the boundary blowup and the blowup rate.

In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

Complex Ginzburg–Landau equations with dynamic boundary conditions

In this paper, we study a quantum Boltzmann equation with a harmonic oscillator for isotropic gases of bosons and fermions, respectively. This model comes from physics literatures (see, e.g., [M. Holland, J. Williams and J. Cooper, Bose–Einstein condensation: Kinetic evolution obtained from simulated trajectories, Phys. Rev. A 55 (1997) 3670–3677]). The distribution function, i.e. the solution, is discrete in the energy variable. We give the classification of equilibria of the equation for bosons and fermions, respectively, and prove the global existence, uniqueness and the strong convergence to equilibrium for solutions of the equation.

UNIFORM BLOWUP PROFILES FOR DIFFUSION EQUATIONS WITH NONLOCAL SOURCE AND NONLOCAL BOUNDARY