UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 9

Teori för differentialekvationer 1998 02 18

9.1. Let f ∈ C^{2}(R^{n}) have the properties f(x) > 0 when |x| < 1 and f(x) = 0 when |x| ⩾ 1. Determine in each of the three cases a, b, and c below the set A_{t} = {x ∈ R^{n}; u(x, t) ≠ 0} for all t > 0.

a) Let n = 1 and let u ∈ C^{2}(R × [0, +∞[) be the solution to

u_{t} − u_{xx} = 0, x ∈ R, t > 0; u(x, 0) = f(x), x ∈ R.

b) Let n = 2 and let u ∈ C^{2}(R^{2} × [0, +∞[) be the solution to

u_{tt} − Δ_{x}u = 0, x ∈ R^{2}, t > 0; u(x, 0) = 0 and u_{t}(x, 0) = f(x), x ∈ R^{2}.

c) Let n = 3 and let u ∈ C^{2}(R^{3} × [0, +∞[) be the solution to

u_{tt} − Δ_{x}u = 0, x ∈ R^{3}, t > 0; u(x, 0) = 0 and u_{t}(x, 0) = f(x), x ∈ R^{3}.

9.2. Let P (ζ) be a polynomial of n complex variables (ζ_{1}, ..., ζ_{n}) of degree m ⩾ n + 1 and with the properties that |P (iξ)| ⩾ (1 + |ξ|)^{m} when ξ is real. Then

E(x) = (2π)^{−n} ∫_{R^{n}} \frac{e^{ix·ξ}}{P (iξ)}dξ

is a well-defined function, for the integral converges. Prove that E is a fundamental solution to the differential operator P (D) which is obtained by substituting ∂/∂x_{j} for the variables ζ_{j}. Prove the formula

E(x) = (−1)^{k}|x|^{−2k}(2π)^{−n} ∫_{R^{n}} e^{−ix·ξ}Δ^{k}(\frac{1}{P (iξ)})dξ

for every k = 1, 2, ... . From this formula we can deduce that E is a C^{∞} function outside the origin, for |x|^{2k}E can be differentiated quite a few times, depending on the fact that Δ^{k}(1/P (iξ)) decreases rather rapidly.

9.3. Let b: R → R be an increasing continuous function with b(0) = 0 and let f: R → R be continuous, bounded and ⩾ 0. The initial-value problem

u_{t} − Δ_{x}u + b(u) = 0, u(x, 0) = f(x),

then has exactly one bounded solution and it is ⩾ 0. Moreover, if we let u and v be the solutions that belong to the initial values f and g, respectively, then 0 ⩽ f ⩽ g implies that 0 ⩽ u ⩽ v. Prove that if the integral

(⊗)  ∫_{0}^{1} \frac{ds}{b(s)}

converges, then u(x, t) will be zero for every bounded function f when t is sufficiently large. Prove that, conversely, if the integral (⊗) diverges, there exist functions f > 0 such that u(x, t) never becomes zero for large t. Hint: As a starter, study solutions which depend on t only.