We consider the Ginzburg-Landau-Chern-Simons model with defocusing or focusing potential. For the model with defocusing potential, the existence of global in time solution is shown for the $H^1(\mathbb{R}^2)$ initial data. For the model with focusing potential, the nonexistence of global solution can be shown for the initial data with negative energy.

The Cauchy problem for the the Klein-Gordon equation with the Hartree type semilinear term is considered in the de Sitter spacetime. The effects of the spatial expansion and contraction on the existence of the solution of the equation are studied.

This paper considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian \begin{align*} \begin{cases} \left ( \vert u'\vert^{p-2}u' \right ) '+ \left ( \lambda+r(x) \right ) \vert u\vert ^{p-2}u=0,\,\, x\in (0,\pi_{p}),\\ u(0)=u(\pi_{p})=0, \end{cases} \end{align*} where $1 < p < \infty$, $\lambda < p-1$, $\pi_{p}$ is the generalized $\pi$ given by $\pi_{p}=2\pi/ \left ( p\sin(\pi/p) \right ) $ and $r\in C[0,\pi_{p}]$. Sharp Lyapunov-type inequalities, which are necessary conditions for the existence of nontrivial solutions of the above problem are presented. Results are obtained through the analysis of variational problem related to a sharp Sobolev embedding and generalized trigonometric and hyperbolic functions.

We investigate the indirect stabilization of Timoshenko systems with only one local memory effect. By the multiplier method and some specially designed analysis approaches for the problem, we obtain the exponential decay rates for the case in which the wave speeds are the same. Of particular notes are: This is a local memory damping problem for coupled systems without complementary frictional damping (or any other type of damping); It is also an indirect damping problem, which has memory damping only in the second equation and no damping in the first equation. The decay results here are established without the usual nonnegative/decreasing assumptions for the kernel, that is, the kernel may be oscillating or sign-varying. As can be seen, we weaken the hypotheses of the kernel and the memory-effect area, but we still establish the exponential decay rates of the energy. Therefore, we have improved and extended the related results to a considerable extent.

We revisit the Cauchy problem of nonlinear massive Dirac equation with Yukawa type potentials $\mathcal F^{-1}\left[(b^2 + |\xi|^2)^{-1}\right]$ in 2 dimensions. The authors of [10, 4] obtained small data scattering and large data global well-posedness in $H^s$ for $s > 0$, respectively. In this paper, we show that the small data scattering occurs in $L_x^2(\mathbb R^2)$. This can be done by combining bilinear estimates and modulation estimates of [12,10].

Carlen showed that the Beckner-Hirschman entropic uncertainty principle, which is an improvement of Heisenberg's uncertainty principle, is equivalent to a quantitative logarithmic Sobolev inequality involving the entropy of the Fourier-Wiener transform. In this work, we identify sharp spaces and prove quantitative and non-quantitative stability results for the logarithmic Sobolev inequality involving Wasserstein and $L^p$ metrics. Inter-alia, we prove a new inequality relating the entropy of the Fourier-Wiener transform with the Fisher information and Wasserstein metric via the Monge-Ampère equation. Moreover, we investigate the dependence on the dimension of the stability constant and show that in certain configurations it can be taken independently and identify spaces in which the dimension-dependence cannot be removed. The techniques are based on optimal transport theory and Fourier analysis. Lastly, we discuss a probabilistic approach to stability estimates based on Cramér's convolution theorem.