A compactness result is revised in order to prove the pointwise convergence of the gradients of a sequence of solutions to a general quasilinear inequality (anisotropic or not, degenerate or not) and for an arbitrary open set. Combining this result with the well-known Brézis-Lieb lemma, we derive simple proofs of Palais-Smale properties in many optimization problems especially on unbounded domains.

This paper provides a systematic analysis of (weak) compactness problems in connection with $L^{1}$-spectral theory of general neutron transport equations.

We study asymptotic properties of solutions of the nonoscillatory half-linear differential equation \[ (a(t)\Phi(x^{\prime}))^{\prime}+b(t)\Phi(x)=0 \] where the functions $a,b$ are continuous for $t\geq0,$ $a(t)>0$ and $\Phi(u)=|u|^{p-2}u$, $p>1$. In particular, the existence and uniqueness of the zero-convergent solutions and the limit characterization of principal solutions are proved when the function $b$ changes sign. An integral characterization of the principal solutions, the boundedness of all solutions, and applications to the Riccati equation are considered as well.

The present paper is concerned with the uniqueness of solutions to the Cauchy problem for the free Schrödinger equation. It is proved by the wellposedness of the associated adjoint problem that the solution to the Cauchy problem is unique in exponential weighted Sobolev spaces.

We study the Cauchy problem for the model nonlinear equation \begin{equation} \left\{ \begin{array}{c} u_{t}+\mathcal{L}u=\lambda \left| u\right| ^{\sigma }u,\text{ }x\in \mathbf{R },\text{ }t>0, \\ u\left( 0,x\right) =u_{0}\left( x\right) \text{, }x\in \mathbf{R}, \end{array} \right. \tag*{(0.1)} \end{equation} where $\sigma >0,$ $\lambda \in \mathbf{R.}$ We are interested in the critical and subcritical powers of the nonlinearity, especially in the case of large initial data $u_{0}$ from $\mathbf{L}^{1,a}\cap \mathbf{L}^{\infty }.$ We prove that the Cauchy problem (0.1) has a unique global solution $ u\in \mathbf{C}\left( [0,\infty );\mathbf{L}^{\infty }\cap \mathbf{L} ^{1,a}\right) $ and obtain the large time asymptotics.

Equations with non-local dispersal have been used extensively as models in material science, ecology and neurology. We consider the scalar model $$ \frac{\partial u}{\partial t}(x,t)=\rho \Big \{\int_\Omega \beta (x,y)u(y,t)dy- u(x,t) \Big \}+f(u(x,t)),$$ where the integral term represents a general form of spatial dispersal and $u(x,t)$ is the density at $x\in\Omega$, the spatial region, and time $t$ of the quantity undergoing dispersal. We discuss the asymptotic dynamics in the bistable case and contrast these with those for the corresponding reaction-diffusion model. First, we note that it is easy to show for large $\rho$ that the behavior is similar to that of the reaction-diffusion system; in the case of the analogue of zero Neumann conditions, the dynamics are governed by the ODE $\dot u=f(u)$. However, for small $\rho$, it is known that this is not the case, the set of equilibria being uncountably infinite and not compact in $L^p~(1\le p\le\infty)$. Our principal aim in this paper is to enquire whether every orbit converges to an equilibrium, regardless of the size of $\rho$. The lack of compactness is a major technical obstacle, but in a special case we develop a method to show that this is indeed true.