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July, 2008 Global infinite energy solutions of the critical semilinear wave equation
Pierre Germain
Rev. Mat. Iberoamericana 24(2): 463-497 (July, 2008).


We consider the critical semilinear wave equation \begin{equation*} (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. \end{equation*} set in $\mathbb{R}^d$, $d \geq 3$, with $2^* = \frac{2d}{d-2} \,\cdotp$ 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 $(u_0,u_1) \in \dot{H}^1 \times L^2$), there exists a global solution such that $(u,\partial_t u)\in \mathcal{C}(\mathbb{R},\dot{H}^1 \times L^2)$. 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 $(u_0,u_1)$ in $\dot{B}^1_{2,\infty} \times \dot{B}^0_{2,\infty}$ is small enough. In this article, we build up global solutions of $(NLW)_{2^*-1}$ 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.


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Pierre Germain . "Global infinite energy solutions of the critical semilinear wave equation." Rev. Mat. Iberoamericana 24 (2) 463 - 497, July, 2008.


Published: July, 2008
First available in Project Euclid: 11 August 2008

zbMATH: 1173.35084
MathSciNet: MR2459200

Primary: 35L05 , 46E35

Keywords: Besov spaces , critical wave equation , global solutions , infinite energy

Rights: Copyright © 2008 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.24 • No. 2 • July, 2008
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