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Local and global results for wave maps. I. (English) Zbl 0914.35083
Summary: We consider the initial value problem for wave-maps corresponding to constant coefficient second order hyperbolic equations in $$n + 1$$ dimensions, $$n \geq 4$$. We prove that this problem is globally well-posed for initial data which is small in the homogeneous Besov space $$\dot B^{2,1}_{n/2} \times\dot B^{2,1}_{n/2-1}$$.Our second result deals with more regular solutions; it essentially says that if in addition the initial data is in $$H^s \times H^{s-1}$$, $$s> n/2$$, then the solutions stay bounded in the same space. In part II of this work we shall prove that the same result holds in dimensions $$n = 2,3$$.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 58J45 Hyperbolic equations on manifolds 35L15 Initial value problems for second-order hyperbolic equations 35B65 Smoothness and regularity of solutions to PDEs
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##### References:
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