Publication

Nonlinear Stability of Self-Similar Solutions for Semilinear Wave Equations

Roland Donninger
2010
Journal Articles

Abstract

We prove nonlinear stability of the fundamental self-similar solution of the wave equation with a focusing power nonlinearity tt-=p for [image omitted] in the radial case. The proof is based on a semigroup formulation of the wave equation in similarity coordinates.

Official source

https://infoscience.epfl.ch/entities/publication/508fd1ce-65fa-4a80-a58b-b2c92ff79d62

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Roland Donninger
2010
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/508fd1ce-65fa-4a80-a58b-b2c92ff79d62

About this result

Related concepts (12)

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

Self-organization

Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when sufficient energy is available, not needing control by any external agent. It is often triggered by seemingly random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system.

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