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Counterexamples to local existence for quasilinear wave equations. (English) Zbl 0932.35149
The author considers the initial value problem for a wave equation in \(\mathbb{R}^{1+3}\); \[ \square u= (D^\ell u)D^{k-\ell} u,\quad D= (\partial_{x^1}- \partial_t), \] where \(0\leq \ell\leq k-\ell\leq 1\), \(\ell= 0,1\). In the semilinear case \(k-\ell\leq 1\), and in the quasilinear case \(k-\ell= 2\). Let \(\gamma= k\). It is shown that there exists initial data \((f,g)\in \dot H^\gamma(\mathbb{R}^3)\times \dot H^{\gamma-1}(\mathbb{R}^3)\) with compact support and with arbitrarily small norm such that the problem does not have any proper solution in \([0,T)\times \mathbb{R}^3\) for any \(T>0\). Here, a proper solution is a distributional solution with the property that it is a weak limit of a sequence of smooth solutions with data \((\phi_\varepsilon* f,\phi_\varepsilon* g)\), where \(\phi_\varepsilon\) is a molifier. The author also shows that there exist a function \(u\) and an open set \(\Omega\subset \mathbb{R}^1_+\times \mathbb{R}^3\) such that \(\Omega\) is a domain of dependence for \(u\), and some norm of \(u(t,\cdot)\) is finite at the initial time \(t=0\), but infinite at any later time \(t>0\). Furthermore, in the quasilinear case \(u\) can be chosen so that \(\|D^\ell u\|_{L^\infty(\Omega)}\) is arbitrarily small.

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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