Hideaki Sunagawa

On global small amplitude solutions to systems of cubic nonlinear Klein–Gordon equations with different mass terms in one space dimension

Abstract We consider the Cauchy problem for systems of cubic nonlinear Klein–Gordon equations with different mass terms in one space dimension. We prove some result concerning the global existence of small amplitude solutions and their asymptotic behavior. As a consequence, we see that the condition for small data global existence is actually influenced by the difference of masses in some cases.

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.

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

We consider the cubic nonlinear Schrodinger equations of derivative type with small initial data. We present a structural condition on the nonlinear terms under which the corresponding Cauchy problem has a dissipative nature.

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

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.

On Schrödinger systems with cubic dissipative nonlinearities of derivative type

Consider the initial value problem for systems of cubic derivative nonlinear Schrodinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under which the small data solution gains an additional logarithmic decay as compared with the corresponding free evolution.

The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension

Consider the initial value problem for cubic derivative nonlinear Schrodinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors [H. Sunagawa: Osaka J. Math. 43 (2006), 771--789], in which the gauge-invariant nonlinearity was treated.