On global existence of solutions to nonlinear wave equations of wave map type
type: Item
instance of: scholarly article
title: On global existence of solutions to nonlinear wave equations of wave map type
zbMATH Open document ID: 0967.35096
DOI: 10.1016/S0362-546X(99)00153-4
author: Kenji Nakanishi
author: Masahito Ohta
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-08-20T00:00:00Z
review text: The authors study the global existence, blow-up and asymptotic behaviour for large time of solutions to the nonlinear wave equation: \[ \begin{aligned} &u_{tt}-\Delta u +f(u)(u_t^2-|\nabla u|^2)=0,\;(t,x)\in \mathbb{R}^{n+1},\quad n>1,\\ &u(0)=\varphi, \quad u_t(0)=\psi. \end{aligned}\tag{1} \] Suppose that \(f\) satisfies the conditions \[ \int_0^{\infty} \exp\Biggl(\int_0^s f(r) dr\Biggr) ds= \infty, \qquad \int_{-\infty}^0\exp\Biggl(\int_0^s f(r) dr\Biggr) ds=\infty. \tag{2} \] The following theorem contains the global existence result. Theorem 1. Let \(f\in C^{\infty}(\mathbb{R}^n)\). Then, if and only if \(f\) satisfies (2), (1) has a global classical solution \(u\in C^{\infty}(\mathbb{R}^{n+1})\) for any \(\varphi\) and \(\psi\in C^{\infty}(\mathbb{R}^n)\). Denote by \({\dot B}^s_{2,1}(\mathbb{R}^n)\) the homogeneous Besov space. Let \(X={\dot B}^{n/2}_{2,1}(\mathbb{R}^n)\bigoplus {\dot B}^{n/2-1}_{2,1}(\mathbb{R}^n)\). Let \(m\) be the smallest natural number greater than \(n/2\). The following theorem contains the global existence result for the small data. Theorem 2. Let \(n\geq 2\) and \(f\in C^m(\mathbb{R})\). Then there exists \(\delta>0\) such that for any \((\varphi,\psi)\in X\) with \(\|(\varphi,\psi)\|_X<\delta\), (1) has a unique global solution \(u\) satisfying \((u,u_t)\in C_B(\mathbb{R};X)\). Moreover, the correspondence \((\varphi,\psi)\to(u,u_t)\) is Lipschitz continuous from \(X\) to \(C_B(\mathbb{R};X)\). The asymptotic behaviour for large time of solutions to the problem (1) is given in the following theorem. Theorem 3. Let \(n\geq 2\), \(f\in C^{m+1}(\mathbb{R})\) and \((\varphi,\psi)\in X\) with \(\|(\varphi,\psi)\|_X<\delta\), where \(\delta\) is as given by Theorem 2. Let \(u\) be the solution for (1) given by Theorem 2. Then there exists unique \((\varphi_+,\psi_+)\in X\) such that \(v=W(\varphi_+,\psi_+)\) satisfies \(\|(u,u_t)(t)-(v,v_t)(t)\|_X\to 0\) as \(|t|\to \infty\). Moreover, the correspondence \((\varphi,\psi) \to (\varphi_+,\…
reviewed by: Andrei Perjan
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1529448
zbMATH Keywords: well-posed problem
zbMATH Keywords: ill-posed problem
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: blow-up of solutions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1529448