On global existence of solutions to nonlinear wave equations of wave map type
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 \((\varphi,\psi) \to (\varphi_+,\psi_+)\) and its inverse are Lipschitz continuous in some neighborhood of \(0\) in \(X\). The following theorem contains the blow-up result for solutions of problem (1). Theorem 4. Let \(f\in C^{\infty}(\mathbb{R})\). Suppose that (2) does not hold. Then for any natural number \(n\), \(r>1\) and \(\varepsilon>0\), there exists \((\varphi,\psi)\in C_0^{\infty}(\mathbb{R}^n)\) with \(\|\varphi\|_{B^{n/2}_{2,r}}+\|\psi\|_{B^{n/2-1}_{2,r}}< \varepsilon\) such that the classical solution of (1) exists for \(0\leq t<T\) with some \(T<\varepsilon\) and blows up at \(t=T\) in the sense that \(\inf _{x\in B}|u(t,x)|\to \infty\) as \(t\to T\), with some ball \(B\subset \mathbb{R}^n\). \((\varphi,\psi)\) may be taken as spherically symmetric.
scientific article; zbMATH DE number 1529448
scientific article