DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN DIFFERENTIAL EQUATIONS Autumn 2004

Contraction Mapping. Consider the initial value problem \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\) satisfies for all \(x,y\in \mathbb {R}^n\) \[ \lvert f(x)-f(y)\rvert \le (1+\lvert x\rvert +\lvert y\rvert )\lvert x-y\rvert . \] Use the contraction mapping principle to find a condition on \(\lvert x_0\rvert \) which guarantees local existence and uniqueness of solutions. How does the local time interval upon which existence is obtained depend on \(\lvert x_0\rvert \)?

Global Existence. 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 \(f\in C^1(E)\), \(E\subset \mathbb {R}^n\) is open, and \(x_0\in E\). Prove that if the system has a Liapunov function \(V\) with \(\dot {V}(x)=\left .\frac {d}{dt}V(\varphi _t(x))\right |_{t=0}\le 0\) for all \(x\in E\), and the set \(S=\{x\in E:V(x)\le 1\}\) is compact, then solutions \(\varphi _t(x_0)\) with \(x_0\in S\) exist for all positive times \(t>0\).

Periodic Orbits. A model of an autocatalytic chemical reaction is given by the Brusselator system \[ \begin {aligned} \frac {dx}{dt} &= 1-4x+x^2y,\\ \frac {dy}{dt} &= 3x-x^2y. \end {aligned} \tag {1} \] Show that the trapezoidal region with vertices \((\frac {1}{4},0)\), \((13,0)\), \((1,12)\), \((\frac {1}{4},12)\), is an invariant set for (1). Show that the system has a nonconstant periodic trajectory in \(\mathbb {R}^2\).

Linear Systems. Suppose \(A\) is a real \(n\times n\) matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i\le 0\). Show that there are constants \(C<\infty \) and \(k\le n-1\) so that any solution \(x(t)\in \mathbb {R}^n\) of the initial value problem \[ \begin {aligned} \frac {dx}{dt} &= Ax,\\ x(0) &= x_0, \end {aligned} \] satisfies \[ \lvert x(t)\rvert \le C(1+\lvert t\rvert ^k)\lvert x_0\rvert \] for all \(t\in \mathbb {R}\) and \(x_0\in \mathbb {R}^n\).

Linearization. Assume \(g\in C^\infty (\mathbb {R}^n)\) has a strict local minimum at the origin. Consider the second-order system \[ \ddot {x}+B\dot {x}+\nabla g(x)=0, \] where \(x(t)\in \mathbb {R}^n\) and \(B\in \mathbb {R}^{n\times n}\) is a symmetric positive definite matrix.

1. Rewrite the system as a first-order system.
2. Write the linearization of the system at the origin.
3. Prove that the origin is an asymptotically stable fixed point.

Stable and Unstable Manifolds. Find the stable and unstable manifolds in the neighborhood of the origin for the system \[ \begin {aligned} x' &= -x,\\ y' &= y+x^2,\\ z' &= z+y^2. \end {aligned} \] How are the stable and unstable manifolds for the linearization of this system related to those of the nonlinear system?

Variational Principles. Suppose \(\Omega \in \mathbb {R}^n\) is a smooth, bounded domain. Show that there is an \(\epsilon >0\) depending on \(n\) and \(\Omega \) such that if \(f,g\in C(\overline {\Omega })\) are continuous functions such that \(f\le \epsilon \) then the equation \[ \begin {aligned} \Delta u+f(x)u &= g(x),\quad \text {for }x\in \Omega ,\\ u(x) &= 0,\quad \text {for }x\in \partial \Omega , \end {aligned} \] has a weak solution \(u\in H^1_0(\Omega )\).

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary \(\partial \Omega \). Let \(f\in C^2(\overline {\Omega })\), \(g\in C^\infty (\partial \Omega )\), and assume \(f=g\) on \(\partial \Omega \). Let \(u(x,t)\) solve \[ \begin {aligned} u_t &= \Delta u,\quad x\in \Omega ,\quad t>0,\\ u(x,0) &= f(x),\quad x\in \overline {\Omega },\\ u(x,t) &= g(x),\quad x\in \partial \Omega ,\quad t>0. \end {aligned} \] Prove that \[ \lim _{t\to \infty }\int _\Omega \lvert u(x,t)-v(x)\rvert ^2\,dx=0 \] where \(v\) solves \[ \begin {aligned} \Delta v &= 0,\quad \text {in }\Omega ,\\ v &= g,\quad \text {on }\partial \Omega . \end {aligned} \]

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be a domain. Let \(\Omega _1\) be a bounded subdomain with \(\overline {\Omega }_1\subset \Omega \).

1. State the Harnack Inequality for harmonic functions.
2. Let \(\{u_n\}\) be a monotone increasing sequence of nonnegative harmonic functions on \(\Omega \). Suppose that for some point \(\xi \in \Omega _1\), the sequence of real numbers \(\{u_n(\xi )\}\) is bounded. Prove that \(\{u_n\}\) converges uniformly on \(\Omega _1\) to a harmonic function. You may use standard theorems about convergence of sequences of harmonic functions without proof.

Conservation Laws. A model for flow of fluid in an infinite pipe says that the density \(\rho (x,t)\) satisfies \[ \frac {\partial \rho }{\partial t}+\rho \frac {\partial \rho }{\partial x}=-\rho ^2,\qquad \rho (x,0)= \begin {cases} 1, & \text {if }x\le 0,\\ \frac {1}{2}, & \text {if }x>0. \end {cases} \]

1. Find a solution \(\rho \).
2. Define what it means for your solution \(\rho \) from part (a) to be a weak solution of the system. (Do not carry out the calculations to prove that \(\rho \) is a weak solution.)

Distributions. Suppose \(f(\xi ),g(\xi )\in C^2_0(\mathbb {R})\) are compactly supported twice continuously differentiable functions. Then \[ u(x,t)=f(x+t)+g(x-t) \tag {2} \] is a classical solution of the wave equation \[ u_{tt}-u_{xx}=0\qquad \text {for }-\infty <x,t<\infty . \tag {3} \] Show that if \(f(\xi ),g(\xi )\in C_0(\mathbb {R})\) are merely compactly supported continuous functions, then (2) still defines a solution to (3) in the sense of distributions.

Wave Propagation. Consider the equation \[ u_{tt}-u_{xx}+\lambda u=0,\quad x\in \mathbb {R},\quad t>0, \] where \(\lambda >0\) is fixed.

1. Explain what is meant by a dispersive solution, and find a dispersion relation for the equation.
2. Define an “energy” functional \(E(t)\) for this equation and show that each solution \(u(x,t)\) with finite initial energy conserves energy. State conditions under which uniqueness of solutions holds.