we rewrite the classical linear heat equation $u_t = u_{xx}$ in the following strictly self-adjoint form: $$\label{Burg:mult.eq7}
 \frac{1}{u}\,[u_t - u_{xx}] = 0.$$ This result can be extended to the heat equation $$\label{Burg:mult.eq8}
  u_t - \Delta u = 0,$$ where $\Delta u$ is the Laplacian with $n$ variables $x = (x^1,
\ldots, x^n).$ Namely, the strictly self-adjoint form of Eq. (\[Burg:mult.eq8\]) is $$\label{Burg:mult.eq9}
 \frac{1}{u}\,[u_t - \Delta u] = 0.$$ Indeed, the formal Lagrangian (\[sa:Lag\]) for Eq. (\[Burg:mult.eq9\]) has the form $${\cal L} = \frac{v}{u}\,[u_t - \Delta u].$$ Substituting it in (\[saeq.3\]) we obtain $$F^* = - D_t\left(\frac{v}{u}\right) - \Delta \left(\frac{v}{u}\right)
 - \frac{v}{u^2}\,[u_t - \Delta u].$$ Upon letting  $v = u$ it becomes $$F^* = - \frac{1}{u}\,[u_t - \Delta u].$$ Hence, Eq. (\[Burg:mult.eq9\]) satisfies the condition (\[saeq.5\]) with $\lambda = - 1.$

Nonlinear heat equation {#samul:3}
-----------------------

Consider the nonlinear heat equation $ u_t - D_x\left(k(u) u_x\right) =
 0,$ or $$\label{Burg:mult.eq10}
  u_t - k(u) u_{xx} - k'(u) u_x^2 = 0.$$ Its adjoint equation has the form $$v_t + k(u) v_{xx} = 0.$$ Therefore it is obvious that (\[Burg:mult.eq10\]) does not satisfy Definition \[saeq:def\]. But it becomes strictly self-adjoint if we rewrite it in the form $$\label{Burg:mult.eq11}
 \frac{1}{u}\,\left[u_t - k(u) u_{xx} - k'(u) 