Partial Differential Equations on Unbounded Domains

Abstract

In this chapter we investigate problems on unbounded spatial domains. We begin with the heat equation, or diffusion equation, on the real line. That is, we consider the initial value problem

$$ u_t = ku_{xx} , x \in R, t > 0, $$

((2.1))

$$ u(x, 0) = \varphi (x), x \in R. $$

((2.2))

Physically, this problem is a model of heat flow in an infinitely long bar where the initial temperature ϕ(x) is prescribed. In a chemical or biological context, the equation governs density variations under a diffusion process. Notice that there are no boundaries in the problem, so we do not prescribe boundary conditions explicitly. However, for problems on infinite domains, conditions at infinity are sometimes either stated explicitly or understood. Such a condition might require boundedness of the solution or some type of decay of the solution to zero as x → ±∞. In mathematics, a pure initial value problem like (2.1)–(2.2) is often called a Cauchy problem.