engleski

Microlocal energy densities for semilinear wave equations

We show that the microlocal energy density for the wave equation in three space dimension $$ \left\{\eqalign{ \rho (x) \partial_t^2 u_n - \hbox{div} \bigl(A(x)\nabla_x u_n\bigr) - u_n^3 &= 0 \cr u_n(0) = \gamma_n \rightharpoonup 0 &\quad \hbox{in} \quad {\rm H}^1({\rm R}^d)\cr \partial_t u_n(0) = \beta_n \rightharpoonup 0 &\quad \hbox{in} \quad {\rm L}^2({\rm R}^d) \cr} \right. $$ is the same as for the linear equation as obtained by Francfort and Murat (CPDE, 1992) The essential tool in these computations are $H$-measures (also known as microlocal defect measures). Some other variants of the above result will be discussed as well.

H-measures; semilinear wave equation

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