Formation of singularities in solutions of a wave equation
The author studies the following Cauchy problem: \[ \begin{aligned} w_{tt}(x,t) &=a( x)(\varphi(w_x(x,t))w_{xx}, x\in \mathbb R, t>0,\\ w(x,0)&=u_0(x), w_t(x,0)=v_0(x,t), x\in \mathbb R, \end{aligned} \] where \(a>0, \varphi >0\). Suppose that \(a\in W^{1,\infty}(\mathbb R)\), \(\varphi \in C^2(\mathbb R)\), \(a(x)\geq\alpha\), \(\varphi(\xi)\geq \alpha'\), \(x,\xi\in \mathbb R\), \(\alpha>0\). If \(u_0,v_0 \in H^2(\mathbb R)\), the Cauchy problem has a unique local solution such that \(u,u_t\in C([0,T); H^2(\mathbb R)) \cap C^1([0,T);H^1(\mathbb R))\). Moreover, the author proves that if \(a''\in L^{\infty}(\mathbb R)\), \(\varphi'(0)<0\), then there exist the initial data \(u_0,v_0\) such that the solution blows up in finite time.
scientific article; zbMATH DE number 1369342