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The Cauchy problem for the coupled Klein-Gordon-Schrödinger system

Changxing Miao, Youbin Zhu (2006)

Annales Polonici Mathematici

We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural...

The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations

Ferruccio Colombini, Guy Métivier (2008)

Annales scientifiques de l'École Normale Supérieure

In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form L u : = j , k = 0 n y j ( a j , k y k u ) + j = 0 n { b j y j u + y j ( c j u ) } + d u = f , when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant...

The critical nonlinear wave equation in two space dimensions

Michael Struwe (2013)

Journal of the European Mathematical Society

Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.

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