It is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear function can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem.
Lions's generalization is an important one since it allows one to tackle boundary value problems beyond the Hilbert space setting of the original Lax–Milgram theory.
Two questions arise immediately: on what domain in spacetime is the heat equation to be solved, and what boundary conditions are to be imposed?
In simple settings, it suffices to consider cylindrical domains: i.e., one fixes a spatial region of interest, Ω, and a maximal time, T ∈(0, +∞], and proceeds to solve the heat equation on the "cylinder" One can then proceed to solve the heat equation using classical Lax–Milgram theory (and/or Galerkin approximations) on each "time slice" {t} × Ω.
Such domains and boundary conditions are beyond the reach of classical Lax–Milgram theory, but can be attacked using Lions's theorem.