Semester Two Examinations 2023 MATH4407 PDF

Semester Two Examinations, 2023 MATH4407 Venue ____________________ Seat Number __________...

Semester Two Examinations, 2023 MATH4407 Venue ____________________ Seat Number __________ Student Number |__|__|__|__|__|__|__|__| Family Name ____________________ This exam paper must not be removed from the venue First Name ____________________ School of Mathematics & Physics Semester Two Examinations, 2023 MATH4407 Advanced Ordinary and Partial Differential Equations This paper is for St Lucia Campus students. Examination Duration: 120 minutes For Examiner Use Only Planning Time: 10 minutes Question Mark 1 Exam Conditions: 2 This is a Closed Book examination - no written materials permitted Casio FX82 series or UQ approved and labelled calculator only 3 During Planning Time - Students are encouraged to review and plan 4 responses to the exam questions This examination paper will be released to the Library 5 6 Materials Permitted in the Exam Venue: (No electronic aids are permitted e.g. laptops, phones) None Materials to be supplied to Students: Additional exam materials (e.g. answer booklets, rough paper) will be provided upon request. None Instructions to Students: If you believe there is missing or incorrect information impacting your ability to answer any question, please state this when writing your answer. Answer all questions. Total marks are 55. Total _________ Page 1 of 14 Semester Two Examinations, 2023 MATH4407 Question 1 (9 marks) Consider the system dx = y + x − 2x(x2 + y 2 ), dt dy = −x + y − 2y(x2 + y 2 ). dt Using the Poincare-Bendixon theorem, show that the system has at least one nonconstant periodic orbit Working space on next page. Page 2 of 14 Semester Two Examinations, 2023 MATH4407 Question 1 (continued) Question 2 on next page. Page 3 of 14 Semester Two Examinations, 2023 MATH4407 Question 2 (9 marks) (a) Consider the system in R2 dx = y − µx3 + x, dt dy = −x, dt where µ is a constant in R. Show that: (i) For any µ < 0, the system has no periodic solution. (ii) For any µ > 0, the system has a non-constant periodic solution. (b) Consider the equation ẋ = A(t)x with x ∈ R2 and   1 − cos t b A(t) = a 2 + sin t with two constants a, b. Show that there exists at least one parameter family of solutions which becomes unbounded as t → ∞. Working space on next page. Page 4 of 14 Semester Two Examinations, 2023 MATH4407 Question 2 (continued) Question 3 on next page. Page 5 of 14 Semester Two Examinations, 2023 MATH4407 Question 3 (9 marks) Assume that U is a bounded open set in Rn and ∂U ∈ C 1. Let p be a real number with 2 ≤ p < ∞. Prove the following inequality: Z Z 1/2 Z 1/2 |Du| dx ≤ C p |u| dx p |D u| dx 2 p U U U for all u ∈ W 2,p (U ) ∩ W01,p (U ), where C is independent of u. Working space on next page. Page 6 of 14 Semester Two Examinations, 2023 MATH4407 Question 3 (continued) Question 4 on next page. Page 7 of 14 Semester Two Examinations, 2023 MATH4407 Question 4 (9 marks) Fix a > 0. Let B be the unit ball in Rn and let p be a real number with 2 ≤ p ≤ n. Show there exists a constant C > 0, depending only on n, p and a, such that Z Z |u| dx ≤ C p |Du|p dx B B for each u ∈ W 1,p (B) satisfying |{x ∈ B : u(x) = 0}| ≥ a > 0. Working space on next page. Page 8 of 14 Semester Two Examinations, 2023 MATH4407 Question 4 (continued) Question 5 on next page. Page 9 of 14 Semester Two Examinations, 2023 MATH4407 Question 5 (9 marks) Let U be an open, bounded set in Rn with ∂U ∈ C 1. Assume that u is a smooth solution of X n Lu = − aij uxi xj = f in U, u = 0 on ∂U, i,j=1 where f is bounded and L is elliptic in Ū. Fix x0 ∈ ∂U. A barrier at x0 is a C 2 function w such that Lw ≥ 1 in U, w(x0 ) = 0, w ≥ 0 on ∂U. Show that if w is a barrier at x0 , there exists a constant C such that ∂w 0 |Du(x0 )| ≤ C (x ) , ∂ν where ν is the outer normal vector of ∂U. Working space on next page. Page 10 of 14 Semester Two Examinations, 2023 MATH4407 Question 5 (continued) Question 6 on next page. Page 11 of 14 Semester Two Examinations, 2023 MATH4407 Question 6 (10 marks) Let U be a bounded, open set of Rn with smooth boundary ∂U. Assume that u is a smooth solution of ut − ∆u = 0 in U × (0, ∞) u = 0 on ∂U × [0, ∞) u=g on U × {t = 0}. Prove the exponential decay estimate: ∥u(·, t)∥L2 (U ) ≤ e−λ1 t ∥g∥L2 (U ) (t ≥ 0), R where λ1 := min{ U |Dv|2 dx : v ∈ H01 (U ), ∥v∥L2 = 1} > 0 is the principle eigenvalue of −∆ (with zero boundary conditions) on U. Working space on next page. Page 12 of 14 Semester Two Examinations, 2023 MATH4407 Question 6 (continued) Working space on next page. Page 13 of 14 Semester Two Examinations, 2023 MATH4407 Question 6 (continued) END OF EXAMINATION. Page 14 of 14