The large time behaviour for a more general prey-axis system is considered. The asymptotically uniform boundedness of solutions in a suitable space is derived to ensure the dissipativity of the system. Based on the dissipativity of the system, the existence of a global attractor is obtained. The main technique used in this paper is the Lp-Lq estimation method.
In this paper, we consider the following nonhomogeneous Schrödinger- Poisson system
where $\eta \neq 0$, λ>0 is a real parameter and $f \in L^{\frac{6}{5}}\left(\mathbb{R}^{3}\right)$ is a nonzero nonnegative function. By using the Mountain Pass theorem and variational method, for λ small, we show that the system with η >0 has at least two positive solutions, the system with η <0 has at least one positive solution. Our result generalizes and improves some recent results in the literature.
This paper is concerned with the well-posedness, uniform asymptotic sta- bility and dynamics for a semilinear thermoelastic system with time-varying delay boundary feedback and nonlinear weight, which can be used to describe the physical procedure of meridian retraction and release therapy. The perturbation theory of linear operators by Kato is used to deal with the invalidity of Lumper-Phillips theorem on non-autonomous PDEs operator, the multiplier approach and quasi-stability method lead to the stability and dynamics for our semilinear problem, which are also true for linear thermoelastic system without weight.
In this paper, we study the following coupled nonlinear logarithmic Hartree system
where β,µi,λi (i = 1,2) are positive constants, ∗ denotes the convolution in $\mathbb{R}^{2}$. By considering the constraint minimum problem on the Nehari manifold, we prove the existence of ground state solutions for β >0 large enough. Moreover, we also show that every positive solution is radially symmetric and decays exponentially.
This paper is devoted to study the existence and multiplicity of nontrivial solutions for the following boundary value problem
where B is the unit ball in $ \mathbb{R}^{N}$, the radial positive weight ω(x) is of logarithmic type function, the functional f(x,u) is continuous in $ B \times \mathbb{R}$ and has double exponential crit-ical growth, which behaves like exp$ \left\{e^{\alpha|u|^{\frac{N}{N-1}}}\right\}$as |u| → ∞ for some α >0. Moreover, ϵ> 0, and the radial function h belongs to the dual space of $ W_{0, \text { rad }}^{1, N}(B), h \neq 0$.
The incompressible limit of nonisentropic ideal magnetohydrodynamic equ- ations with general initial data in the whole space $ \mathbb{R}^{3}$ is proved in this paper. The uni- form estimates of solutions with respect to the Mach number are obtained by using energy estimate. Strong convergence results of the smooth solutions are established by using Strichartz’s estimates in the whole space.
We study a system of equations arising in the Chern-Simons model on finite graphs. Using the iteration scheme and the upper and lower solutions method, we get existence of solutions in the non-critical case. The critical case is dealt with by priori estimates. Our results generalize those of Huang et al. (Journal of Functional Analysis 281(10) (2021) Paper No. 109218).