We consider the $k$-dispersion generalized Benjamin-Ono equation in the supercritical case. We establish sharp conditions on the data to show global well-posedness in the energy space for this family of nonlinear dispersive equations. We also prove similar results for the generalized Benjamin-Ono equation.

We study a class of one parameter (denoted by $\kappa$) family of quasi-linear Schrӧdinger equations arising in the theory of superfluid film in plasma physics. Using variational techniques, we prove the orbital instability of solitary waves for small values of the parameter $\kappa$, which gives an answer to a question raised in [9].

In this note, we consider the motion of a solid body in an incompressible perfect fluid filling the plane. We prove the global existence of solutions in the case where the initial vorticity belongs to $L^p$ with $p\gt 1$ and is compactly supported. We do not assume that the energy is finite.

This paper studies the evolution $p$-Laplacian equation with inner absorption $u_t=(|u_x|^{p-2}u_x)_x-\lambda u^{\kappa}$ in $\mathbb{R}^+\times \mathbb{R}^+$ subject to nonlinear boundary flux $-|u_x|^{p-2}u_x(0,t)=u^q(0,t)$. First, we determine the critical boundary flux exponent $q^*=\max\{\frac{2(p-1)}p$, $\frac{(\kappa+1)(p-1)}{p}\}$ to identify global and nonglobal solutions, relying on or independent of initial data. It is interesting to find that, for the critical case $q=q^*$ (the balanced case between inner absorption and boundary flux) with the absorption coefficient $\lambda > 0$ small, there is the so-called critical Fujita absorption exponent $\kappa_c=2p-1 > \kappa_0=1$ related to a Fujita type conclusion that (i) when $\kappa\le \kappa_0$, all solutions are global; (ii) if $\kappa_0 < \kappa < \kappa_c$, the solutions blow up in finite time under any nontrivial nonnegative initial data; (iii) if $\kappa > \kappa_c$, there are both global and nonglobal solutions, respectively, determined by the sizes of initial data. Obviously, when $\lambda > 0$ large for the balanced case $q=q^*$, the absorption would overcome the boundary flux to yield global solutions. Furthermore, we will show how and in what ways the inner absorption affects the evolution of the support of solutions, where a complete classification for all the nonlinear parameters is given to distinguish localization and non-localization behavior of the global solutions.

The normalized or game-theoretic $p$-Laplacian operator given by $$ -\Delta_p^Nu:=-\frac{1}{p}|\nabla u|^{2-p}\Delta_p(u) $$ for $p\in(1,\infty)$ with $\Delta_pu=\rm{div}(|\nabla u|^{p-2}\nabla u)$ has no apparent variational structure. Showing the existence of a first (positive) eigenvalue of this fully nonlinear operator requires heavy machinery as in [6]. If it is restricted to the class of radial functions, however, the normalized $p$-Laplacian transforms into a linear Sturm-Liouville operator. We investigate radial eigenfunctions to this operator under homogeneous Dirichlet boundary conditions and come up with an explicit complete orthonormal system of Bessel functions in a suitably weighted $L^2$-space. This allows us to give a Fourier-series representation for radial solutions to the corresponding evolution equation $u_t-\Delta_p^Nu=0$.

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some implicit differential equations exhibiting I--singularities (in the sense given in [16, p. 166]). In particular, we show persistence of such I-singularities and orbits connecting them in finite time provided a Melnikov like condition holds. We start from a concrete example where, we prove that this Melnikov condition actually holds. Then, we extend our results to more general implicit differential equations.

This paper analyzes the dynamics of a spatially heterogeneous parabolic system, which is a refinement of a classical prototype model in Nuclear Engineering proposed by W. E. Kastenberg and P. L. Chambré [13], introduced to study the interactions between the density of fast neutrons and the temperature in the reactor. The dynamics is completely characterized within the ranges of values of the parameters where the model does not admit a positive steady state, as well as in the region where it admits a positive steady state and the system can be transformed, through an appropriate change of variable, into an irreducible cooperative system, as discussed by J. López-Gómez and M. Molina-Meyer [19], where it is shown that the coexistence state is a global attractor.

We consider a class of functional differential equations subject to perturbations, which vary in time, and we study the exponential stability of solutions of these equations using the theory of generalized ordinary differential equations and Lyapunov functionals. We introduce the concept of variational exponential stability for generalized ordinary differential equations and we develop the theory in this direction by establishing conditions for the trivial solutions of generalized ordinary differential equations to be exponentially stable. Then, we apply the results to get corresponding ones for impulsive functional differential equations. We also present an example of a delay differential equation with Perron integrable right-hand side where we apply our result.

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and are defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. The main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ being the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].

We study the Cauchy problem for the non-linear Schrӧdinger equation with singular potentials. For the point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-$L^{p}$ spaces. Specific interest is given to the point-like $\delta$ and $\delta'$ impurity and to two $\delta$-interactions in one dimension. We also consider the periodic case, which is analyzed in a functional space based on Fourier transform and local-in-time well-posedness is proved.