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

Picard’s Existence Theorem.

1. Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Show that there is an \(\epsilon >0\) so that the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0 \end {cases} \] has a unique solution \(y\in C^1([t_0-\epsilon ,t_0+\epsilon ],\mathbb {R}^n)\).
2. Suppose that the hypothesis on \(f\) was replaced by \(f(x,t)\in C^0(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\). Would all or part of the conclusions in (a.) continue to hold?

Constant Coefficients. Suppose that \(A\) is a real \(n\times n\) matrix all of whose eigenvalues \(\lambda \) have strictly negative real part \(\operatorname {Re}\lambda <0\). Suppose \[ \frac {dx}{dt}=Ax,\qquad x(0)=x_0. \tag {1.} \]

1. Show that there is a decreasing function \(\delta (t)>0\) such that \(\delta (t)\to 0\) as \(t\to \infty \) and such that every solution of (1.) satisfies \[ \lvert x(t)\rvert \le \delta (t)\lvert x_0\rvert \qquad \text {as }t\to \infty . \]
2. Define: \(z\) is Liapunov Stable. Define: \(z\) is Asymptotically Stable. Show that the zero solution \(z(t)=0\) of the autonomous system is asymptotically stable.

Periodic Orbits. Show that the following translated Fitzhugh-Nagumo system modelling a nerve axon has a nonconstant periodic trajectory in \(\mathbb {R}^2\). \[ \begin {cases} \dfrac {dx}{dt}=0.9x(1-x^2)-y,\\ \dfrac {dy}{dt}=0.5x-0.5y. \end {cases} \]

Nonhomogeneous Equations. Let \(v_0,x_0\in \mathbb {R}^n\) and \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i\le -\epsilon _0\) for all \(i=1,\ldots ,n\) and for some \(\epsilon _0>0\). Show that solution of the initial value problem (2.) is bounded for \(t\ge 0\) as long as it exists. Argue that therefore it exists for all time. \[ \begin {cases} \dfrac {dx}{dt}=Ax+e^{-t}(x+v_0),\\ x(0)=x_0. \end {cases} \tag {2.} \]

Periodic Coefficients. Prove that if \(\lvert \epsilon \rvert \ne 0\) is small enough, then all solutions are bounded \[ \ddot {u}+[1+\epsilon \sin (3t)]u=0. \] [Convert to a system of equations. Give a condition for the boundedness of all solutions. Verify that the condition is satisfied in this case.]

Three Species Predator-Prey Model. Consider the following Lotka-Volterra model for populations of three species \(x,y,z\ge 0\). \[ \begin {cases} \dot {x}=x(4-y-z),\\ \dot {y}=y(1-z),\\ \dot {z}=z(x+y-5);\\ x(0)=x_0,\quad y(0)=y_0,\quad z(0)=z_0. \end {cases} \] The only stationary point for this system with all positive coordinates is at \(P=(2,3,1)\). Show that there is an open \(U\subset \mathbb {R}^2\) such that \((2,3)\in U\), a function \(f(x,y)\in C^1(U)\) and \(\epsilon >0\) so that for initial points which are \(\epsilon \)-close to \(P\), that is for \(\lvert (x_0,y_0,z_0)-P\rvert <\epsilon \), if \(z_0=f(x_0,y_0)\) then the solution converges to \(P\) as \(t\to \infty \), but if \(z_0\ne f(x_0,y_0)\) then the solution eventually goes away from \(P\), that is for some \(t_0>0\), \(\lvert (x(t_0),y(t_0),z(t_0))-P\rvert \ge \epsilon \).

Elliptic Equations. Suppose \(\Omega \subset \mathbb {R}^n\) is a smooth bounded domain. Let \(a^i(x)\in C^\infty (\overline {\Omega })\) be smooth functions. Give conditions on \(a^i(x)\) so that for all \(f\in L^2(\Omega )\) there exists a weak solution to the boundary value problem \[ \begin {cases} \Delta u+\displaystyle \sum _{i=1}^n a^i(x)D_i u=f(x), & \text {for }x\in \Omega ,\\ u(x)=0, & \text {for }x\in \partial \Omega . \end {cases} \] [Here \(\Delta =\sum _{i=1}^n\frac {\partial ^2}{\partial x_i^2}\) is the Laplacian and \(D_i=\frac {\partial }{\partial x_i}\) is the partial derivative.]

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain, and \(0<T<\infty \), \(c\in \mathbb {R}\) be constants. Let \(\psi (x)\in C^\infty _0(\overline {\Omega })\). Let \(u\in C^{2,1}(\overline {\Omega }\times [0,T])\) be a solution to \[ \begin {cases} \dfrac {\partial u}{\partial t}=\Delta u+cu, & \text {if }(x,t)\in \Omega \times [0,T],\\ u(x,0)=\psi (x), & \text {if }x\in \Omega ,\\ u(x,t)=0, & \text {if }(x,t)\in \partial \Omega \times [0,T]. \end {cases} \tag {3.} \]

1. Show that the solution \(u(x,t)\) is bounded on \(\overline {\Omega }\times [0,T]\) in terms of \(c\), \(\sup _{x\in \Omega }\lvert \psi (x)\rvert \) and \(t\). [You may assume \(c\le 0\) but it is true for any \(c\).]
2. Show that such solutions of (3.) depends continuously on \(\psi \).

Vibrating Plate. Let \(\Omega \subset \mathbb {R}^n\) be a smooth bounded domain. Let \(\varphi ,\psi \in C^\infty _0(\Omega )\). Suppose that there exists a solution \(u(x,t)\in C^4(\overline {\Omega }\times [0,T])\) to the initial-boundary value problem for the vibrating clamped plate equation \[ \begin {cases} \dfrac {\partial ^2u}{\partial t^2}+\Delta \Delta u=0, & \text {if }x\in \Omega \text { and }t\ge 0,\\ u(x,0)=\varphi (x),\\ \dfrac {\partial u}{\partial t}(x,0)=\psi (x), & \text {for }x\in \Omega ,\\ u(x,t)=\dfrac {\partial u}{\partial n}(x,t)=0, & \text {if }x\in \partial \Omega \text { and }0\le t\le T. \end {cases} \tag {4.} \]

1. Show that the total energy \[ \mathcal {E}(t)=\int _\Omega \left \{\left (\frac {\partial u}{\partial t}(x,t)\right )^2+ \left (\Delta u(x,t)\right )^2\right \}\,dx \] is bounded in terms of \(\varphi \), \(\psi \) and \(t\).
2. Show that such a solution of (4.) is unique.

Conservation Laws. Find a solution to (5.) and prove that it is a weak solution. \[ \begin {cases} \dfrac {\partial u}{\partial t}+(1+u^2)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)= \begin {cases} 1, & \text {if }x<0,\\ 0, & \text {if }x\ge 0, \end {cases} & \text {for }x\in \mathbb {R}. \end {cases} \tag {5.} \]

Distributions. Let \(k(x,y)=\frac {1}{2\pi }\log \lvert x-y\rvert \), where \(x,y\in \mathbb {R}^2\).

1. Show that \(k(\cdot ,y)\) is a distribution on \(\mathbb {R}^2\) for each \(y\in \mathbb {R}^2\).
2. Show that as distributions, \(\Delta _x k(x,y)=\delta _y(x)\) for all \(y\in \mathbb {R}^2\). [\(\delta _y(x)=\delta (x-y)\) and \(\delta \) is the Dirac delta function.]

Fundamental Elliptic Estimate. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain and let \(H^k(\Omega )\) denote the Sobolev space of functions whose distributional derivatives up to the \(k\)-th order are in \(L^2(\Omega )\). Show that if for some constant \(\lambda \), \(u\in H^1(\Omega )\) is a weak solution to \[ \Delta u+\lambda u=0 \] then \(u\in C^\infty (\Omega )\). [You may assume the fundamental elliptic interior estimate: If \(\Omega '\Subset \Omega \) is a compactly contained subdomain, then there is a finite constant \(C=C(\Omega ,\Omega ',k,n)\) so that \[ \lVert u\rVert _{k+2,\Omega '}\le C\bigl (\lVert \Delta u\rVert _{k;\Omega }+\lVert u\rVert _{0;\Omega }\bigr ) \] for all \(u\in H^{k+2}(\Omega )\). Moreover, if \(f\in H^k(\Omega )\) and \(u\in H^1(\Omega )\) is a weak solution to \(\Delta u=f\), then \(u\in H^{k+2}(\Omega ')\).]