We prove the existence of infinitely many periodic solutions, as well as the presence of chaotic dynamics, for a periodically perturbed planar Liénard system of the form $\dot{x} = y - F(x) + p(\omega t),\; \dot{y} = - g(x)$. We consider the case in which the perturbing term is not necessarily small. Such a result is achieved by a topological method, that is by proving the presence of a horseshoe structure.

In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial _t^2u-\Delta u + \partial _tu = f(u) & \text{in}\ \Omega\times (0,T), \\ u=0 & \text{on}\ \partial \Omega\times (0,T), \\ u(0)=u_0, \partial _tu(0)=u_1 & \text{in}\ \Omega \end{cases} \end{equation*} in an exterior domain $\Omega$ in $\mathbb R^N$ $(N\geq 2)$, where $f:\mathbb R\to \mathbb R$ is a smooth function which behaves like $f(u)\sim |u|^p$. From the view point of weighted energy estimates given by Sobajima--Wakasugi [26], the existence of global-in-time solutions with small initial data in the sense of $\langle{x}\rangle^{\lambda}u_0, \langle{x}\rangle^{\lambda}\nabla u_0, \langle{x}\rangle^{\lambda}u_1\in L^2(\Omega)$ with $\lambda\in (0,\frac{N}{2})$ is shown under the condition $p\geq 1+\frac{4}{N+2\lambda}$. The lower and upper bounds for the lifespan of blowup solutions with small initial data $(\epsilon u_0,\epsilon u_1)$ are also given.

In this paper, we study the initial-boundary value problem for a class of infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity $$ u_{tt}-\triangle_{X} u=u\log | u | , $$ where $X= (X_1,X_2,...,X_m)$ is an infinitely degenerate system of vector fields, and $$ {\triangle_X} = \sum\limits_{j = 1}^m {X_j^2} $$ is an infinitely degenerate elliptic operator. By using the logarithmic Sobolev inequality and a family of potential wells, we first prove the invariance of some sets. Then, by the Galerkin method, we obtain the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy.

The critical constant $\mu$ (see (1.1)) of time-decaying damping in the scale-invariant case is recently conjectured. It also has been expected that the lifespan estimate is the same as for the associated semilinear heat equations if the constant is in the “heat-like” domain. In this paper, we point out that this is not true if the total integral of the sum of initial position and speed vanishes. In such a case, we have a new type of the lifespan estimates which is closely related to the non-damped case in shifted space dimensions.

The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation $$ { \partial_t} u - { \partial_x}^5 u -{ \partial_x}^{-1} { \partial_y}^2u + u{ \partial_x} u = 0, ~(t,x,y)\in\mathbb R\times\mathbb T^2 . $$ We prove global well-posedness for constant $x$ mean value initial data in the space $\mathbf E = \{u\in L^2,~{ \partial_x}^2 u \in L^2, ~{ \partial_x}^{-1} { \partial_y} u \in L^2\}$ which is the natural energy space associated with this equation.

We analyze the limiting problem for the anisotropic $p$-Laplacian ($p\rightarrow\infty$) on convex sets, with the mean of the viscosity solution. We also prove some geometric properties of eigenvalues and eigenfunctions. In particular, we show the validity of a Szegö-Weinberger type inequality.