A `super-critical' nonlinear wave equation in 2 space dimensions
This work is concerned with the initial value problem of a wave equation with an exponential nonlinearity in two space dimensions. Initial values for the displacement and the velocity fields are prescribed smooth functions. The form of the nonlinearity allows the author to derive a conservation law leading to a global energy identity evaluated over the entire plane and implying that the global energy is time-independent, so that it is determined by the initial values. The global energy is assumed to be larger than a particular number. Therefore, the so-called supercritical case is investigated. The principal aim of the work is to show that this nonlinear wave equation possesses a smooth radially symmetric unique solution for all time for all radially symmetric smooth Cauchy data. To this end, the author employs extensively techniques of the functional analysis and analyses carefully the blow-up conditions at the origin characterised by the concentration of energy so that they can be eliminated to reach a smooth solution.
scientific article; zbMATH DE number 5960503
scientific article