Caustics for dissipative semilinear oscillations

This paper describes focusing effects near caustics for high frequency solu tions of first order semilinear dissipative hyperbolic systems Lu + F(u) = 0 or analogous semilinear wave equations. The dissipitivity assumption is m ade because for general equations nonlinear interactions can amplify the gr owth of amplitudes of focusing solutions and lead to noncontinuation of sol utions, as shown in [JMR 1] [JMR 2]. For the dissipative equations we consi der, global solutions exist and we study their behavior beyond caustics. As in the linear theory, we use Lagangian integrals to describe the oscillati ons near the caustics. The nonlinear solutions are well approximated by osc illatory integrals on domains which may include caustics. The approximation is valid in L-p, for an exponent p related to the growth at infinity of th e nonlinear dissipative terms. Away from the caustics, phase-amplitude expa nsions are recovered by the stationary phase theorem. A key difficulty (and novelty) is to analyse the action of nonlinear functi ons on oscillatory integrals. This relies on new sharp uniform bounds far t he L-p norm of oscillatory integrals. This analysis reveals a critical expo nent p(c), associated to each point of the caustic. For generic caustic poi nts, this exponent is 3. For radial focusing in R-d, d greater than or equa l to 2, it is 1 + 2/(d - 1). When the nonlinearity F(u) grows at infinity f aster than \u\(pc) the dissipative mechanism is strong and oscillations are absorbed at caustics so there are no oscillations past the focusing point. This extends the results in [JMR 2] from focusing spherical wavefronts to general caustics. When the nonlinear term grows at infinity at a strictly s lower rate than \u\(pc), oscillations cross caustics and amplitudes can be computed after focusing. This generalizes the results of [JMR 3] to superli near dissipative equations.