linguistic:voice_change
Solutions of heat equation

Let Ω be a subdomain (bounded or not) of R n {\displaystyle \mathbb {R} ^{n}} (with n an integer). Denote by Γ its boundary (assumed smooth). Consider the following heat equation on Ω × (0, T), for T > 0,

with the following initial value condition u (0) = u in Ω.

The equation u = 0 on Γ × (0, T) corresponds to the Homogeneous Dirichlet boundary condition. The mathematical setting for this problem can be the semigroup approach. To use this tool, we introduce the unbounded operator Δ defined on L 2 (Ω) {\displaystyle L^{2}(\Omega)} by its domain

(see the classical Sobolev spaces with H k (Ω) = W k, 2 (Ω) {\displaystyle H^{k}(\Omega)=W^{k,2}(\Omega)} and

is the closure of the infinitely differentiable functions with compact support in Ω for the H 1 (Ω) − {\displaystyle H^{1}(\Omega)-} norm).

For any v ∈ D (Δ D) {\displaystyle v\in D(\Delta _{D})}, we have

With this operator, the heat equation becomes u ′ (t) = Δ D u (t) {\displaystyle u'(t)=\Delta _{D}u(t)} and u (0) = u. Thus, the flow corresponding to this equation is (see notations above)

where exp(t Δ) is the (analytic) semigroup generated by Δ.