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A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations. (English) Zbl 0797.35123

The paper is concerned with the question: what is the smallest \(s\) such that one has local existence and uniqueness in \(H_ s\) of the problem: \(\square u = G(u,u')\) for \(0 \leq t<T\); \(u(0,x) = u_ 0(x) \in H_ s (\mathbb{R}^ 3)\); \(\partial_ t u(0,x) = u_ 1(x) \in H_{s-1} (\mathbb{R}^ 3)\), where \(\square = \partial^ 2_ t - \Delta\) and \(G(u,u')\) is a smooth function of \(u\) and \(u' = (u_ t, \nabla_ xu)\). The main result is the following theorem: The initial value problem \(\square u = u^ 2\), \(0 \leq t<T\); \(u(0,x) = u_ 0(x) \in H_ s (\mathbb{R}^ 3)\), \(\partial_ t u(0,x) = u_ 1(x) \in H_{s-1} (\mathbb{R}^ 3)\), has a unique solution in \(H_ s\) for some \(T>0\) if \(s>0\). If \(u_ 0\) and \(u_ 1\) have radial symmetry, the same statement is true also with \(s=0\). In general, the problem is ill posed in \(H_ 0\). Moreover, there are data of compact support \(u_ 0 \in L^ 2\) and \(u_ 1 \in H_{-1} \cap L^ p\), for any \(p<6/5\), such that the problem does not have a solution in \(L^ 2([0,T] \times \mathbb{R}^ 3)\) for any \(T>0\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35R25 Ill-posed problems for PDEs
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