Soichiro Katayama

Global existence for coupled systems of nonlinear wave and Klein–Gordon equations in three space dimensions

We consider the Cauchy problem for coupled systems of wave and Klein–Gordon equations with quadratic nonlinearity in three space dimensions. We show global existence of small amplitude solutions under certain condition including the null condition on self-interactions between wave equations. Our condition is much weaker than the strong null condition introduced by Georgiev for this kind of coupled system. Consequently our result is applicable to certain physical systems, such as the Dirac–Klein–Gordon equations, the Dirac–Proca equations, and the Klein–Gordon–Zakharov equations.

Null Structure in a System of Quadratic Derivative Nonlinear Schrödinger Equations

We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.

ASYMPTOTIC POINTWISE BEHAVIOR FOR SYSTEMS OF SEMILINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker sufficient condition for the small data global existence, and we investigate the asymptotic pointwise behavior of global solutions for systems satisfying this condition. As an application, the asymptotic behavior of global solutions under the Alinhac condition is also derived.

LIFESPAN OF SOLUTIONS FOR TWO SPACE DIMENSIONAL WAVE EQUATIONS WITH CUBIC NONLINEARITY

We study the lifespan of classical solutions to the Cauchy problem for nonlinear wave equations of the type with initial data u = εf, u t = ε g at t = 0, where D x = (∂ x 1 , ∂ x 2 ), D = (∂ t , D x ) and ε is a small parameter. Estimates of the lifespan from below are derived for some cases where F is a function of cubic order in a neighborhood of the origin, and ∂3 u F(0, 0, 0) = 0, but not necessarily ∂4 u F(0, 0, 0) = 0.

Decay Estimates of a Tangential Derivative to the Light Cone for the Wave Equation and Their Application

We consider wave equations in three space dimensions, and obtain new weighted

Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions

L^\infty

Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions

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Global Existence for Quadratically Perturbed Massless Dirac Equations Under the Null Condition

L^\infty

A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions†

estimates for a tangential derivative to the light cone. As an application, we give a new proof of the global existence theorem, which was originally proved by Klainerman and Christodoulou, for systems of nonlinear wave equations under the null condition. Our new proof has the advantage of using neither the scaling nor the pseudo-rotation operators.

Global and almost-global existence for systems of nonlinear wave equations with different propagation speeds

We consider the Cauchy problem for systems of semilinear wave equations in two space dimensions. We present a structural condition on the nonlinearity under which the energy decreases to zero as time tends to infinity if the Cauchy data are sufficiently small, smooth and compactly-supported.