HYPERBOLIC SMOOTHING EFFECT FOR SEMILINEAR WAVE EQUATIONS AT A FOCAL POINT

Abstract

For semi-linear dissipative wave equation □u + |ut|p - 1ut = 0, we consider finite energy solutions with singularities propagating along a focusing light cone. At the tip of cone, the singularities are focused and partially smoothed out under strong nonlinear dissipation, i.e. the solution gets up to 1/2 more L2 derivative after the focus. The smoothing phenomenon is in fact the result of simultaneous action of focusing and nonlinear dissipation.