课程名称︰偏微分方程导论 课程性质︰数学系大三必修 课程教师︰林绍雄 开课学院：理学院 开课系所︰数学系 考试日期︰2013年05月31日(五)，08:10-09:10 考试时限：60分钟 是否需发放奖励金：是 试题 : 　　　　　　Math 2206 (Intorduction to PDE) Quiz No.7 (5/31/2013) Solve the following problems. Please write down your computational or proof steps clearly on the answer sheets. A. (20 points) Solve u_t = u_xx for x > 0, t > 0 satisfying the initial 　　condition u(x,0) = g(x) for x≧0, and the boundary condition u_x(0,t) = 0 　　for t≧0, where g(x) is a bounded continuous function defined in x≧0. B. (25 points) Let G⊂R^n be a bounded domain so that ∂G is a C^2 boundary. 　　Consider the initial-boundary value problem 　　　u_t　　= △u　for x∈G, t > 0 　　　u(x,0) = f(x) for x∈bar(G) 　　　u(x,t) = g(x) fpr x∈∂G, t > 0 　　where f∈C^2(bar(G)), and g∈C(∂G). Prove that lim u(x,t) = v(x) so that 　　　　　　　　　　　　　　　　　　　　　　　　　　t→∞ 　　△v(x) = 0 for x∈G, and v(x) = g(x) for x∂G. C. Consider the Porous media equation u_t = △(u^γ), where γ>1 is a constant. 　　(a) (5 points) Find necessary conditions on a,b,c∈|R such that the 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　a　　b　　c　 　　　　equation has a solution u(x,t) satisfying u(x,t) = λ u(λ x,λ t) for 　　　　all λ > 0. 　　(b) (20 points) Find an example of such solution in (a) which is also 　　　　rotationally invariant. --

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