We generalize the phase-plane method approach introduced in [9] to the case of higher dimensions. To this aim, the phase-space is assumed to be decomposable as a product of planes, where the corresponding components of the solutions can be controlled by means of suitable plane curves. We then apply our general result to the periodic problem associated to a system of coupled oscillators, with retraction forces having a linear growth, or with one-sided superlinear nonlinearities.

In this paper, we consider the well-posedness for the Cauchy problem of the Kawahara equation with low regularity data in the periodic case. We obtain the local well-posedness for $s \geq -3/2$ by a variant of the Fourier restriction norm method introduced by Bourgain. Moreover, these local solutions can be extended globally in time for $s \geq -1$ by the I-method. On the other hand, we prove ill-posedness for $s < -3/2$ in some sense. This is a sharp contrast to the results in the case of $\mathbb{R}$, where the critical exponent is equal to $-2$.

We discuss the existence of a regular map $\varphi$ satisfying $$ \left\{ \begin{array}{cl} \det\nabla \varphi=f & \text{in $\Omega$}\\ \varphi=0 & \text{on $\partial \Omega,$} \end{array} \right. $$ where $\Omega$ is a bounded smooth domain and $f$ is a regular function satisfying $ \int_{\Omega}f=0$.

In this paper we consider the Cauchy problem for a higher-order modified Camassa--Holm equation. By using some dyadic bilinear estimates and the fixed-point theorem, we establish the local well-posedness of the higher-order modified Camassa--Holm equation for the small initial data in $H^{-n+\frac{5}{4}}({{\mathbf R}}),$ $n\geq 2,$ $ n\in {{\mathbf N}}$. We also prove that the Cauchy problem for the higher-order modified Camassa--Holm equation is ill-posed for the initial data in homogeneous Sobolev spaces $\dot{H}^{s}({{\mathbf R}})$ with $s < -n+\frac{5}{4},$ $n\in {{\mathbf N}},$ $ n\geq 2$. Our result partially answers the open problem which is proposed below in Theorem 1.2 by Erika A. Olson in the Journal of Differential Equations, 246 (2009), 4154--4172.

We study the scattering and blowup problem for a class of nonlinear Schrödinger equations with general nonlinearities in the spirit of Kenig and Merle [17]. Our conditions on the nonlinearities allow us to treat a wider class of those than ever treated by several authors, so that we can prove the existence of a ground state (a standing-wave solution of minimal action) for any frequency $\omega > 0$. Once we get a ground state, a so-called potential-well scenario works well: for the nonlinear dynamics determined by the nonlinear Schrödinger equations, we define two invariant regions $A_{\omega, +}$ and $A_{\omega,-}$ for each $\omega > 0$ in $H^1(\mathbb{R}^d)$ such that any solution starting from $A_{\omega,+}$ behaves asymptotically free as $t\to\pm\infty$, one from $A_{\omega, -}$ blows up or grows up, and the ground state belongs to $\overline{A_{\omega, +}}\bigcap \overline{A_{\omega,-}}$. Our weaker assumptions as to the nonlinearities demand that we argue in a subtle way in proving the crucial properties of the solutions in the invariant regions.

We consider a class of coupled systems with past history acting only in one equation. We show in the abstract setting that the dissipation given by the history term is not strong enough to produce exponential stability. We show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data. Some examples are given.

In this paper it is proved that, when a regular functional is bounded from below, the mountain pass geometry and the existence of at least two distinct local minima are equivalent conditions. As a consequence, the classical mountain pass theorem, under the additional assumption of boundedness from below of the functional, ensures actually three distinct critical points. Moreover, as application, the existence of three solutions to Hamiltonian systems is established.

We consider traveling waves related to shocks for a non-local scalar conservation law \[ u_{t}+(f(u))_{x}=K\ast u-u, \] where $f$ is an arbitrary convex function, $K\ast u$ stands for the convolution in the spatial variable $x$, and $K$ is an arbitrary non-negative kernel with unit mass centered at the origin. Given a pair of values at $\pm\infty$ with $u_{-} > u_{+}$, we establish here the existence and uniqueness of a traveling-wave solution $u=\phi\left( x-ct\right) $ such that the wave speed \[ c=\frac{f\left( u_{+}\right) -f\left( u_{-}\right) }{u_{+}-u_{-}} \] and $\phi$, satisfying $\phi\left( \pm\infty\right) =u_{\pm}$, is a smooth decreasing function that can have at most one jump discontinuity at the origin. Our approach here is to consider a family of ``truncated" problems on $\left[ -n,n\right] $ which can be solved using Schauder's fixed-point theorem and then sending $n\rightarrow\infty$.

We study the existence of positive solutions to the singular problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda f(u)-\frac{1}{u^{\alpha}} & \mbox{ in } \Omega, \\ u = 0 & \mbox{ on } \partial \Omega, \end{cases} \end{equation*} where $\lambda$ is a positive parameter, $\Delta_p u =\operatorname{div}(|\nabla{u}|^{p-2}\nabla{u})$, $p > 1$, $\Omega $ is a bounded domain in $\mathbb{R}^{n}, n \geq 1$ with smooth boundary $\partial\Omega$, $0 < \alpha < 1$, and $f:[0,\infty) \rightarrow \mathbb{R}$ is a continuous function which is asymptotically $p$-linear at $\infty$. We prove the existence of positive solutions for a certain range of $\lambda$ using the method of sub-supersolutions. We also extend our study to classes of systems which have forcing terms satisfying a combined asymptotically p-linear condition at $\infty$ and to corresponding problems on exterior domains.

We consider the Dirichlet problem $ - \Delta u = \mu u + f$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb R^N$. Let $\hat \lambda$ be an eigenvalue with $\hat \phi$ an associated eigenfunction. We study the following question $(*)$: Assuming $ \int_{\Omega} f \hat \phi \neq 0$, has $u$ the same number of nodal domains as $\hat \phi$ if $\mu $ is sufficiently close to $\hat \lambda$? The answer to $(*)$ is known to be affirmative in various cases; see [1], [5], and [6]. Here we study a specific situation where, on the contrary, the answer to $(*)$ is not always affirmative: $\Omega=$ the unit disk in $\mathbb R^2$ and $\hat \lambda = \lambda_4 = \lambda_5$.