We consider a system of quasilinear parabolic type equations involving operator curl associated with the Maxwell equations in a multi-connected domain. The paper is a continuation of the author's previous paper. We deal with a variational inequality with curl constraint. It is an extension of the results of Miranda et al. for $p$-curl system.

Via a variational approach involving Concentration-Compactness principle, we show the existence of $x$-periodic travelling wave solutions for a general 2D-Boussinesq system that arises in the study of the evolution of long water waves with small amplitude in the presence of surface tension. We also establish that $x$-periodic travelling waves have almost the same shape of solitons as the period tends to infinity, by showing that a special sequence of $x$-periodic travelling wave solutions parameterized by the period converges to a solitary wave in a appropriate sense.

The boundary value problems for coupled systems of parabolic and ordinary differential equations, where all equations contain time depended delay and degenerate at initial moment, are considered. Existence and uniqueness of classical solutions of these problems are proved. A priori estimates are obtained.

This article is devoted to the study of the nonlinear eigenvalue problem $$-\Delta_{p} u \quad=\quad \lambda |u|^{p-2}u \;\mbox{in}\; \Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\quad+\quad\beta |u|^{p-2}u=\lambda |u|^{p-2}u \;\mbox{on}\quad\partial\Omega,$$ where $ν$ denotes the unit exterior normal, $1 \lt p \lt ∞ \,\mathrm {and} ∆_{p}u = div(|∇u|^{p−2}∇u)$ denotes the p-laplacian. $Ω ⊂ \mathbb{R}^{N}$ is a bounded domain with smooth boundary where $N ≥ 2$ and $β \in L^{∞}(∂Ω) \,\mathrm{with}\, β^{−} := \mathrm{inf}_{x∈∂Ω}β(x) > 0$. Using Ljusternik-Schnirelman theory, we prove the existence of a nondecreasing sequence of positive eigenvalues and the first eigenvalue is simple and isolated. Moreover, we will prove that the second eigenvalue coincides with the second variational eigenvalue obtained via the Ljusternik-Schnirelman theory.

We prove a weighted Poincaré inequality in a subspace of $BV_\text{loc}$ whose elements have variation measure in a Wiener amalgam space of Radon measures.