Midterm Exam for Partial Differential Equations | MATH 444, Exams of Mathematical Methods for Numerical Analysis and Optimization

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Math 444, Partial Differential Equations May 1999

Math 444, Partial Differential Equations May 1999 Partial Differential Equations Comprehensive Exam, May 1999 Mathematics Department, UIUC 1. Let 2 be a smooth, bounded subset of R" with outward unit normal v. Consider the equation ue + A?u =0 on 2 x (0,00) with u = 4 = 6 on A x (0,00) and u(z,0) = f(z), where f is some given function. Prove that, if a solution exists that is smooth up to the boundary, then it is unique. Here A?u = A(Au) = Te je1 Ve veieses 2. Discuss the local solvability of the initial value problem ru — tu, = 0 for (x,t) in a neighborhood of the line ¢ = 0 u(2, 0) given. In particular, say whether the problem is locally solvable for arbitrary smooth initial data. If so, indicate how to construct a solution. If not, explain why not, and give an example of initial data for which a solution does not exist. 3. Suppose that u solves the initial-boundary value problem tt = ee + u(1 ~ u?) for (x, 4) € (0, L) x (0, 0) u(z,0) given, u(0,t) = u(L,t) = OVE > 0, and that u is smooth on (0, L] x [0, 00). Show that if L < #, then for any initial data, fy ule, t)?de + 0 ast + 00. Do you expect it to be true if L > 7?