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PDE Qualifying Exam Fall 2015
1. (20 points) Let u be a smooth function that solves. the wave equation
U t t -L:.u
=
0 in !Rn x (O,oo),· where the integer n ;?: 2. Let x E !Rn, t
>
0, r>
0. Define U(x; r, t) :=t
u(y, t)dS(y),8B x,r)
I
the average of u(-, t) over 'the sphere 8B(x, r). Show that for fixed x E JR, U satisfies
Utt-Urr- --Ur n-1
=
0r in JR+ x (O,oo).
2. (20 points) State and prove the Lax-Milgram Theorem.
3. (20 points) Let U c JRN be a smooth and bounded open set and suppose
f
E L2(U), where N is a positive integer. Show that there exists a unique weak solution u E HJ(U) of the following boundary-value problem· { L:.u
= f
· u=O
in U, on 8U.
4. (20 points) Do you think the following second order differential equations have periodic solutions ? Prove or disprove it.
x"(t)
+
x(t)-x2(t)=
0,x"(t)
+
x3(t)+
x4(t)=
0.5. (20 points) Solve the follo'wing partial differential equation
, { UUx1
+
Ux2=
1,u(x1,x1)
= ~x1.
1
in JR2,
(0.1) (0.2)
