Threshold Behavior for Nonlinear Wave Equations

In this brief contribution, which is based on my talk at the conference, I discuss the dynamics of solutions of nonlinear wave equations near the threshold of singularity formation. The heuristic picture of threshold behavior is first presented in a general setting and then illustrated with three examples. Consider an initial value problem du dt = A(u), u(0) = φ ∈ D, (1) where A is a nonlinear operator. We can say that we understand qualitatively the dynamics of solutions of (1) if we can map the space of initial data D into a space of all possible final states (attractors). To construct such a mapping one needs first to find all generic (that is stable) attractors. If there are two or more stable attractors and the space D is connected, then there arises a problem of determining the boundaries between the basins of attraction of generic attractors. Here I discuss this problem in the case when (1) is a nonlinear wave equation whose solutions admit only two generic asymptotic behaviors, namely dispersion or blowup. By dispersion I mean that the solution exists globally in time and the energy density asymptotically decays to zero in any compact region. By blowup I mean that either the solution or some of its derivatives become unbounded in some norm (usually in finite time). A canonical example of equations I consider here is the nonlinear Klein–Gordon equation