On squeezing and flow of energy for nonlinear wave equations
The paper is devoted to a general analysis of the energy flow along the spectrum in the nonlinear scalar hyperbolic equation: \[ u_{tt}- \delta^2 \Delta u+ mu+ f(u)=0, \tag{1} \] where \(u\) is a real scalar function of time \(t\) and \(n\) spatial coordinates, \(\Delta\) stands for the \(n\)-dimensional Laplacian, \(m\) is a constant, and the function \(f(u)\) accounts for the nonlinear part of the equation. The parameter \(\delta\) is treated as a small one, which makes it natural to compare solutions of equation (1) to those of the corresponding ODE obtained in the limit \(\delta=0\); it is assumed that the latter equation has a family of time- periodic solutions. Estimating a deviation of the solution to equation (1) from the ODE solutions, and using a technique similar to that employed for the proof of the known Gromov's theorem which establishes squeezing properties of the phase flow in the nonlinear PDE's, the author proves that (a) an absolute value of the Fourier transform of the solution at a fixed wave vector vanishes as \(t\to 0\), and (b) the norm of the function \(u_t\) diverges at \(t\to \infty\). Qualitatively, these two fundamental properties of the evolution of the wave field may be constructed as direct cascade of the energy, i.e., its flow from small to large wave numbers. An important fact is that the proof is not possible for the solutions with a low smoothness (generalized solutions), but only for the classical solutions with a sufficiently high smoothness. As a particular application, the well-known model with \(f(u)= \lambda u^3\) is considered in more detail.
scientific article; zbMATH DE number 800443
scientific article