Decay of solutions to a water wave model with a nonlocal viscous dispersive term
Min Chen S. Dumont Louis Dupaigne Olivier Goubet
Discrete & Continuous Dynamical Systems - A 2010, 27(4): 1473-1492 doi: 10.3934/dcds.2010.27.1473
In this article, we investigate a water wave model with a nonlocal viscous term

$ u_t+u_x+\beta $uxxx$+\frac{\sqrt{\nu}}{\sqrt{\pi}}\int_0^t \frac{u_t(s)}{\sqrt{t-s}}ds+$uux$=\nu $uxx$. $

The wellposedness of the equation and the decay rate of solutions are investigated theoretically and numerically.

keywords: long-time asymptotics. nonlocal viscous asymptotic models Nonlinear water waves
A new critical curve for the Lane-Emden system
Wenjing Chen Louis Dupaigne Marius Ghergu
Discrete & Continuous Dynamical Systems - A 2014, 34(6): 2469-2479 doi: 10.3934/dcds.2014.34.2469
We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\mathbb{R}^N$, $-\Delta v=u^q$ in $\mathbb{R}^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
keywords: singular solutions. stable solutions critical curve radially symmetric solutions Lane-Emden system
The extremal solution of a boundary reaction problem
Juan Dávila Louis Dupaigne Marcelo Montenegro
Communications on Pure & Applied Analysis 2008, 7(4): 795-817 doi: 10.3934/cpaa.2008.7.795
We consider

$\Delta u = 0$ in $ \Omega$, $\qquad \frac{\partial u}{\partial \nu} =\lambda f(u)$ on $\Gamma_1, \qquad u = 0$ on $\Gamma_2$

where $\lambda>0$, $f(u) = e^u$ or $f(u) = (1+u)^p$, $\Gamma_1$, $\Gamma_2$ is a partition of $\partial \Omega$ and $\Omega\subset \mathbb R^N$. We determine sharp conditions on the dimension $N$ and $p>1$ such that the extremal solution is bounded, where the extremal solution refers to the one associated to the largest $\lambda$ for which a solution exists.

keywords: Kato's inequality. Nonlinear Neumann problem stability extremal solution

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