Parabolic equations

Abstract

In this chapter, we consider parabolic equations of the form

$$ \frac{{\partial u}} {{\partial t}} + Lu = f, x \in \Omega , t > 0, $$

(5.1)

where Ω is a domain of ℝ^d, d = 1, 2, 3, f = f(x, t) is a given function, L = L(x) is a generic elliptic operator acting on the unknown u = u(x,t). When solved only for a bounded temporal interval, say for 0 < t < T, the region QT = Ω × (0, T) is called cylinder in the space ℝ^d × ℝ⁺ (see Fig. 5.1). In the case where T = +∞, Q={(x,t):x ∈ Ω, t > 0} will be an infinite cylinder.