<p style='text-indent:20px;'>We define the notions of impulsive evolution processes and their pullback attractors, and exhibit conditions under which a given impulsive evolution process has a pullback attractor. We apply our results to a nonautonomous ordinary differential equation describing an integrate-and-fire model of neuron membrane, as well as to a heat equation with nonautonomous impulse and a nonautonomous 2D Navier-Stokes equation. Finally, we introduce the notion of tube conditions to impulsive evolution processes, and use them as an alternative way to obtain pullback attractors.

We will extend a recent result of B. Choi, P. Daskalopoulos and J. King [5]. For any $ n\ge 3 $, $ 0&lt;m&lt;\frac{n-2}{n+2} $ and $ \gamma&gt;0 $, we will construct subsolutions and supersolutions of the fast diffusion equation $ u_t = \frac{n-1}{m}\Delta u^m $ in $ \mathbb{R}^n\times (t_0, T) $, $ t_0&lt;T $, which decay at the rate $ (T-t)^{\frac{1+\gamma}{1-m}} $ as $ t\nearrow T $. As a consequence we obtain the existence of unique solution of the Cauchy problem $ u_t = \frac{n-1}{m}\Delta u^m $ in $ \mathbb{R}^n\times (t_0, T) $, $ u(x, t_0) = u_0(x) $ in $ \mathbb{R}^n $, which decay at the rate $ (T-t)^{\frac{1+\gamma}{1-m}} $ as $ t\nearrow T $ when $ u_0 $ satisfies appropriate decay condition.

In this paper, nearly integrable system under almost periodic perturbations is studied \begin{document}$ \left\{\begin{array}{l} \dot{x} = \omega_0+y+f(t, x, y), \\ \dot{y} = g(t, x, y), \end{array}\right. $\end{document} where $ x\in\mathbb{T}^n, \, y\in\mathbb{R}^n $, $ \omega_0\in\mathbb{R}^n $ is the frequency vector, and the perturbations $ f, g $ are real analytic almost periodic functions in $ t $ with the infinite frequency $ \omega = (\cdots, \omega_\lambda, \cdots)_{\lambda\in\mathbb{Z}} $. We also assume that the above system is reversible with respect to the involution $ \mathcal{M}_0:(x, y)\rightarrow (-x, y) $. By KAM iterative method, we prove the existence of invariant tori for the above reversible system. As an application, we discuss the existence of almost periodic solutions and the boundedness of all solutions for a second-order nonlinear differential equation.

For transformations with regularly varying property, we identify a class of moduli of continuity related to the local behavior of the dynamics near a fixed point, and we prove that this class is not compatible with the existence of continuous sub-actions. The dynamical obstruction is given merely by a local property. As a natural complement, we also deal with the question of the existence of continuous sub-actions focusing on a particular dynamic setting. Applications of both results include interval maps that are expanding outside a neutral fixed point, as Manneville-Pomeau and Farey maps.

We study the regularity of exceptional actions of groups by $ C^{1, \alpha} $ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $ \alpha $. Let $ G $ be a finitely generated group admitting a $ C^{1, \alpha} $ action $ \rho $ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $ G $ has spherical growth bounded by $ c n^{d-1} $ and if the function $ 1/\alpha^d $ is integrable near zero, then under some mild technical assumptions on $ \alpha $, there is a sequence of exceptional $ C^{1, \alpha} $ actions of $ G $ which converge to $ \rho $ in the $ C^1 $ topology. As a consequence for a single diffeomorphism, we obtain that if the function $ 1/\alpha $ is integrable near zero, then there exists a $ C^{1, \alpha} $ exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus $ \alpha $. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $ C^1 $ diffeomorphisms of the circle.

Let $ (h_t)_{t\in {\mathbb{R}}} $ be the horocycle flow acting on $ (M,\mu) = (\Gamma \backslash \operatorname{SL}(2,{\mathbb{R}}), \mu) $, where $ \Gamma $ is a co-compact lattice in $ \operatorname{SL}(2,{\mathbb{R}}) $ and $ \mu $ is the homogeneous probability measure locally given by the Haar measure on $ \operatorname{SL}(2,{\mathbb{R}}) $. Let $ \tau\in W^6(M) $ be a strictly positive function and let $ \mu^{\tau} $ be the measure equivalent to $ \mu $ with density $ \tau $. We consider the time changed flow $ (h_t^\tau)_{t\in {\mathbb{R}}} $ and we show that there exists $ \gamma = \gamma(M,\tau)&gt;0 $ and a constant $ C&gt;0 $ such that for any $ f_0, f_1, f_2\in W^6(M) $ and for all $ 0 = t_0&lt;t_1&lt;t_2 $, we have \begin{document}$ \left|\int_M \prod\limits_{i = 0}^{2} f_i\circ h^\tau_{t_i} {\rm{d}} \mu^\tau -\prod\limits_{i = 0}^{2}\int_M f_i {\rm{d}} \mu^\tau \right|\leq C \left(\prod\limits_{i = 0}^{2} \|f_i\|_6\right) \left(\min\limits_{0\leq i&lt;j\leq 2} |t_i-t_j|\right)^{-\gamma}. $\end{document} With the same techniques, we establish polynomial mixing of all orders under the additional assumption of $ \tau $ being fully supported on the discrete series.

In this paper, we study the quasi-shadowing property for partially hyperbolic flows. A partially hyperbolic flow $ \varphi_{t} $ has the quasi-shadowing property if for any $ (\delta,T) $-pseudoorbit $ g(t) $ of $ \varphi_{t} $ there exist a sequence of points $ \{y_{k}\}_{k\in\mathbb{Z}} $ and a reparametrization $ \alpha $ such that $ \varphi_{\alpha(t)-\alpha(kT)}(y_k) $ trace $ g(t) $ in which $ y_{k} $ is obtained from $ \varphi_{\alpha(kT)-\alpha((k-1)T)}(y_{k-1}) $ by a motion along the central direction. We prove that any partially hyperbolic flow $ \varphi_{t} $ has the quasi-shadowing property. We also investigate the limit quasi-shadowing properties for flows. That is, a partially hyperbolic flow has the $ \mathcal{L}^p $, limit and asymptotic quasi-shadowing properties.

In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean $ R^{n+1} $ with speed $ u^\alpha\sigma_k^\beta $ firstly, where $ u $ is support function of the hypersurface, $ \alpha, \beta \in R^1 $, and $ \beta&gt;0 $, $ \sigma_k $ is the $ k $-th symmetric polynomial of the principal curvature radii of the hypersurface, $ k $ is an integer and $ 1\le k\le n $. For $ \alpha\le1-k\beta $, $ \beta&gt;\frac{1}{k} $ we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $ \alpha\le1-k\beta $, $ \beta&gt;\frac{1}{k} $, we prove that the flow with the speed $ fu^\alpha\sigma_k^\beta $ exists for all time and converges smoothly after normalisation to a soliton which is a solution of $ fu^{\alpha-1}\sigma_k^{\beta} = c $ provided that $ f $ is a smooth positive function on $ S^n $ and satisfies that $ (\nabla_i\nabla_jf^{\frac{1}{1+k\beta-\alpha}}+\delta_{ij}f^{\frac{1}{1+k\beta-\alpha}}) $ is positive definite. When $ \beta = 1 $, our argument provides a proof to the well-known $ L_p $ Christoffel-Minkowski problem for the case $ p\ge k+1 $ where $ p = 2-\alpha $, which is identify with Ivaki's recent result. Especially, we obtain the same result for $ k = n $ without any constraint on smooth positive function $ f $. Finally, we also give a counterexample for the two anisotropic expanding flows when $ \alpha&gt;1-k\beta $.

In this paper, we study the long-time existence and uniqueness (solvability) for the initial value problem of the 2D inviscid dispersive SQG equation. First we obtain the local solvability with existence-time independent of the amplitude parameter $ A $. Then, assuming more regularity and using a blow-up criterion of BKM type and a space-time estimate of Strichartz type, we prove long-time solvability of solutions in Besov spaces for large $ A $ and arbitrary initial data. In comparison with previous results, we are able to consider improved cases of the regularity and larger initial data classes.

<p style='text-indent:20px;'>We consider a system of coupled cubic Schrödinger equations in one space dimension <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \text{i}\partial_t u + \partial_x^2 u +(|u|^2 + \omega |v|^2) u = 0\\ \text{i}\partial_t v + \partial_x^2 v+ (|v|^2 + \omega |u|^2) v = 0 \end{cases}\quad (t,x)\in \mathbb{R}\times \mathbb{R}, \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>in the non-integrable case <inline-formula><tex-math id="M1">\begin{document}$ 0 &lt; \omega &lt; 1 $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>First, we justify the existence of a symmetric 2-solitary wave with logarithmic distance, <i>i.e.</i> a solution of the system satisfying <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \lim\limits_{t\to +\infty}\left\| \begin{pmatrix} u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix} e^{ \text{i} t}Q (\cdot - \frac{1}{2} \log (\Omega t) - \frac{1}{4} \log \log t) \\[4pt] e^{ \text{i} t}Q (\cdot + \frac{1}{2} \log (\Omega t) + \frac{1}{4} \log \log t)\end{pmatrix}\right\|_{H^1\times H^1} = 0 $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ Q = \sqrt{2} $\end{document}</tex-math></inline-formula>sech is the explicit solution of <inline-formula><tex-math id="M3">\begin{document}$ Q'' - Q + Q^3 = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \Omega&gt;0 $\end{document}</tex-math></inline-formula> is a constant. This result extends to the non-integrable case the existence of symmetric 2-solitons with logarithmic distance known in the integrable case <inline-formula><tex-math id="M5">\begin{document}$ \omega = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \omega = 1 $\end{document}</tex-math></inline-formula> ([<xref ref-type="bibr" rid="b15">15</xref>,<xref ref-type="bibr" rid="b33">33</xref>]). Such strongly interacting symmetric <inline-formula><tex-math id="M7">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-solitary waves were also previously constructed for the non-integrable scalar nonlinear Schrödinger equation in any space dimension and for any energy-subcritical power nonlinearity ([<xref ref-type="bibr" rid="b20">20</xref>,<xref ref-type="bibr" rid="b22">22</xref>]). <p style='text-indent:20px;'>Second, under the conditions <inline-formula><tex-math id="M8">\begin{document}$ 0&lt;c&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ 0&lt;\omega &lt; \frac 12 c(c+1) $\end{document}</tex-math></inline-formula>, we construct solutions of the system satisfying <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \lim\limits_{t\to +\infty}\left\| \begin{pmatrix}u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix}e^{ \text{i} c^2 t}Q_c (\cdot - \frac{1}{(c+1)c} \log (\Omega_c t) )\\[4pt] e^{ \text{i} t} Q (\cdot + \frac{1}{c+1} \log (\Omega_c t))\end{pmatrix} \right\|_{H^1\times H^1} = 0 $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M10">\begin{document}$ Q_c(x) = cQ(cx) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ \Omega_c&gt;0 $\end{document}</tex-math></inline-formula> is a constant. Such logarithmic regime with non-symmetric solitons does not exist in the integrable cases <inline-formula><tex-math id="M12">\begin{document}$ \omega = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ \omega = 1 $\end{document}</tex-math></inline-formula> and is still unknown in the non-integrable scalar case.