# zbMATH — the first resource for mathematics

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.

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