PhD Qualify Exam, PDE, Mar. 04, 2011 Show all works
E: easy, M: moderate, D: difficult
1.(M) Find the solution for the problem [15%]
ut− uxx = 0, u(x, 0) = 0, u(0, t) = h(t),
x > 0, t > 0, (1)
where h(0) = 0.
2.(M) Use the Fourier transform method to solve the initial value problem
( ut= uxx, x ∈ R, t > 0, u(x, 0) = f (x), x ∈ R.
And prove that u satisfies the following inequality, for 1 ≤ q ≤ p ≤ ∞ and t > 0, [15%]
ku(·, t)kLp(R)≤ (4πt)−12(1q−1p)kf kLq(R).
3.(M) Let λ ∈ R, a > 0, and u be a smooth function defined on a neighborhood of D = {(x, y) ∈ R2| x2 + y2 ≤ 1} such that
(
∆u + λu = 0, in x2+ y2 < 1,
∂u/∂n = −au, on x2+ y2 = 1, where n is the unit outward normal vector to ∂D. Prove that if u is not identically zero in x2+ y2 < 1, then λ > 0. [15%]
4.(E) Let D = {x ∈ R3| z > 0} and x0 = (x0, y0, z0) ∈ D. Let u(x0) = z0 2π
Z Z
∂D
h(x)
|x − x0|3 dS. Show
that u(x0, y0, z0) → h(x0, y0) as z0 → 0. [15%]
5.(E) Let F : Rn × R × Rn be a smooth function of (p, z, x). Assume that u is a smooth solution to F (Du, u, x) = 0 and x(s) is a smooth curve. Prove that if p(s) = Du(x(s)), z(s) = u(x(s)), and x0i(s) = ∂F
∂pi, then p0i(s) = ∂F
∂xi − ∂F
∂z pi. [15%]
6.(E) By using the method of characteristics, find an explicit local solution to ut+1 2
(ux)2+ x2
= 0
if t > 0, x ∈ R, with initial condition u(x, 0) = x2/2. [15%]
7.(D) Solve the wave equation for infinite vibrating string utt = c2(x)uxx, where c(x) =
( c1, x < 0.
c2, x > 0.
Let a wave u(x, t) = f (x − c1t) come in from the left, see the figure below. Thus the initial conditions are u(x, 0) = f (x) and ut(x, 0) = −c1f0(x). Assume that u(x, t) and ux(x, t) are continuous everywhere. Also
give an interpretation for the solution you find. [10%]
Figure 1:
1
