A note on a theorem of Jörgens
A note on a theorem of Jörgens
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关于 Jörgens 定理的注释
DOI：
10.1007/bf02571902
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发表时间：
1995
影响因子：
0.8
通讯作者：
P. D’Ancona
中科院分区：
文献类型：
--
作者：
P. D’Ancona
1 Introduction in 1961, in a well known paper J6rgens [J] proved the existence of global smooth solutions on R+ x R 3 to the Cauchy problem for the semilinear wave equation [] u=-IulP-lu(1) u (0, x)= uo (x), ut (O, x)= Ul (X), uO, Ul E C~(R 3)(2) provided p< 5.(Actually, he considered a more general semilinear term of the form-F'(IuI2) u, substantially equivalent to (1)). His result was extended to the case p= 5 by Struwe (radial solutions,[S]) and Grillakis (general case,[G]). Moreover, Eq.(1) has been the object of thorough investigations in any space dimension (see eg [Ka, SS] and the references therein). Our aim here is to extend J6rgens' result to a weakly hyperbolic Cauchy problem of the form utt-a (t) Au=-f (u)(3) u (O, x)= uo (x), u1 (O, x)= u1 (x),(4) where a (t) is a nonnegative (possibly vanishing) smooth function. As it is known from the theory of weakly hyperbolic equations, even a simple linear equation such as utt= a (t) Uxx may fail to be locally solvable in C, if the coefficient a (t) is only C and nonnegative [CS]. Various sufficient conditions ensuring the global solvability of second order weakly hyperbolic equations are known, see [CDS, O, K, Ni]. One of these is the requirement that the coefficients be real analytic functions of time only. Thus we will assume that a (t)> O, a (t) is real analytic(5)
