We give a refined blow-up criterion for the solution for the compressible Navier-Stokes system with a Yukawa-potential in the critical Besov space [13]. The result may be considered as a compressible counterpart of the results for the incompressible Navier-Stokes system.

In this paper, we study the multiplicity results of solutions for a class of quasi-linear problems involving singular weights and Hardy terms on unbounded domains. With the help of the variational method and the theory of genus, we prove that the problem admits infinitely many solutions under different conditions.

This article focuses on almost global existence for quasilinear wave equations with small initial data in 4-dimensional exterior domains. The nonlinearity is allowed to depend on the solution at the quadratic level as well as its first and second derivatives. In the boundaryless setting, Hörmander proved that the lifespan $T_\epsilon \gtrsim \exp(c/\epsilon)$, where $\epsilon\gt 0$ denotes the size of the Cauchy data. Later Du, the second author, Sogge, and Zhou showed that this inequality also holds for star-shaped obstacles. Following up on the authors' work in the 3-dimensional case, we only require that the obstacle allow for a sufficiently rapid decay of local energy for the linear homogeneous wave equation. The key innovation of this paper is the combination of the boundary term estimates of the second author and Sogge with a variant of an estimate of Klainerman and Sideris, which is obtained via a Sobolev inequality of Du and Zhou.

The Kuramoto-Sakaguchi (or simply Kuramoto) equation is considered when the "frequency distribution", the frequency being an independent variable in the model equation, has unbounded support. This equation is a nonlinear, Fokker-Planck-type, parabolic integro-differential equation, and arises from the statistical description of the dynamical behavior of populations of infinitely many nonlinearly coupled random oscillators. The space-integral term in the equation accounts for mean-field interaction occurring among these oscillators. Existence, uniqueness, and regularity of solutions are established here, taking suitable limits in the formulation of the previously studied problem, where the aforementioned support was assumed to be bounded.

We study the inviscid limit problem for the incompressible Navier-Stokes equation on a half-plane with a Navier boundary condition depending on the viscosity. On one hand, we prove the $L^{2}$ convergence of Leray solutions to the solution of the Euler equation. On the other hand, we show the nonlinear instability of some WKB expansions in the stronger $L^{\infty}$ and $\dot{H}^{s}$ ($s>1$) norms. These results are not contradictory, and in the periodic setting, we provide an example for which both phenomena occur simultaneously.

This paper is devoted to the long-time behavior of solutions for a class of plate equations with nonlocal weak damping $$ u_{tt} + \Delta^2 u + g(u) + M \Big (\int_{\Omega}|\nabla u|^2 dx \Big )u_t =f\quad \mbox{in} \quad \Omega\times\mathbb{R}^{+}, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$. Under suitable conditions on the nonlinear forcing term $g(u)$ and Kirchhoff damping coefficient $M (\int|\nabla u|^2 ),$ the existence of a global attractor with finite Hausdorff and fractal dimensions is proved.

In this paper, we give existence and regularity results for nonlinear parabolic problems with degenerate coercivity and singular lower order terms, whose simplest example is \begin{eqnarray*} \begin{cases} u_t-\Delta_p u= {\frac{f(x,t)}{u^\gamma}} & \mbox{in}\;\Omega\times (0,T)\\ u(x,t)=0 & \mbox{on}\;\partial\Omega\times(0,T)\\ u(x,0)=u_0 (x) & \mbox{in}\;\Omega\; \end{cases} \end{eqnarray*} with $\gamma>0$, $p\geq 2$, $\Omega$ a bounded open set of $\mathbb{R}^{\mathrm{N}}$ ($N\geq 2$), $0 < T < +\infty$, $f\geq 0$, $f\in L^m(Q_T)$, $m\geq 1$ and $u_0\in L^\infty(\Omega)$ such that $$ \forall \, \omega\subset\subset\Omega\; \exists\;d_{\omega} > 0\,:\,u_{0}\geq d_{\omega}\;\mbox{in}\,\;\omega\,. $$ The aim of the paper is to extend the existence and regularity results recently obtained for the associated singular stationary problem. One of the main difficulties that arises in the parabolic case is the proof of the strict positivity of the solution in the interior of the parabolic cylinder, in order to give sense to the weak formulation of the problem. The proof of this property uses Harnack's inequality.

The purpose of this paper is to study the non-uniformly elliptic equations with non-uniformly $p$-superlinear nonlinearities satisfying sets of other nonuniform conditions. In particular, we weaken the well-known Ambrosetti-Rabinowitz condition. By introducing the notion of pseudo-uniform convexity, we complete the studies in [5], [10] for non-uniformly p-Laplacian problems with $1 < p < 2$. We are able to prove the existence of nontrivial, non-negative, non-positive solutions.