課程名稱︰偏微分方程式一 課程性質︰必選 課程教師︰陳逸昆 開課學院：理學院 開課系所︰數學研究所 考試日期︰2020年11月10日(二) 考試時限：10:20-12:10，共計110分鐘 試題 : 　　　　　　　　　　　　　　　　PDE, Fall 2020 Midterm Exam DEP ＿＿＿＿＿＿＿＿＿＿＿　　NAME ＿＿＿＿＿＿＿＿＿ ID NUMBER ＿＿＿＿＿＿＿ 1. Solve the following initial-value problem 　　　　　　　　　　u_t - u_x + 2u = x in |R × (0,∞) u = exp on |R × {t=0}. (25%) 2. Prove that r - |x| r + |x| r --------- u(0) ≦ u(x) ≦ r -------- u(0) r + |x| r - |x| whenever u is positive and harmonic in B(0,r)∈|R^3. (25%) n 3. Let Ω be an open bounded set in |R with smooth boundary. Define 1 I(w) = ∫ ---(▽w．▽w) + w^2 - wf dx, Ω 2 where w belongs to 2＿ A = {w∈C(Ω) | w = g on ∂Ω}. (a.) Suppose u is a minimizer of I over A. Derive a P.D.E. and boundary value problem that u satisfies. (b.) Show that a solution to boundary value problem obtained in (a.) is a minimizer of I. (25%) 1 4. (a) Suppose f(x)∈C([0,1]) and f(0) = 0. Prove that 1 2 1 1 2 ∫|f(x)| dx ≦ ---∫|f'(x)| dx. 0 2 0 2 Suppose u∈C([0,1] × [0,∞)) is a solution to 1 u_t = u_xx in (0,1) × (0,∞) u(x,0) = g(x) u(0,t) = u(1,t) = 0. 1 2 (b.) Show that ∫|u(x,t)| dx is decreasing. 0 1 (c.) Show that ∫|u(x,t)|dx decays to 0 exponentially. (25%) 0 --

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