Suppose that V is open in R2, that H = [a, b] [0, c] V, that : V R is C2 on

Suppose that V is open in R2, that H = [a, b] × [0, c] ⊂ V, that μ: V → R is C2 on V, and that μ(x0, t0) > 0 for all (x0, t0) ∈ ϑH.
a) Show that, given ε > 0, there is a compact set K ⊂ such that u(x, t) > -ε for all (x, t) ∈ H\K.
b) Suppose that μ(x1. t1) = - < 0 for some (x1, t1) ∈ Ho, and choose r > 0 so small that 2rt1 < . Apply part a) to ε: = /2 - rt1 to choose the compact set K, and prove that the minimum of
w(x, t) := u(x, t) + r(t - t1)
on H occurs at some (x2, t2) ∈ K.
c) Prove that if u satisfies the heat equation (i.e., uxx - ut = 0 on V), and if μ (x0, t0) > 0 for all (x0, t0) ∈ ϑH, then u(x, t) > 0 for all (x, t) ∈ H.