Differential Equations Preliminary Examination

Let \(E\) be an open subset of \(\mathbb {R}^n\) containing \(x_0\) and assume that \(\mathbf {f}\) is a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\).

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

Let \(E\) be an open subset of \(\mathbb {R}^n\) containing the origin and \(x_0\). Assume that \(\mathbf {f}\) is a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\). Suppose \(\mathbf {f}(0)=0\) and \(A=D\mathbf {f}(0)\). Let \(U\) and \(V\) be subsets of \(E\) containing the origin.

1. Define the maximal interval of existence \(I(x_0)\).
2. Define the flow \(\phi _t(x_0)\).
3. Define a homeomorphism \(H\) of \(U\) onto \(V\).
4. Suppose \(x_0\in U\). State the Hartman-Grobman theorem in terms of \(\phi _t\), \(A\), \(H\) and \(x_0\). Be sure to include all necessary conditions.
5. Consider \(x=(x_1,x_2)\) with \[ \begin {aligned} \dot {x}_1&=-x_1,\\ \dot {x}_2&=x_2-x_1^2, \end {aligned} \] and \(x(0)=x_0\). Write down the explicit solution to this equation. Give the linearized system \(\dot {y}=Ay\), \(y(0)=y_0\) and its explicit solution. Give the homeomorphism \(H\) which relates the two solutions.

1. Prove Bendixson’s Criterion: Let \(\mathbf {f}\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(\mathbf {f}\) is not identically zero and is either nonnegative in \(E\) or nonpositive in \(E\). Then \(\dot {x}=\mathbf {f}(x)\) has no closed orbit which is entirely in \(E\).
2. Extend the above proof to cover Dulac’s Criterion: Let \(B(x_1,x_2)\) and \(\mathbf {f}\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(B\mathbf {f}\) is not identically zero and is either nonnegative in \(E\) or nonpositive in \(E\). Then \(\dot {x}=\mathbf {f}(x)\) has no closed orbit which is entirely in \(E\).
3. Consider the nondimensionalized Lotka-Volterra competition equations \[ \begin {aligned} \dot {u}&=ru(1-u-\alpha _1v),\\ \dot {v}&=v(1-v-\alpha _2u), \end {aligned} \] where \(r\), \(\alpha _1\) and \(\alpha _2\) are positive parameters. Apply Dulac’s criterion to this system for \(u,v>0\). (HINT: try \(B(u,v)=1/uv\).) Does Bendixson’s criteria give a similar result?
4. Draw a biological conclusion.

Consider \(\dot {x}=\mathbf {f}(x)\) where \(x\in \mathbb {R}^n\) and \(\mathbf {f}\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

1. Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
2. Consider the nonlinear spring equation \[ x''+\mu x'+g(x)=0 \] where \(g(x)\) is an odd function of \(x\), \(g'(x)>0\) and \(\mu \ge 0\). Translate this equation to the \(\dot {x}=\mathbf {f}(x)\) form and linearize about the trivial equilibrium. What conclusions can be made about the stability of this equilibrium?
3. Define what is meant by a Lyapunov function for the above equation; derive one and verify that it is a Lyapunov function. Describe the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\) for the case \(\mu =0\) and for the case \(\mu >0\). Make a conclusion about the stability of the trivial equilibrium for each of these two cases.

Consider the system \[ \begin {aligned} \dot {x}_1&=x_2,\\ \dot {x}_1&=-x_1+x_2(1-x_1^2-2x_2^2). \end {aligned} \] Prove, by means of the Poincaré-Bendixson theorem, that there exists at least one periodic orbit in the annulus \(1/2<x_1^2+x_2^2<1\).

Consider the equation \[ a(x,y)u_x+b(x,y)u_y=c(x,y,u) \] where \(a\), \(b\) and \(c\) are suitably smooth.

1. Define what is meant by an integral surface for the above equation.
2. Write down the characteristic equations and show how they can be used to construct an integral surface that contains the curve \(x=\phi (s)\), \(y=\psi (s)\), \(u=\eta (s)\), where \(\phi \), \(\psi \) and \(\eta \) are \(C^1\).
3. What additional conditions, if any, must be placed on the data to guarantee a locally unique solution? Explain carefully.
4. Demonstrate the above method by solving \[ \begin {aligned} u_x+2u_y&=-u,\\ u(0,y)&=f(y) \end {aligned} \] where \(f\) is of class \(C^1\).

Let \(\Omega \) be a bounded open set. Define the Sobolev space \(H^{1,2}_0(\Omega )\). Indicate why this is a Hilbert space. State and prove Poincaré’s inequality for functions \(u\in H^{1,2}_0(\Omega )\) and thus verify that \(H^{1,2}_0(\Omega )\) is also a Hilbert space with respect to the inner product \[ (u,v)=\int _\Omega \nabla u\cdot \overline {\nabla v}\,dx. \]

Let \(\Omega \) be a bounded domain in \(\mathbb {R}^n\). Consider \[ \begin {aligned} \Delta u&=f &&\text {in }\Omega ,\\ u&=0 &&\text {on }\partial \Omega \end {aligned} \] where \(f\in L^2(\Omega )\).

1. Define the concept of a weak \(H^{1,2}_0(\Omega )\) solution to the above problem.
2. Use Poincaré’s inequality to prove existence of a weak \(H^{1,2}_0(\Omega )\) solution.
3. Assume that \(\Omega \) is smooth. Prove that any two solutions in \(C^2(\Omega )\cap C^1(\overline {\Omega })\) must agree. How would you prove this result for any two solutions in \(C^2(\Omega )\cap C(\overline {\Omega })\)?

Give the heat equation on a semi-infinite rod with initial temperature \(g(x)\) and Dirichlet boundary conditions. Apply the method of odd extension of the heat kernel formula for \((-\infty ,\infty )\) to derive a formula for the solution.

1. State d’Alembert’s formula for the solutions \(u\) of the wave equation \(u_{tt}=u_{xx}\).
2. Consider the initial boundary value problem \[ \begin {aligned} u_{tt}&=u_{xx} &&\text {for }0<x<\pi ,\ t>0,\\ u(x,0)&=f(x),\qquad u_t(x,0)=0 &&\text {for }0<x<\pi ,\\ u(0,t)&=0,\qquad u(\pi ,t)=0 &&\text {for }t\ge 0. \end {aligned} \] Use d’Alembert’s formula and the parallelogram rule to define a weak solution solution to this problem. HINT: divide the region \(R=\{(x,t):0<x<\pi ,\ t>0\}\) into subregions. What compatibility conditions are needed upon \(f\) to ensure that the above solution is \(C\), is \(C^1\) and is \(C^2\)?
3. Show that the energy \[ E(t)=\frac {1}{2}\int _0^\pi (u_t^2+u_x^2)\,dx \] is conserved, and use this to prove that a \(C^2\) solution to the above initial boundary value problem is unique.