DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN DIFFERENTIAL EQUATIONS Winter 2005

Contraction Mapping Theorem. Show that the problem \[ \begin {aligned} \dot {x}(t) &= f(x(t))+\frac {1}{a}\int _0^a x(s)\,ds,\quad t\in (0,a),\\ x(0) &= 0, \end {aligned} \] admits a unique solution \(x\in C([0,a];\mathbb {R}^n)\) for \(a>0\) sufficiently small. Assume \(f:\mathbb {R}^n\to \mathbb {R}^n\) satisfies \(\lvert f(x)-f(y)\rvert \le M\lvert x-y\rvert \) for all \(x,y\in \mathbb {R}^n\).

Limit Sets. Let \(\varphi _t(x_0)\) be the flow of the differential equation \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\in C^1(\mathbb {R}^n)\) satisfies a global Lipschitz condition, and \(x_0\in \mathbb {R}^n\).

1. Given a trajectory \(\Gamma \) of \(\varphi _t\), define the \(\omega \)-limit set \(\omega (\Gamma )\).
2. Prove that \(\omega (\Gamma )\) is closed and invariant with respect to the flow \(\varphi _t\).

Periodic Orbits. Consider the system \[ \begin {aligned} \dot {x} &= -y+x(1-x^2-y^2),\\ \dot {y} &= x+y(1-x^2-y^2),\\ \dot {z} &= z. \end {aligned} \]

1. Find the invariant manifolds for this system.
2. Find the unique periodic orbit \(\Gamma \) for this system.
3. Find the global stable manifold \(W^s(\Gamma )\) and global unstable manifold \(W^u(\Gamma )\) with respect to the periodic orbit \(\Gamma \).

Linear Systems. Consider the linear initial value problem \[ \begin {aligned} \dot {x} &= Ax,\\ x(0) &= x_0, \end {aligned} \] where \(A\in \mathbb {R}^{n\times n}\) is constant, and \(x_0\in \mathbb {R}^n\) is given.

1. Prove that the matrix exponential \(e^{At}\) exists and satisfies \(\lVert e^{At}\rVert \le e^{\lVert A\rVert t}\), where \(\lVert \cdot \rVert \) is the operator norm.
2. Prove directly that \(x(t)=e^{At}x_0\) is the unique solution to the initial value problem above.

Floquet Theory. Consider the system \[ \dot {x}(t)=A(t)x(t), \] where \(A:\mathbb {R}\to \mathbb {R}^{n\times n}\) is continuous and \(T\)-periodic, i.e., \(A(t+T)=A(t)\) for all \(t\in \mathbb {R}\). Let \(\Phi (t)\) be the fundamental matrix solution for this system. Assume that there exists a complex-valued matrix \(B\) such that \(e^{BT}=\Phi (T)\).

1. Explain in detail how solutions \(x(t)\) to the periodic system above are related to solutions \(y(t)\) of the constant-coefficient problem \(\dot {y}=By\), and prove your assertions.
2. Explain what the eigenvalues of \(A(t)\) or \(B\) have to do with stability of the solutions \(x(t)\).

Dependence on Initial Conditions. Consider the scalar initial value problem \[ \begin {aligned} y'' &= y-y^3,\\ y(0) &= a,\\ y'(0) &= b. \end {aligned} \] Denote the dependence of \(y(t)\) on its initial conditions by \(y(t;a,b)\). Without explicitly solving for \(y(t;a,b)\), find \(\frac {\partial y}{\partial a}(t;0,0)\) and \(\frac {\partial y}{\partial b}(t;0,0)\), then write down a first-order approximation for \(y(t;a,b)\) which is valid for small \(\lvert a\rvert ,\lvert b\rvert \).

Variational Principles. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary \(\partial \Omega \) and unit outward normal \(\eta \). Consider the problem \[ \begin {aligned} \Delta u-\epsilon u &= f,\quad \text {in }\Omega ,\\ \frac {\partial u}{\partial \eta } &= 0,\quad \text {on }\partial \Omega , \end {aligned} \] where \(f\in L^2(\Omega )\) satisfies \(\int _\Omega f=0\).

1. Prove that for each real constant \(\epsilon >0\), there exists a unique weak solution \(u_\epsilon \in H^1(\Omega )\).
2. In the case \(\epsilon =0\), find an additional condition on \(u\) which guarantees a unique weak solution \(u_0\), such that the solutions \(u_\epsilon \) from part (a) converge to \(u_0\) as \(\epsilon \to 0^+\). Prove existence and uniqueness of \(u_0\) (with the extra condition), and prove \(u_\epsilon \to u_0\) in \(H^1(\Omega )\) as \(\epsilon \to 0^+\).

Heat Equation. Given the constant \(\beta \in \mathbb {R}\), consider the initial value problem \[ \begin {aligned} u_t &= \beta \Delta u,\quad x\in \mathbb {R}^n,\quad t>0,\\ u(x,0) &= g(x),\quad x\in \mathbb {R}^n. \end {aligned} \]

1. Assume \(\beta >0\) and let \(g\in C(\mathbb {R}^n)\) be bounded. Prove that there exists a bounded solution \(u(x,t)\) which is \(C^\infty \) for all \(t>0\). Under what condition are solutions unique?
2. Assume \(\beta <0\) and let \(g\in C^\infty (\mathbb {R}^n)\) be bounded. Are solutions \(u(x,t)\) necessarily bounded and continuous for \(t>0\)? Prove, or provide a counterexample.

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be open. Suppose \(u\in C(\Omega )\) satisfies the mean value property: for every ball \(B_r(\xi )=\{x\in \mathbb {R}^n:\lvert x-\xi \rvert <r\}\) with \(B_r(\xi )\subset \Omega \), \[ u(\xi )=\frac {1}{\omega _n}\int _{\lvert x\rvert =1}u(\xi +rx)\,dS_x, \] where \(\omega _n\) is the measure of the \((n-1)\)-dimensional unit sphere in \(\mathbb {R}^n\). Prove that \(\Delta u=0\) in \(\Omega \).

Conservation Laws. Consider the first-order equation \[ (G(u))_x+u_y=0, \] where \(G\in C^\infty \), with initial condition \(u(x,0)=h(x)\).

1. Using the method of characteristics, derive the general solution of the initial value problem. Assuming \(h\in C^\infty \), find a condition on \(G'(h(x))\) which implies the solution \(u(x,y)\) will develop a singularity in finite time \(y>0\).
2. Set \(G(u)=\frac {1}{2}u^2\). Given \(u_0\in \mathbb {R}\), find explicit weak solution(s) for \[ h(x)= \begin {cases} u_0, & x<0,\\ 0, & x>0. \end {cases} \] Show that your solutions are weak solutions. Which properties do your solutions exhibit: shocks, rarefaction, nonuniqueness?

Distributions. Let \(\Omega \subset \mathbb {R}^n\) be an open set.

1. Let \(\{f_n\}\subset C(\Omega )\), with \(f_n\) converging to \(f\in C(\Omega )\) pointwise, i.e., \(f_n(x)\to f(x)\) for all \(x\in \Omega \). Does it follow that \(f_n\) converges to \(f\) in the sense of distributions? Prove or give a counterexample.
2. Let \(\{\mu _n\}\subset \mathcal {D}'(\Omega )\) with \(\mu _n\) converging to \(\mu \in \mathcal {D}'(\Omega )\) in the sense of distributions. Let \(\alpha \) be a fixed multi-index. Does it follow that the derivatives \(D^\alpha \mu _n\) converge to \(D^\alpha \mu \) in the sense of distributions? Prove or give a counterexample.
3. Find a sequence of \(L^1(\mathbb {R})\) functions which do not converge in \(L^1(\mathbb {R})\) but which do converge in the sense of distributions.

Wave Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary. Consider the initial boundary value problem \[ \begin {aligned} u_{tt} &= \Delta u,\quad x\in \Omega ,\quad t>0,\\ u(x,0) &= g(x),\quad x\in \Omega ,\\ u_t(x,0) &= 0,\quad x\in \Omega ,\\ u(x,t) &= 0,\quad x\in \partial \Omega ,\quad t>0. \end {aligned} \] Show that there exists a sequence of initial conditions \(\{g_j\}\subset C^2_0(\overline {\Omega })\) and a sequence of frequencies \(\{\omega _j\}\subset \mathbb {R}\) with \(\omega _j^2\to \infty \), such that each solution \(u^{(j)}(x,t)\) corresponding to initial condition \(g_j\) satisfies \[ u^{(j)}_{tt}(x,t)=-\omega _j^2u^{(j)}(x,t),\quad x\in \Omega ,\quad t>0. \] Moreover, prove that there is at most a countable set of \(\omega \) for which \(u_{tt}=-\omega ^2u\), no matter what initial conditions \(g\) are chosen.