Showing that the map that takes $u_0$ to solution $u(t)$ is self-adjoint

Question

Let $u$ and $v$ be the solution of the heat equation $$w'(t) - \Delta w(t) =0$$ with initial data $u_0$ and $v_0$ respectively, and with either homogeneous Dirichlet or Neumann BCs on a bounded domain $\Omega$.

Define $S(t):L^1(\Omega) \to L^2(\Omega)$ by $S(t)u_0 = u(t)$ be the mapping taking initial data to the solution evaluated at time $t$.

How can I show that $S(t)$ is self-adjoint without using Green's kernel? It amounts to showing that $$(u(t), v_0)_{L^2} = (v(t), u_0)_{L^2}$$ but I cannot prove it.

Answer 1

For $0 \le s \le t$, the selfadjointness of $\Delta$ gives \begin{align} \frac{\partial}{\partial s}(u(t-s),v(s)) & =-(u'(t-s),v(s))+(u(t-s),v'(s)) \\ & =-(\Delta u(t-s),v(s))+(u(t-s),\Delta v(s)) =0. \end{align} Therefore, $$ (u(t-s),v(s)|_{s=0} = (u(t-s),v(s))|_{s=t} \\ (u(t),v_0) =(u_0,v(t)). $$