New answers tagged

1

$$ \tilde{u}_{tt} - \frac{2}{t}\tilde{u}_{t}-\Delta \tilde{u} = g_t -\frac{2}{t^2}\tilde{u} $$ $$ \dot{E} = 2 \int \tilde{u}_t (2 t^{-1} \tilde{u}_t + g_t - 2 t^{-2} \tilde{u} ) $$ $$ \dot{E} = 2 \int \tilde{u}_t (2 t^{-1} \tilde{u}_t + g_t) - 2 t^{-2} \frac{d}{dt} \int \tilde{u}^2 $$ $$ \dot{E} + \frac{d}{dt} (2 t^{-2} \int \tilde{u}^2 ) = 2 \int \tilde{u}...


2

Define using $H(t) = \int (u_t)^2 + |\nabla u|^2 ~dx $ the standard energy. Taking the time derivative you find $$ \frac{d}{dt}H(t) = 2 \int u_t( g + \frac{2}{t} u_t)$$ Writing $\|\cdot \|$ for the $L^2$ integral, you have then $$ \frac{d}{dt} H(t) \geq - 2 \|u_t\|\cdot \|g\| + \frac{4}{t} \|u_t\|^2 \tag{A}$$ by Cauchy-Schwarz, and then by AM-GM you get $$ \...


Top 50 recent answers are included