DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN DIFFERENTIAL EQUATIONS Autumn 2003

Peano Existence Theorem. Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Consider the system of ODE’s \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \qquad (\spadesuit ) \]

1. Find an example that shows that the solution of \((\spadesuit )\) may not be unique. Give another example that shows that the solution may not exist for all time.
2. Prove the Peano Existence theorem: Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R})\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Then there is an \(\epsilon >0\) and a function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) which solves \((\spadesuit )\).

Picard’s Existence Theorem. Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Suppose that there is a continuous function \(m(t)\in C(\mathbb {R})\) such that \(\lvert f(x,t)\rvert \le m(t)(1+\lvert x\rvert )\) for all \((x,t)\in \mathbb {R}^n\times \mathbb {R}\). Show that the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \] has a unique solution \(y\in C^1(\mathbb {R},\mathbb {R}^n)\) which exists for all time.

Periodic Orbits. Show that the following system has a nonconstant periodic trajectory. \[ \begin {cases} \dfrac {dx}{dt}=\bigl [3-\sin (x^2+y^2)\bigr ]y,\\ \dfrac {dy}{dt}=-3x+\cos (x^2+y^2)y. \end {cases} \]

Stability of Solutions. The following questions have to do with the stability of the zero solution \(z(t)=0\) to a system of ODE’s \[ y'=f(t,y), \] where \(f(t,x)\in C^\infty (\mathbb {R}\times \mathbb {R}^n,\mathbb {R}^n)\) such that \(f(t,0)=0\) for all \(t\).

1. Define: \(z\) is Liapunov Stable. Define: \(z\) is asymptotically stable.
2. Give an example of an ODE system such that the zero solution is Liapunov stable but not asymptotically stable.
3. Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i<0\) for all \(i=1,\ldots ,n\). Let \(g(x)\in C^\infty (\mathbb {R}^n,\mathbb {R}^n)\) such that there are constants \(1<\beta ,c<\infty \) so that \(\lvert g(x)\rvert \le c\lvert x\rvert ^\beta \) for all \(x\in \mathbb {R}^n\). Show that the zero solution \(z(t)\) is asymptotically stable in the equation \[ y'=Ay+\sin (t)g(y). \]

Periodic Coefficients. Prove that the equation \[ \ddot {u}+(\sin ^2 t)\dot {u}+(1+\sin 2t)u=0 \] does not have a fundamental set of periodic solutions.

Two Species Competition Model. Consider the simplified competition model for two populations \(x(t)\ge 0\) and \(y(t)\ge 0\), \[ \begin {cases} \dot {x}=x(1-x-3y),\\ \dot {y}=3y(1-y-2x). \end {cases} \] Its equilibrium points are \((0,0)\), \((0,1)\), \((1,0)\) and \((.4,.2)\). The equilibria \((0,1)\) and \((1,0)\) are stable, but \((0,0)\) is unstable. This system has a seperatrix, a curve which divides the first quadrant of the plane into two basins of attraction corresponding to the two stable equilibria. Thus one species or the other wins out, depending which has the starting advantage. As a first step in showing that \((.4,.2)\) lies on the seperatrix, prove that in an open neighborhood \(O\) of \((.4,.2)\) there is a \(C^1\) curve \(\sigma \) through \((.4,.2)\) with the property that trajectories starting on one side of the \(\sigma \) in \(O\), stay on that side of \(\sigma \) while in \(O\).

Conservation Laws. Consider the first order equation \[ \begin {cases} \dfrac {\partial u}{\partial t}+(\tan u)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)=\varphi (x), & \text {for }x\in \mathbb {R}. \end {cases} \qquad (\diamond ) \]

1. Find a solution \(u(x,t)\) to \((\diamond )\) where \(\varphi (x)=\operatorname {Arctan}(x)\).
2. For \[ \varphi (x)= \begin {cases} 0, & \text {if }x<0,\\ \frac {\pi }{4}, & \text {if }x>0, \end {cases} \] find two weak solutions to \((\diamond )\). Prove that one of them is a weak solution.
3. State the entropy (Rankine-Hugoniot) condition. Which weak solution of part (b.) satisfies this condition?

Distributions. Let \(\Omega \subset \mathbb {R}^n\) be an open set and \(\mathcal {D}'(\Omega )\) the space of distributions on \(\Omega \).

1. Let \(\delta \in \mathcal {D}'(\mathbb {R}^n)\) be the Dirac delta function. Fix a multiindex \(\beta \). Show that the derivative \(D^\beta \delta \) is a distribution.
2. Let \(C^\infty _0(\Omega )\) denote the space of compactly supported smooth functions on \(\Omega \). Briefly explain why there exist nonzero \(\varphi \in C^\infty _0(\Omega )\).
3. Suppose \(\rho \in C^\infty _0(\mathbb {R}^n)\) is such that \(\rho \ge 0\), the support is contained in the unit ball around zero, \(\operatorname {spt}\rho \subset B_1(0)\), and \(\int _{\mathbb {R}^n}\rho \,dx=1\). Let \(\eta _h=h^{-n}\rho (x/h)\). Show that as distributions, \(\eta _h\to \delta \) as \(h\to 0\).

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain, \(0<T<\infty \) be a constant, \(c(x)\in C^\infty (\overline {\Omega })\) such that \(c(x)\le 0\) for all \(x\) and \(\psi (x,t)\in C^\infty (\overline {\Omega }\times [0,T])\). Let \(u\in C^{2,1}(\overline {\Omega }\times [0,T])\) be a solution to \[ \begin {cases} \dfrac {\partial u}{\partial t}=\Delta u+c(x)u, & \text {if }(x,t)\in \Omega \times [0,T],\\ u(x,t)=\psi (x,t), & \text {if }(x,t)\in \Omega \times \{0\}\text { or if }(x,t)\in \partial \Omega \times [0,T]. \end {cases} \qquad (\heartsuit ) \]

1. Prove the Maximum Principle: If \(\psi (x,t)\ge 0\) then the solution satisfies \(u(x,t)\ge 0\).
2. Show that such solutions of \((\heartsuit )\) are unique.

Wave Equation. Let \(f(x,t)\in C^\infty (\mathbb {R}^3\times \mathbb {R})\) and \(R>0\) be fixed. Suppose that the support of \(f\) is contained in the cylinder \[ \operatorname {spt} f\subset \{(x,t)\in \mathbb {R}^3\times \mathbb {R}:\lvert x\rvert \le R\text { and }t\ge 0\}. \] Show that there exists a solution \(u(x,t)\) to the inhomogeneous wave equation \((\spadesuit )\). Show that the solution satisfies \(u(x,t)=0\) whenever \(\lvert x\rvert \ge ct+R\). \[ \begin {cases} \dfrac {\partial ^2u}{\partial t^2}=c^2\Delta u+f(x,t), & \text {if }x\in \mathbb {R}^3\text { and }t\ge 0,\\ u(x,0)=0,\\ \dfrac {\partial u}{\partial t}(x,0)=0, & \text {for }x\in \mathbb {R}^3. \end {cases} \qquad (\spadesuit ) \]

Laplace’s Equation. Let \(\varphi (y_1,y_2)\in C_0(\mathbb {R}^2)\) be continuous with compact support. Let \[ u(x)=\frac {x_3}{2\pi }\int _{\mathbb {R}^2} \frac {\varphi (y_1,y_2)\,dy_1\,dy_2} {\bigl [(y_1-x_1)^2+(y_2-x_2)^2+x_3^2\bigr ]^{3/2}}, \] for \(x\in \mathbb {R}^3_+=\{(x_1,x_2,x_3)\in \mathbb {R}^3:x_3>0\}\), the open upper three space. Show that \(u\in C^2(\mathbb {R}^3_+)\cap C(\overline {\mathbb {R}^3_+})\) and solves the boundary value problem for Laplace’s equation \[ \begin {cases} \Delta u=0, & \text {if }x\in \mathbb {R}^3_+,\\ u(x_1,x_2,0)=\varphi (x_1,x_2), & \text {if }(x_1,x_2)\in \mathbb {R}^2. \end {cases} \] [You may assume \[ \int _{\mathbb {R}^2}\frac {x_3\,dy_1\,dy_2} {\bigl [(y_1-x_1)^2+(y_2-x_2)^2+x_3^2\bigr ]^{3/2}}=2\pi . \] ]

Fundamental Elliptic Estimate. Let \(\T ^n=S^1\times \cdots \times S^1\) be the \(n\)-dimensional torus. Let \(u\in C^\infty (\T ^n)\), which is the same as saying \(u\) is a complex valued smooth function on \(\mathbb {R}^n\) which is \(2\pi \)-periodic in each coordinate. Then for \(k\in \mathbb {R}\), the Sobolev norm is given by \[ \lVert u\rVert _k^2=\sum _\alpha (1+\lvert \alpha \rvert ^2)^k\lvert u_\alpha \rvert ^2 \] where \(u\sim \sum _\alpha u_\alpha e^{i\alpha \cdot x}\) is the Fourier series expansion of \(u\) such that the sum runs through all multiindices \(\alpha \in \mathbb {Z}^n\). Let \(H^k(\T ^n)\) be the \(\lVert \cdot \rVert _k\)-completion of the trigonometric polynomials on \(\T ^n\) (finite sums of the form \(p(x)=\sum _{\lvert \alpha \rvert \le N}c_\alpha e^{i\alpha \cdot x}\).)

1. Show that for \(k=2\), there is a constant \(c(k,n)<\infty \) so that whenever \(u\in H^{k+2}(\T ^n)\) then \[ \lVert u\rVert _{k+2}\le c\bigl (\lVert \Delta u\rVert _k+\lVert u\rVert _0\bigr ). \tag {\dagger } \]
2. Supposing that \((\dagger )\) holds for all \(k\), prove that if \(f\in C^\infty (\T ^n)\) such that \(\int _{\T ^n} f\,dx=0\) then there is \(u\in C^\infty (\T ^n)\) such that \[ \Delta u=f\qquad \text {on }\T ^n. \]