2005 PH.D PRELIM UU-MATH DIFFERENTIAL EQUATIONS

Problem.

1. Let \(D\) be an open set in \(\mathbb {R}\times \mathbb {R}^N\) and let \[ f:D\to \mathbb {R}^N \] be a continuous function such that \(f(t,x)\) satisfies a local Lipschitz condition with respect to the \(x\) variable. Consider the initial value problem \[ x'=f(t,x),\quad x(t_0)=x_0,\quad (t_0,x_0)\in D. \] State what is known about the existence and continuability of solutions of this problem. Be particularly detailed about the behavior of solutions as the time variable approaches the endpoints of existence intervals.
2. Apply the above results by sketching the solution picture for the scalar differential equation \[ x'=\frac {1}{tx}. \] (I.e. find \(D\) and then consider typical cases.)

Problem. Consider the nonlinear system \[ \begin {aligned} x' &= 2\sin y,\\ y' &= 3\sin x-\sin y. \end {aligned} \] Provide a complete phase plane analysis of this system and give justifications for your reasoning.

Problem. Let \(A\) be an \(N\times N\) matrix, none of whose eigenvalues have zero real part. Show that if \(b:(-\infty ,\infty )\to \mathbb {R}^N\) is a continuous function which is periodic of period 1, \((b(t+1)=b(t),\ t\in \mathbb {R})\), then the equation \[ x'=Ax+b(t), \] has a unique periodic solution of period 1.

Problem. In an isolated region of the Canadian Northwest Territories, a population of white wolves, \(x\), and one of silver foxes, \(y\), compete for survival (for each population one unit represents 100 individuals). They have a common, limited food supply, which consists mainly of mice. A mathematical biologist models the dynamics of the competing species by the nonlinear system \[ \begin {aligned} x' &= x-x^2-xy,\\ y' &= \frac {3}{4}y-y^2-\frac {1}{2}xy. \end {aligned} \] Can the two species survive together according to this model?

Problem. Let \(\Omega \) be an open subset of \(\mathbb {R}^N\).

1. Define what is meant by a distribution on \(\Omega \).
2. Show that all functions in \(L^1_{\mathrm {loc}}(\Omega )\), are, by an appropriate identification, distributions.
3. Give an example of a distribution which is not of this type.
4. Give the derivative formula for differentiating distributions.
5. Give a definition of the \(L^2\) Sobolev spaces \(H^m(\Omega )\) and \(H^m_0(\Omega )\). (Other notations used are the \(L^2\) Sobolev spaces \(W^{m,2}(\Omega )\) and \(W^{m,2}_0(\Omega )\).)

Problem. State and proof Poincaré’s inequality for \(H^1_0(\Omega )\) and use it to obtain two equivalent inner products for this space.

Problem. State the Lax-Milgram theorem and give a brief sketch of a proof. Use it to establish the existence and uniqueness of solutions (in the sense of distributions) of the boundary value problem \[ \begin {aligned} -\Delta u &= h,\quad \text {in }\Omega ,\\ u &= 0,\quad \text {on }\partial \Omega , \end {aligned} \] where \(h\in L^2(\Omega )\). Also, in your discussion, tell, in what sense the boundary data are assumed.

Problem. State the Phillips version of the Hille-Yosida theorem and illustrate it by a discussion of the heat equation \[ u_t-\Delta u=h(t,x). \]