課程名稱︰偏微分方程式一
課程性質︰數學研究所基礎課
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︰2015年01月13日(二)，10:20-12:10
考試時限：110分鐘
試題 :
　　　　　　　　　　　　　Final Examination　　　　　　　　　　　　1/13/2015
1. 20 pts
　Assume F(0) = 0, u is a continuous integral solution of the conservation law
　　　　　　　　　　　　u_t + F(u)_x = 0 in |R ×(0,∞)
　　　　　　　　　　　　　　　　　 u = g on |R ×{t=0}.
　and u has compact support in |R ×[0,∞]. Prove
　　　　　　　　　　　　 ∞　　　　　　 ∞
　　　　　　　　　　　　∫ u(‧,t)dx = ∫ gdx
　　　　　　　　　　　　-∞　　　　　　-∞
　for all t > 0.
2. 20 pts
　Let u be a solution of
　　　　　　1 　　　　　　　a(x,y) u_x + b(x,y) u_y = -u
　of class C in closed unit disk Ω in the xy-plane. Let a(x,y)x + b(x,y)y > 0
　on the boundary of Ω. Prove that u vanishes identically.
3. 20 pts
　Which of the following statements is (are) true?
　(A) Burger's equation has no regularity theorem,
　(B) Wave equation has regularity theorem,
　(C) Poisson equation has no regularity theorem.
　Find and justify your answer.
4. 20 pts
　Let u be the solution of the following initial value problem of heat equation
　　　　　　　　　　　　　　　u_t = Δu　for x∈Ω, t > 0,
　　　　　　　　　　　　　　　　u = 0　　on x∈∂Ω, t > 0,
　　　　　　　n 　　　　　　　　u = δ_0 at t = 0,
　where Ω⊆|R is a bounded smooth domain with 0 ∈Ω and δ_0 is the standard
　Dirac measure concentrating at the origin. Let G be the standard heat kernel
　on |R^n ×(0,∞). Define w = u - G. Which of the following statements is
　(are) true?
　(A) sup |w(x,t)| < ∞,
　　x∈Ω,t>0
　(B) sup |w(x,t)| = ∞,
　　x∈Ω,t>0
　(C) w is smooth on Ω ×(0,∞)m
　(D) There exists x0 ∈Ω such that |w(x,t)|→∞ as (x,t)→(x0,t).
　Find and justify your answer.
5. 20 pts　　　　　　　　　　　　　　　　n　　　　　　　　　　　　　　　　_+
　Let B+ denote the open half-ball {x∈|R : |x|<1, xn > 0}. Assume u ∈C(B) is
　harmonic in B+ with u = 0 on ∂B+∩{xn = 0}. Set v(x) = u(x) for
　x = (x1,x2,...,xn)∈B, xn ≧ 0, and v(x) = - u(x1,...,x(n-1),-xn) for
　x = (x1,...,xn)∈B, xn < 0. Here B = {x∈|R^n : |x|<1}.
　Which of the following statements is (are) true?
　(A) v is smooth in B,　　　　　　 1
　(B) v is only continuous but not of C,
　(C) v is only C1 but not of C2.
　Find and justify your answer.