JournalsrmiVol. 24, No. 2pp. 463–497

Global infinite energy solutions of the critical semilinear wave equation

  • Pierre Germain

    New York University, USA
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Abstract

We consider the critical semilinear wave equation

(NLW)21      {u+u22u=0ut=0=u0tut=0=u1,(NLW)_{2^*-1} \;\;\; \left\{ \begin{aligned} \square u + |u|^{2^*-2} u & = 0 \\ u_{|t=0} & = u_0 \\ \partial_t u_{|t=0} & = u_1 \, \,, \end{aligned} \right.

set in Rd\mathbb{R}^d, d3d \geq 3, with 2=2dd22^* = \frac{2d}{d-2} \,\cdotp Shatah and Struwe [Shatah, J. and Struwe, M.: Geometric wave equations. Courant Lecture Notes in Mathematics 2. New York University, Courant Institute of Mathematical Sciences. American Mathematical Society, RI, 1998] proved that, for finite energy initial data (ie if (u0,u1)H˙1×L2(u_0,u_1) \in \dot{H}^1 \times L^2), there exists a global solution such that (u,tu)C(R,H˙1×L2)(u,\partial_t u)\in \mathcal{C}(\mathbb{R},\dot{H}^1 \times L^2). Planchon [Planchon, F.: Self-similar solutions and semi-linear wave equations in Besov spaces. J. Math. Pures Appl. (9) 79 (2000), no. 8, 809-820] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u0,u1)(u_0,u_1) in B˙2,1×B˙2,0\dot{B}^1_{2,\infty} \times \dot{B}^0_{2,\infty} is small enough. In this article, we build up global solutions of (NLW)21(NLW)_{2^*-1} for arbitrarily big initial data of infinite energy, by using two methods which enable to interpolate between finite and infinite energy initial data: the method of Calderón, and the method of Bourgain. These two methods give complementary results.

Cite this article

Pierre Germain, Global infinite energy solutions of the critical semilinear wave equation. Rev. Mat. Iberoam. 24 (2008), no. 2, pp. 463–497

DOI 10.4171/RMI/543