arXiv subjects

G. Raugel

Publications and source records attributed to G. Raugel.

4 recordsLinked to original sources

arXiv2015

Long time dynamics for damped Klein-Gordon equations

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, $1\textless{}p\textless{}(d+2)/(d-2)$ as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).

math.AP↗

arXiv1998

Scaling Variables and Stability of Hyperbolic Fronts

We consider the damped hyperbolic equation (1) εu_{tt} + u_t = u_{xx} + F(u), x \in R, t \ge 0, where εis a positive, not necessarily small parameter. We assume that F(0) = F(1) = 0 and that F is concave on the interval [0,1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This family is indexed by a parameter c \ge c_* related to the speed of the front. In the critical case c=c_*, we prove that the travelling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space. In addition, we show that the perturbations decay to zero like t^{-3/2} as t \to +\infty and approach a universal self-similar profile, which is independent of ε, F and of the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting ε= 0 in Eq.(1). The proof of our results relies on careful energy estimates for the equation (1) rewritten in self-similar variables x/\sqrt{t}, \log t.

patt-sol↗

arXiv1998

Stability of Propagating Fronts in Damped Hyperbolic Equations

We consider the damped hyperbolic equation in one space dimension $εu_{tt} + u_t = u_{xx} + F(u)$, where $ε$ is a positive, not necessarily small parameter. We assume that $F(0)=F(1)=0$ and that $F$ is concave on the interval $[0,1]$. Under these assumptions, our equation has a continuous family of monotone propagating fronts (or travelling waves) indexed by the speed parameter $c \ge c_*$. Using energy estimates, we first show that the travelling waves are locally stable with respect to perturbations in a weighted Sobolev space. Then, under additional assumptions on the non-linearity, we obtain global stability results using a suitable version of the hyperbolic Maximum Principle. Finally, in the critical case $c = c_*$, we use self-similar variables to compute the exact asymptotic behavior of the perturbations as $t \to +\infty$. In particular, setting $ε= 0$, we recover several stability results for the travelling waves of the corresponding parabolic equation.

patt-sol↗

arXiv1996

Stability of Travelling Waves for a Damped Hyperbolic Equation

We consider a nonlinear damped hyperbolic equation in $\real^n$, $1 \le n \le 4$, depending on a positive parameter $ε$. If we set $ε=0$, this equation reduces to the well-known Kolmogorov-Petrovski-Piskunov equation. We remark that, after a change of variables, this hyperbolic equation has the same family of one-dimensional travelling waves as the KPP equation. Using various energy functionals, we show that, if $ε>0$, these fronts are locally stable under perturbations in appropriate weighted Sobolev spaces. Moreover, the decay rate in time of the perturbed solutions towards the front of minimal speed $c=2$ is shown to be polynomial. In the one-dimensional case, if $ε< 1/4$, we can apply a Maximum Principle for hyperbolic equations and prove a global stability result. We also prove that the decay rate of the perturbated solutions towards the fronts is polynomial, for all $c > 2$.

patt-sol↗