No Arabic abstract
We consider the so-called prion equation with the general incidence term introduced in [Greer et al., 2007], and we investigate the stability of the steady states. The method is based on the reduction technique introduced in [Gabriel, 2012]. The argument combines a recent spectral gap result for the growth-fragmentation equation in weighted $L^1$ spaces and the analysis of a nonlinear system of three ordinary differential equations.
We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as beta-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a double null form that annihilates interactions between spatially coherent waves and a lemma for Fourier integral operators which allows us to control a strong weighted norm.
In this paper we consider the problem: $partial_{t} u- Delta u=f(u),; u(0)=u_0in exp L^p(R^N),$ where $p>1$ and $f : RtoR$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $exp L^p_0(R^N)$ for $f(u)sim mbox{e}^{|u|^q},;0<qleq p,; |u|to infty.$ However, if for some $lambda>0,$ $displaystyleliminf_{sto infty}left(f(s),{rm{e}}^{-lambda s^p}right)>0,$ then non-existence occurs in $exp L^p(R^N).$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|sim |u|^{m}$ as $uto 0,$ ${N(m-1)over 2}geq p$, we show that the solution is global. In particular, $p-1>0$ sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m$.
We study the Lipschitz stability in time for $alpha$-dissipative solutions to the Hunter-Saxton equation, where $alpha in [0,1]$ is a constant. We define metrics in both Lagrangian and Eulerian coordinates, and establish Lipschitz stability for those metrics.
We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.
The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa-Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line. By constructing a Liapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations.