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Solutions globales de l’équation des ondes semi-linéaire critique à coefficients variables. (Global solutions for the critical nonlinear wave equation in variable coefficients). (French) Zbl 1024.35077

Summary: In this work, we study the existence of both global smooth and Shatah-Struwe’s solutions of the critical wave equation in variable coefficients in dimension \(d\) of space \[ \square_Au+ u ^{{4}/{(d-2)}}u=\partial^2_t u- \text{div}(A(x)\cdot \nabla_xu)+ u ^{{4}/{(d-2)}}u=0,\quad \mathbb{R}_t\times \mathbb{R}^d_x, \] where \(A\) is a regular function valued in the space of \(d\times d\) positive definite matrix and which is the identity outside a fixed compact.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B33 Critical exponents in context of PDEs
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