課程名稱︰偏微分方程式一
課程性質︰研究所基礎課
課程教師︰林太家
開課學院：理學院
開課系所︰數學系、數學研究所、應用數學科學研究所
考試日期︰2014年11月11日(二)，10:20-12:10
考試時限：110分鐘
試題 :
Test 2 11/11/2014
1.
(20 pts) Write down an explict formula for a solution of
u_t - Δu + cu = f in R^n ×(0,∞),
u = g on R^n ×{t = 0},
where c in R is a constant.
2. (20 pts)
We say v in C^2_1(U_T) is a subsolution of the heat equation if
u_t - Δu ≦ 0 in U_T.
(a) Prove for a subsolution v that
1 |x-y|^2
v(x,t) ≦ -------- ∫∫ v(y,s) ---------dyds
4r^n E(x;t,r) (t-s)^2
for all E(x;t,r) contained U_T.
(b) Prove that therefore max v = max v
U_T Γ_T
(c) Let ψ: R → R be smooth and convex. Assume u solves the heat equation
and v := ψ(u). Prove v is a subsolution.
(d) Prove v := |Du|^2 + (u_t)^2 is a subsolution, whenever u solves the
heat equation.
3. (20 pts)
Let f(x) be bounded and continuous for x in R^n and satisfy
∫|f(y)|dy < ∞.
Show that there exists a solution u(x,t) of (1.8a,b) for which
lim u(x,t) = 0.
t→∞
u_t - Δu = 0 for x in R^n, t>0 (1.8a)
u = f(x) for x in R^n, t=0. (1.8b)
4. (20 pts)
Let
1/(4πt)^(n/2) ×exp(-|z|^2 / 4t) (x in R^n, t > 0)
Φ(x,t) :=
0 (x in R^n, t < 0)
Prove that
lim ∫ Φ(x,t)p(x) = p(0)
t→0+ R^n
∞
for p in C (R^n).
0
5. (20 pts)
2
Find a function f in C (R^2) such that ∫ f(y) log|x-y|dy is unbounded as
0 R^2
|x| → ∞.
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