Department of Mathematics University of Utah Written Qualifying Examination in DIFFERENTIAL EQUATIONS September 18, 1992

Let \(E\) be an open subset of \(\mathbb {R}^n\). Let \(f\) be a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\), and let \(x_0\) belong to \(E\).

1. Use the method of successive approximations to prove that the initial value problem \(x'=f(x)\), \(x(0)=x_0\) has a unique solution on some interval \([-a,a]\), and give an estimate for \(a\).
2. Suggest how the result in a) can be used to prove a similar result for the nonautonomous case.

Let \(A\) be a constant \(n\times n\) matrix, and define the matrix exponential by \[ e^A=\sum _{k=0}^{\infty }A^k/k!. \]

1. Prove that \(e^{At}\) is differentiable (you may use known theorems about scalar power series), and that it satisfies the initial value problem \(X'=AX\), \(X(0)=I\) (the identity matrix). Justify your steps.
2. Let \(A\) have distinct eigenvalues. Prove that for the origin to be a globally asymptotically stable point for \(X'=AX\) it is sufficient that all of the eigenvalues of \(A\) have negative real parts.
3. Is the condition in b) that \(A\) have distinct eigenvalues required in order to prove the result? (no proof necessary)
4. Is the sufficient condition in b) also necessary? Justify your answer.

DE Qualifying Examination 1992

1. What does Floquet’s Theorem say about solutions of linear \(n\times n\) systems \(X'=A(t)X\) (\(A\) is a continuous \(n\times n\) matrix)?
2. Show that, given any \(n\times n\) matrices \(B\) (constant) and \(A(t)\) (continuous), there is a change of variables \(Y=P(t)X\) which reduces the system \(X'=A(t)X\) to the system \(Y'=BY\).
3. Show that there is a particular choice of \(P(t)\) and \(B\) that is related to Floquet’s Theorem. Why might you be interested in that choice?

1. Show that the only equilibrium point of the system \[ x'=-2y-4x^3-4xy^2,\qquad y'=x-2x^2y-2y^3 \] is the origin.
2. Draw what conclusions you can about stability, asymptotic stability, and regions of attraction, using first i) linearization and then ii) Lyapunov functions. Make clear the reasons for your conclusions.
3. What if the signs of the nonlinear terms in a) were changed?
4. What kinds of functions \(f\) and \(g\) would give similar results for systems of the form \(x'=-2y+f(x,y)\), \(y'=x+g(x,y)\)?

The quasilinear Cauchy problem \[ xu_x+yuu_y+xy=0,\qquad u=5\quad \text {when }xy=1, \] has the solution \[ u(x,y)=-1+\sqrt {38-2xy}. \]

1. Derive this solution by the method of characteristics. (Hint: Along a characteristic \(x(s)\), \(y(s)\), \(u(s)\) find \(d(xy)/ds\) and then \(du/dz\) where \(z=xy\).)
2. Verify that \(u(x,y)\) is a solution in the plane domain defined by \(xy<19\).

Consider the Dirichlet problem in a bounded plane domain \(\Omega \): \[ u_{xx}+u_{yy}=0\quad \text {in }\Omega , \] \[ u=f\quad \text {on }\partial \Omega , \] where \(f\) is continuous on \(\partial \Omega \).

1. Prove that this Dirichlet problem has at most one solution. Define clearly what you mean by a solution.

2. Prove that if \(\Omega \) is a bounded domain for which the Dirichlet problem is solvable for all \(f\in C(\partial \Omega )\) then the mapping \(f\in C(\partial \Omega )\longrightarrow u\in C(\overline {\Omega })\) is continuous in the uniform norms. (\(\overline {\Omega }\) denotes the closure of \(\Omega \).)

DE Qualifying Examination 1992

Consider the Dirichlet problem \[ \Delta u=f\quad \text {in }\Omega , \] \[ u=0\quad \text {on }\partial \Omega , \] where \(\Omega \subset \mathbb {R}^n\) is a bounded open set and \(f\in L_2(\Omega )\).

1. Define the concept of a weak \(H_0^1(\Omega )\) solution of the above problem.
2. Use the Poincaré inequality, \[ \int _\Omega u^2\,dx\le c\int _\Omega \lvert \nabla u\rvert ^2\,dx \quad \text {for all }u\in H_0^1(\Omega ), \] to prove the existence of a weak \(H_0^1(\Omega )\) solution.

Consider the integral \[ u(x,t)=\frac {1}{2\sqrt {\pi t}}\int _{-\infty }^{\infty } f(\xi )e^{-(x-\xi )^2/4t}\,d\xi . \]

1. Prove that if \(f(x)\) is continuous and bounded for all \(x\) then \(u\) is a solution of the initial value problem \[ u_t=u_{xx}\quad \text {for all }-\infty <x<\infty ,\ t>0. \] \[ u(x,0)=f(x)\quad \text {for all }-\infty <x<\infty . \] Formulate clearly what you mean by a solution.
2. Show how the results of part a) can be used to solve the initial-boundary value problem \[ u_t=u_{xx}\quad \text {for all }0<x<1,\ t>0, \] \[ u(x,0)=g(x)\quad \text {for }0\le x\le 1, \] \[ u(0,t)=u(1,t)=0\quad \text {for }t\ge 0. \] Define what you mean by a solution and state what conditions on \(g\) guarantee that such a solution exists.

DE Qualifying Examination 1992

Consider the wave equation \(u_{tt}=u_{xx}\).

1. State and derive the D’Alembert formula for the solutions \(u\) of this equation.
2. Use the method of separation of variables to solve the initial-boundary value problem \[ u_{tt}=u_{xx}\quad \text {for }0<x<1,\ t>0, \] \[ u(x,0)=f(x),\quad u_t(x,0)=0\quad \text {for }0\le x\le 1, \] \[ u(0,t)=u(1,t)=0\quad \text {for }t\ge 0. \] Define what you mean by a solution.
3. Show how the solution of b) can be written in the D’Alembert form.

Let \(A\) be a selfadjoint operator in a Hilbert space \(\mathcal {H}\) and let \(\{E_\lambda \}\) be the spectral family of \(A\). Prove that if a real number \(\mu \) is in the resolvent set of \(A\) then there exists an interval \(I=\{\lambda :\lvert \lambda -\mu \rvert <\delta \}\) such that \(E_\lambda =\text {const.}\) for all \(\lambda \in I\). (You may use without proof the Stieltjes-Hellinger-Stone theorem: \[ R_z=(A-z)^{-1}. \] \[ \bigl ((E_{b+}+E_{b-}-E_{a+}-E_{a-})u,u\bigr ) =\lim _{\epsilon \to 0+}\frac {1}{\pi i} \int _a^b\bigl ((R_{\lambda +i\epsilon }-R_{\lambda -i\epsilon })u,u\bigr )\,d\lambda . \]

Consider the regular Sturm-Liouville operator \[ \mathcal {L} u=-e^{-2x}(e^{2x}u')'-u \] on \(0\le x\le \pi \) with boundary conditions \(u(0)=0\), \(u(\pi )=0\).

1. Show how \(\mathcal {L} u\) and the boundary conditions can be used to construct a selfadjoint operator \(S\) in a suitable Hilbert space \(\mathcal {H}\).
2. Calculate the eigenvalues and the corresponding eigenfunctions of \(S\).

The singular Sturm-Liouville operator \(Lu=-u''\) on \(-\infty <x<\infty \) is known to have a unique selfadjoint realization \(T\) in the Hilbert space \(H=L_2(-\infty ,\infty )\). It is defined by \(D(T)=H\cap \{u:u''\in H\}\) and \(Lu=-u''\) for all \(u\in D(T)\). (Do not prove this). Calculate the Green’s function (resolvent kernel) of \(T\).