A class of nonlinear hyperbolic problems with global solutions
The paper is concerned with a global in time solvability of smooth solutions in the sense of \(L_2\) Sobolev spaces to the Cauchy problem of a nonlinear strictly hyperbolic equation of the Kirchhoff type for small and smooth initial data. More precisely, the authors consider the following equations: \[ u_{tt} + (-1)^m \sum_{|\alpha |= 2m} f_\alpha \biggl( |D^{\beta_1}_x u |^2, \ldots, |D^{\beta_N}_xu |^2 \biggr) D^\alpha_xu = 0\tag{C} \] \[ u(0,x) = \varepsilon u_0 (x),\;u_t (0,x) = \varepsilon u_1(x) \] where \(|v |^2 = \int_{\mathbb{R}^n} |v(x) |^2 dx\), while \(f_\alpha (A_1, \ldots, A_N)\) are \(C^2\) real functions on \(\mathbb{R}^N\) satisfying the strict hyperbolicity condition: \[ \sum_{|\alpha |= 2m} f_\alpha (A_1, \ldots, A_N) \xi^\alpha \geq \nu^2 |\xi |^{2m} \quad (\nu > 0). \] The main theorem is the following. Define \[ b_1 = \text{Min}_{1 \leq j \leq N} |\beta_j |, \quad b_2 = \text{Max}_{1 \leq j \leq N} |\beta_j | \] with \(b_1 > m - n/2\). Assume that the initial data \(u_0\) and \(u_1\) satisfy the following conditions: \(u_0 \in H^m (\mathbb{R}^n)\), \(u_1 \in L_2 (\mathbb{R}^n)\), \(x^\alpha u_0 \in W^{m + s,1} (\mathbb{R}^2)\), \(x^\alpha u_1 \in W^{s,1} (\mathbb{R}^n)\) \(^\forall |\alpha |\leq 2\) for some \(s > b_2+n/2-m/2\). Then, there exists an \(\overline \varepsilon = \overline \varepsilon (u_0, u_1) > 0\) such that the Cauchy problem (C) admits a unique solution \(u\in C(\mathbb{R}^+;H^m(\mathbb{R}^n)\cap H^{b_2}(\mathbb{R}^n))\cap C^1(\mathbb{R}^+;L_2 (\mathbb{R}^n))\) for \(\varepsilon \leq \overline \varepsilon\). The following three equations are typical examples where the theorem can be applied: \[ \begin{aligned} u_{tt} - f \Bigl( \int_{\mathbb{R}^n} |\nabla_x u |^2 dx \Bigr) \Delta_x u = 0 \quad & \text{with} \quad f \geq \nu^2 > 0.\\ u_{tt} - \sum^n_{j = 1} f_j \Bigl( \int_{\mathbb{R}^n} |u_{x_j} |^2 dx \Bigr) u_{x_jx_j} = 0 \quad & \text{with} \quad f_j \geq \nu^2 > 0.\\ u_{tt} + (- 1)^m f \Bigl( \sum_{b_1 \leq |\alpha |\leq b_2} \int_{\mathbb{R}^n} |D^\alpha_x u |^2 dx \Bigr) \Delta^m_x u = 0 \quad & \text{with} \quad f \geq \nu^2 > 0. \end{aligned} \] As a related topic, [\textit{J. M. Greenberg} and \textit{S. C. Hu}, Q. Appl. Math. 38, 289-311 (1980; Zbl 0487.73006)]was the first paper to prove a global in time solvability of small and smooth solutions in the Sobolev space framework of the one-dimensional nonlinear wave equation of the Kirchhoff type: \[ u_{tt} - \left( 1 + \gamma \int^\infty_{- \infty} |u_x |^2 dx \right) u_{xx} = 0\;(\gamma > 0). \] When initial data are real analytic functions, a global in time existence of solutions for (C) which are real analytic with respect to space variables was already proved by Bernstein, Pohozaev, Arosio-Spagnolo, D'Ancona-Spagnolo, Kajitani-Yamaguchi and so on (cf. references cited therein).
nonlinear strictly hyperbolic equation of the Kirchhoff type
scientific article; zbMATH DE number 484797