On transitions to stationary states in 1D nonlinear wave equations
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Submission date: 23. Jan. 1999
published in: Archive for rational mechanics and analysis, 149 (1999) 3, p. 213-228
DOI number (of the published article): 10.1007/s002050050173
with the following different title: On transitions to stationary states in one-dimensional nonlinear wave equations
MSC-Numbers: 35L70, 37K40, 37K45
Keywords and phrases: attractor, stationary state, fréchet topology, energy scattering to infinity, goursat problem
We develop the theory of attractors for finite energy solutions to conservative nonlinear wave equations in a whole space. For ``nondegenerate'' equations the attractor coincides with the set of all finite energy stationary states. The convergence to the attractor holds as in the Fréchet topology defined by local energy seminorms. The proof of the attraction is based on the investigation of energy scattering to infinity. The results give a mathematical model of N.Bohr's transitions to quantum stationary states (``quantum jumps'').