Picture below is from 74th page of Topping's Lectures on the Ricci flow. The (6.4.8) is $$ -\partial_t u -\Delta u + Ru =0 \tag{6.4.8} $$ where $R$ is scalar curvature. And the $N$ is $$ N= -\int_M u \ln u dV $$ where $M$ is closed manifold.
I want to show $\tilde N \rightarrow 0$. But I don't know how to solute (6.4.8) from $u(T)$ is a delta function (I know what is backward solution). In fact, I feel it can't be solved by analytic solution since $M$ is not especial.
