We study the large time behavior of the solution to the Cauchy problem for the one-dimensional, cubic nonlinear Klein-Grodon equation with complex-valued initial data. We show that the small amplitude solution decays like $t^{-1/2}$ as $t$ tends to infinity. Several remarks are also given on the large time asymptotics.

Given $(M,g),$ a compact Riemannian manifold of dimension $n \ge 8,$ we consider positive solutions $u_{\alpha} $ of ${\Delta}^2_gu - div_g(A_{\alpha} du) + a_{\alpha} u = u^{2^\sharp-1}$, where $ A_{\alpha}$ is a smooth, symmetric (2,0) tensor and $a_{\alpha}$ a smooth function. Assuming that $ A_{\alpha}$ and $a_{\alpha}$ converge in a suitable sense as ${\alpha} \rightarrow \infty$, we obtain conditions under which the weak limit of $u_{\alpha}$ is nontrivial.

Smooth Finsler metrics are a natural generalization of Riemannian ones and have been widely studied in the framework of differential geometry. The definition can be weakened by allowing the metric to be only Borel measurable. This generalization is necessary in view of applications, such as, for instance, optimization problems. In this paper we show that smooth Finsler metrics are dense in Borel ones, generalizing the results obtained in [15]. The case of degenerate Finsler distances is also discussed.

We consider a class of second-order degenerate elliptic operators. Continuing the study started in [3], we prove Schauder estimates for the distributional solutions of the nonhomogeneous elliptic equation $\lambda u-{\mathcal A}u=f$ and the Cauchy problem $D_tu={\mathcal A}u+g$, $u(0,\cdot)=f$.

In this paper we obtain existence, nonexistence, and multiplicity results for the Dirichlet boundary-value problem $-\Delta u=f_{\alpha}(u+c)$ in a bounded domain $\omega\subset\mathbb R^d,$ with a nonlocal condition $\int_{\omega}f_{\alpha}(u+c)=M.$ The solutions of this BVP are steady states for some evolution system describing self-gravitating Fermi-Dirac particles.

In this article we study the hyperbolic problem \begin{align*} & u_{tt} - \Delta u + F(x,t,u, \nabla u)= 0 \quad \mbox{in}\quad \Omega\times {\mathbb R}_+, \\ & u=0 \mbox{ on } \Gamma_0,\quad u +\int_0^t g(t-s) \frac {\partial u}{\partial \nu}(s) ds = 0 \mbox{ on } \Gamma_1 \times {\mathbb R}_+, \\ & u(x,0)=u^0 (x), \quad u'(x,0) = u^1 (x)\quad \mbox{in} \quad \Omega , \end{align*} where $\Omega$ is a bounded region in ${\mathbb R}^n$ whose boundary is partitioned into disjoint sets $\Gamma_0, \Gamma_1$. We prove that the dissipation given by the memory term is strong enough to assure stability of our system. The general decay estimates we obtain depend on the relaxation function. In particular, if the relaxation function decays exponentially (or polynomially), then the solution also decays exponentially (or polynomially) and with the same decay rate. Indeed, the main result of this paper is to give general relations between the decay of the solution and the decay of the relaxation function, under weaker hypotheses on the resolvent kernel function (defined in Section 2),and the potential functions $\phi (x)$ and $\varphi (t),$ which represent (in some sense) the linear and the nonlinear part of $F$ with respect to $\nabla u$, respectively. We assume only that $\varphi$ is bounded and small enough at $\infty$ which means that, in fact, $\varphi$ has no real influence on the stability of our system. In the case where the relaxation function decays exponentially or polynomially, we obtain the same decay for the solution, and then the results of [3,4] become just a particular case of ours. We also distinguish the case where the first data $u^0$ vanishes on $\Gamma_1$ and prove that, in this case, we have exponential or polynomial decay of solution, even if the relaxation function does not converge to 0 at $\infty$.