UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations August 16, 2017.

Let \(D\subset \mathbb {R}\times \mathbb {R}^n\) be an open set, \((t_0,x_0)\in D\) a point and \(f(t,x)\in C(D,\mathbb {R}^n)\) be a map.

1. Let \(X\) be a Banach Space with norm \(\norm {\bullet }\). Let \(C\subset X\) be a closed subset. State the Contraction Mapping Theorem (otherwise known as the Banach Fixed Point Theorem) for a map \(K:C\to X\).
2. Define what it means for a map \(f(t,x)\in C(D,\mathbb {R}^n)\) to be locally Lipschitz with respect to \(x\)
3. Carefully reformulate the Initial Value Problems as a fixed point problem. Prove that the fixed point problem is equivalent to the local existence problem for the (IVP) \[ \begin {cases} \dfrac {dx}{dt}=f(t,x),\\ x(t_0)=x_0. \end {cases} \tag {IVP} \]
4. Assume \(f(t,x)\in C(D,\mathbb {R}^n)\) is locally Lipschitz with respect to \(x\). Using the Contraction Mapping Theorem, show that there is a \(\epsilon >0\) and a unique function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem (IVP) for \(t_0\le t\le t_0+\epsilon \).

1. Let \(A\) be an \(n\times n\) real matrix. Find necessary and sufficient conditions on \(A\) so that for all \(x_0\in \mathbb {R}^n\), the solution \(\varphi (t;x_0)\) of (1) remains bounded for \(t\ge 0\). Prove your result. \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \tag {1} \]
2. Suppose that \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) is such that \(f(0)=0\) and the Jacobian matrix \(A=df(0)\) satisfies the condition in (a). Do all solutions \(\psi (t;x_0)\) of the nonlinear equation (2) still remain bounded for \(t\ge 0\), at least if \(x_0\) is close to \(0\)? Explain. \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \tag {2} \]

1. Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) satisfy \(f(0)=0\). State the definitions: the zero solution of \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0, \end {cases} \] is stable in the sense of Liapunov and asymptotically stable.
2. Suppose that all eigenvalues of the real \(n\times n\) matrix \(A\) have negative real part \(\Reop \,\lambda _i<0\). State an estimate for solutions of \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0, \end {cases} \] and use it to prove that the zero solution is asymptotically stable.
3. Let \(A\) as in (b) and \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) be a smooth function such that for some \(L<\infty \), \(\abs {f(x)}\le L\abs {x}^2\) for all \(x\). Consider the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax+f(x),\\ x(0)=x_0. \end {cases} \tag {3} \] Prove that the zero solution of (3) is stable in the sense of Liapunov. [Don’t just quote a theorem. Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and \(c_0\ge 0\) is a constant that satisfy \(u(t)\le c_0+\int _0^t g(s)u(s)\dd s\) for all \(t\ge 0\), then Gronwall’s Inequality implies \(u(t)\le c_0\exp \left (\int _0^t g(s)\dd s\right )\) for all \(t\ge 0\).]

1. Let \(A(t)\) be a continuous, non-constant, real matrix-valued, \(T>0\) periodic function \(A(t+T)=A(t)\). For the system \[ \dot {x}=A(t)x \tag {4} \] define: Monodromy Matrix, Floquet Multipliers and Floquet Exponents.
2. If \(e^{T\gamma }=\rho \) is a Floquet multiplier, then there is a solution of (4) of the form \[ x(t)=e^{\gamma t}p(t) \] where \(p(t)=\pm p(t+T)\).
3. Prove that this equation does not have a fundamental set of bounded solutions. \[ \ddot {x}-(\cos ^2t)\dot {u}+(\sin ^2t)u=0 \]

Consider the Brusselator system for an autocatalytic reaction. Show that it has a non-constant periodic solution. \[ \begin {cases} \dot {x}=2-13x+x^2y,\\ \dot {y}=12x-x^2y. \end {cases} \] [Hint: show the trapezoid with sides \(y=0\), \(x=\frac {2}{13}\), \(y=78\) and \(x+y=80\) is forward invariant.]

Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) have a non-constant, \(T>0\) periodic trajectory \(\gamma (t)\) satisfying \(\dot {\gamma }(t)=f(\gamma (t))\).

1. Define: the periodic solution \(\gamma (t)\) is orbitally stable.
2. Define: the Poincaré Map for the orbit \(\gamma \).
3. \(\gamma (t)=(2\cos 2t,\sin 2t)\) is a periodic solution to \[ \begin {cases} \dot {x}=-4y+x\left (1-\dfrac {x^2}{4}-y^2\right ),\\ \dot {y}=x+y\left (1-\dfrac {x^2}{4}-y^2\right ). \end {cases} \] Determine the orbital stability of \(\gamma (t)\) by computing the derivative of its Poincaré Map.

Consider the problem of a thin layer of paint of thickness \(h(x,t)\) and speed \(u(x,y,t)\) flowing down a wall. The paint is assumed to be uniform in the \(z\)-direction. The balance between gravity and viscosity (fluid friction) means that the velocity satisfies the equation \[ \frac {\partial ^2u}{\partial y^2}=-c, \] where \(c\) is a positive constant. This is supplemented by the boundary conditions \[ u(x,0,t)=0,\qquad \left .\frac {\partial u(x,y,t)}{\partial y}\right |_{y=h}=0. \] The density of paint per unit length in the \(x\)-direction is \(\rho _0h(x,t)\) where \(\rho _0\) is a constant, and the corresponding flux is \[ q(x,t)=\rho _0\int _0^h u(x,y,t)d y. \]

1. Sketch a figure illustrating the physical problem.
2. Using conservation of paint, and solving for \(u(x,y,t)\) in terms of \(h(x,t)\) and \(y\), derive the following PDE for the thickness \(h\): \[ \frac {\partial h}{\partial t}+ch^2\frac {\partial h}{\partial x}=0. \]
3. Set \(c=1\). Show that the characteristics are straight lines and that the Rankine-Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {[h^3/3]^+_-}{[h]^+_-}. \]
4. A stripe of paint is applied at \(t=0\) so that \[ h(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \] Show that, for small enough \(t\), \[ h= \begin {cases} 0, & x<0,\\ (x/t)^{1/2}, & 0<x<t,\\ 1, & t<x<S(t),\\ 0, & S(t)<x, \end {cases} \] where the shock is \(x=S(t)=1+t/3\). Explain why this solution changes at \(t=3/2\), and show that thereafter \[ \frac {dS}{dt}=\frac {S}{3t}. \]

Consider the following inhomogeneous initial-Neumann problem: \[ \begin {aligned} u_t&=Du_{xx}+\alpha tx, &&0<x<\pi ,\ 0<t,\\ u(x,0)&=1, &&0<x<\pi ,\\ u_x(0,t)&=u_x(\pi ,t)=0, &&0<t; \end {aligned} \] where \(\alpha \) is a constant.

1. Determine the eigenfunctions of the homogeneous equation \[ u_{xx}=\lambda u,\qquad 0\le x\le \pi ,\qquad u_x(0)=u_x(\pi )=0. \]
2. Solve the inhomogeneous initial-Neumann problem by carrying out an eigenfunction expansion of \(u(x,t)\) in terms of the eigenfunctions obtained in part (a). That is, denoting the eigenfunctions by \(v_k\), integer \(k\), set \[ u(x,t)=\sum _{k\ge 0}c_k(t)v_k(x), \] and determine the time-dependent coefficients \(c_k(t)\).
3. Give a physical interpretation of the solution in the limit \(\alpha \to 0\).

1. Consider the velocity potential \(\phi \) for linear water waves. Let \(h_0\) denote the depth of water in the absence of waves, and \(g\) denote the gravitational constant. The corresponding boundary value problem on \(\mathbb {R}\times [-h_0,0]\) is given by Laplace’s equation \[ \frac {\partial ^2\phi }{\partial x^2} +\frac {\partial ^2\phi }{\partial y^2}=0, \] with \[ \frac {\partial ^2\phi }{\partial t^2} +g\frac {\partial \phi }{\partial y}=0 \qquad \text {on } y=0, \] and \[ \frac {\partial \phi }{\partial y}=0 \qquad \text {on } y=-h_0. \] The linear wave solution takes the form \[ \phi (x,y,t)=Y(y)e^{ikx-i\omega t}. \] Derive the transcendental dispersion relation \[ \omega ^2=gk\tanh (kh_0). \] Determine the group and phase velocities for (i) deep water waves (large \(h_0\)) and (ii) shallow water waves (small \(h_0\)). Which case is non-dispersive?
2. Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \). Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _nr^n}\int _{B_r(x)}u(y)\dd y. \]

Consider a string clamped at the end points \(a\) and \(b\), say, with \(u(a,t)=0\) and \(u(b,t)=0\), where \(u(x,t)\) is the string’s deviation from the horizontal rest position. The kinetic energy of the string is \[ E_k(t)=\frac {1}{2}\int _a^b \rho u_t^2d s, \] where \(\rho (x,t)\) is the mass density and \(ds=\sqrt {1+u_x^2}d x\) is an infinitesimal arc length. The potential energy consists of the sum of the energy due to the stretching of the string, and the work done against a load: \[ E_p(t)=\int _a^b \left (d(\sqrt {1+u_x^2}-1)-lu\sqrt {1+u_x^2}\right )d x. \] Here \(d(x,t)\) is the string’s elastic coefficient and \(l(x,t)\) is the load on the string. The classical action for the continuum model is \[ J=\int _{t_1}^{t_2}[E_k(t)+E_p(t)]d t. \]

1. Substitute the integral expressions for \(E_k\) and \(E_p\) into the action \(J\), and consider the variations \(u+\epsilon \psi \) such that \(\psi \) vanishes at the end points \(a,b\) and at the initial and final times \(t_1,t_2\). Calculate the first variation \(\delta J\).
2. After integrating by parts terms of the form \(u_t\psi _t\) and \(u_x\psi _x\), and using the boundary conditions for the variation \(\psi \), show that Hamilton’s principle \(\delta J=0\) yields the following PDE for \(u\): \[ (\rho u_t)_t-(1+u_x^2)^{-1/2} \left [d(1+u_x^2)^{-1/2}u_x\right ]_x-l=0. \]
3. Show that if \(\rho ,d\) are constants, there is no load (\(l=0\)), and the small slope approximation \(\abs {u_x}\ll 1\) holds, then the PDE of part (b) reduces to the classical one-dimensional wave equation. Write down an expression for the wave speed.