## 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

### Keywords:

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