Chapter 6 Partial Differential Equations

6.1. INTRODUCTION 7
t
u=0
u=1/2
u=1/2
u=0 u=1
x
u=0
The propagation of a disturbance in dimension 1.
In general, for hyperbolic equations
1. solutions are no smoother than data,
2. there is a finite speed of propagation,
3. solutions exhibit a strong dependence on spatial dimension,
4. many quantities are preserved, and
5. the Cauchy problem is well-posed.
Parabolic Equations
The heat equation,
u t (x, t) =∆u(x, t), x ∈ R n ,t>0
is an example of a parabolic equation. If we think of u(x, t) as being the temperature at a
point x at time t, this equation describes the flow or diffusion of heat. In view of our earlier
discussion of the interpretation of the Laplacian, we see that if say, ∆u(x, t) < 0, then the
temperature at position x is greater than that at surrounding points. From Fourier’s Law of
Cooling, heat would ‘flow’ away from the position x, and from the differential equation we
see that u t < 0, corresponding to the decrease in temperature at that point.
The Cauchy problem for the heat equation on R n is
u t =∆u(x, t), |x| < ∞, t>0.
u(x, 0) = f(x).