Showing papers by "Hajer Bahouri published in 1997"

Concentration effects in critical nonlinear wave equation and scattering theory

Abstract: This paper is devoted to generalized geometrical optics for the following nonlinear wave equation with critical exponent, $$ \square u + \left| u \right|^4 u = 0 $$ (1) where \(\square = \partial _t^2 - \Delta _x ,t \in {\text{R,}}\,x \in {\text{R}}^{\text{3}} \). The global well-posedness of equation (1) in the energy space was proved rather recently by Shatah-Struwe [16]. Let us recall precisely this result: given \(\phi \in \dot H^1 \left( {{\text{R}}^{\text{3}} } \right),\psi \in L^2 \left( {{\text{R}}^{\text{3}} } \right)\) there exists a unique solution u to (1) satisfying \(u\left| {_{t = 0} = \phi ,\,\partial _t u} \right.\left| {_{t = 0} = \psi } \right.\)and \(u \in L_{{\text{loc}}}^5 \left( {{\text{R}},L^{10} \left( {{\text{R}}^{\text{3}} } \right)} \right)\). Observe that this latter property means exactly that the nonlinear term \(\left| u \right|^4 u\) in (1) belongs to \(L_{{\text{loc}}}^1 \left( {{\text{R,}}\,L^2 \left( {{\text{R}}^{\text{3}} } \right)} \right)\), which allows us to consider it as a source term in the energy method. In particular, this implies \(\left( {u,\partial _t u} \right) \in C\left( {{\text{R}}_t ,\dot H^1 \left( {{\text{R}}^{\text{3}} } \right) \times L^2 \left( {{\text{R}}^{\text{3}} } \right)} \right)\) with conservation of energy $$\int_{{\text{R}}^{\text{3}} } {\left[ {\frac{1} {2}\left| {\partial _t u\left( {t,x} \right)} \right|^2 + \frac{1} {6}\left| {u\left( {t,x} \right)} \right|^6 } \right]} \,dx = {\text{cst}}\,{\text{: = }}\,E\left( u \right) $$ (2) .