Math 742 Heat Equation and Kernel

Second-order parabolic equations are natural generalizations of the heat equation and we will study in this section the existence, uniqueness, and regularity of appropriately defined weak solutions. 1.1. Formulation of Weak Solutions. 1.1.1. Notations. In this note, we assume Ω to be an open, bounded domain in R n , and set Ω T = Ω × (0, T ]. We study the following initial/boundary-value problem (1.1)      u t + Lu = f, in Ω T u = 0, on ∂Ω × [0, T ] u = g, on Ω × {t = 0} where f (x, t) : Ω T → R and g(x) : Ω → R are given with u(x, t) : Ω T → R the unknown function. L here is a time-independent second order differential operator in divergence form (1.2) Lu = −∂ j (a ij ∂ i u) + b i ∂ i u + cu for given coefficients a ij , b i , c. Note that we assume the summation convention for upper and lower indices. We require that the differential operator L to be uniformly elliptic, i.e. there exists a constant θ > 0 such that (1.3) a ij (x)ξ i ξ j ≥ θ|ξ| 2 for all x ∈ Ω, ξ ∈ R n. Also, we assume self-adjointness of L by requiring a ij = a ji. 1.1.2. Weak Solutions. In order to find appropritate notion of weak solution to initial/boundary-value problem (1.1), we first assume that 1 0 (Ω), we have the following time-independent bilinear form (1.4) B[u, v] := Ω a ij ∂ i u∂ j v + b i ∂ i uv + cuvdx Further more, to better accomodate this evolution problem, we consider u(x, t),f (x, t), u (x, t) as mappings from [0, T ] into the functional triplet H