放物型偏微分方程式
{\displaystyle {\begin{cases}u_{t}=\Delta u&{\textrm {on}}\ \ \Omega \times (0,T),\\u=0&{\textrm {on}}\ \ \partial \Omega \times (0,T),\\u=f&{\textrm {on}}\ \ \Omega \times \left\{T\right\}.\end{cases}}}
これは次の後退双曲型方程式と本質的には同じである:
{ u t = − Δ u on Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u = f on Ω × { 0 } . {\displaystyle {\begin{cases}u_{t}=-\Delta u&{\textrm {on}}\ \ \Omega \times (0,T),\\u=0&{\textrm {on}}\ \ \partial \Omega \times (0,T),\\u=f&{\textrm {on}}\ \ \Omega \times \left\{0\right\}.\end{cases}}}