We consider complex-valued solutions $u_{\epsilon }$ of the Ginzburg–Landau equation on a smooth bounded simply connected domain $\Omega$ of ${ℝ}^N$, $N\ge 2$, where $\epsilon \>0$ is a small parameter. We assume that the Ginzburg–Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where $M_0$ is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of $u_{\epsilon }$, as $\epsilon \to 0$, is to establish uniform $L^p$ bounds for the gradient, for some $p\>1$. We review some recent techniques developed in...

We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of ${ℝ}^N$, N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where M0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some...

En esta nota estimamos la tasa máxima de crecimiento en la frontera de las soluciones de viscosidad de -Δ∞u + λ|u|m-1u = f en Ω (λ > 0, m > 3).De hecho, mostramos que sólo hay una única tasa de explosión en la frontera para esas soluciones explosivas. También obtenemos una versión del Teorema de Liouville para el caso Ω = RN.

In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the nonlinear heat equation $u_t-\Delta u+{\left|u\right|}^{\alpha }u=0$ on ${ℝ}^N$ (where $\alpha \>0$) and the asymptotic behavior as $\left|x\right|\to \infty$ of its initial value $u_0$. In particular, we show that if the sequence of dilations ${\lambda }_n^{2/\alpha }{u}_0({\lambda }_n·)$ converges weakly to $z(·)$ as ${\lambda }_n\to \infty$, then the rescaled solution $t^{1/\alpha }u(t,·\sqrt{t})$ converges uniformly on ${ℝ}^N$ to $𝒰\left(1\right)z$ along the subsequence $t_n={\lambda }_n^2$, where $𝒰\left(t\right)$ is an appropriate flow. Moreover, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this...