Well-posedness in the energy space for semilinear wave equations with critical growth. (English) Zbl 0830.35086

We give a proof of global existence and uniqueness of solutions to the critical wave equation with finite energy initial data on \(\mathbb{R}^n\), \[ u_{tt} - \Delta u + |u |^{2^*- 2} u = 0, \text{ on } \mathbb{R}^n \times \mathbb{R}, \tag{1} \]
\[ u |_{t = 0} = u_0, \;\partial_tu |_{t = 0} = u_1 \text{ on } \mathbb{R}^n, \quad (u_0, u_1) \in \dot H^1 \times L^2. \tag{2} \] Theorem. Problem (1)–(2) has a unique global solution \(u\) in the space \((u,u_t) \in C (\mathbb{R}, \dot H^1 \times L^2) \cap L^q_{\text{loc}} (\mathbb{R}, \dot B_q^{1/2} \times \dot B_q^{-1/2})\) where \(q = 2(n + 1)/(n - 1)\). The proof hinges on showing that the energy and scaling identities hold for weak solutions and is divided into four parts: local existence, uniqueness, energy and scaling identities, and global existence.


35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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