Differential Equations Preliminary Examination

Consider the system of ordinary differential equations \[ u'=-\lvert u\rvert ^2u+h(t), \tag {1} \] where \[ h:[0,1]\to \mathbb {R}^N \] is a continuous function and \(\lvert \cdot \rvert \) is the Euclidean norm in \(\mathbb {R}^N\). Use Brouwer’s fixed point theorem to show that (1) has a solution \(u\in C^1[0,1]\) such that \[ u(0)=u(1). \] Use this result to conclude that if \[ h:\mathbb {R}\to \mathbb {R}^N \] is continuous and periodic of period 1, then (1) has a solution of period 1.

(Hint: Consider equation (1) subject to initial data \(u_0:\lvert u_0\rvert \le r\), where \(r\) is large (in relation to \(h\)) and examine solution curves by computing \(\frac {d}{dt}\lvert u\rvert ^2\) for solutions \(u\).) In relation to the above, what can you say about the equation \[ u'=\lvert u\rvert ^2u+h(t). \tag {2} \]

Let \(A\) be a real \(N\times N\) matrix, all of whose eigenvalues have negative real parts. Let \[ h:\mathbb {R}\to \mathbb {R}^N \] be a continuous function which is periodic of period \(T\). Let \[ \begin {aligned} (t,v)&\longmapsto u(t,v),\\ \mathbb {R}\times \mathbb {R}^N&\longrightarrow \mathbb {R}^N \end {aligned} \] denote the unique solution of \[ u'=Au+h(t),\qquad u(0)=v. \tag {3} \] For given \(T>0\), define \[ S_T:\mathbb {R}^N\to \mathbb {R}^N \] by \[ S_T(v)=u(T,v). \]

1. Use the variation of constants formula (and what you know about stability theory for the linear system \(u'=Au\)) to show that \(S_{nT}\) is a contraction mapping, for some positive integer \(n\), sufficiently large.
2. Show that \[ S_{nT}=\underbrace {S_T\circ \cdots \circ S_T}_{n}, \] the \(n\)-fold composition of \(S_T\) with itself.
3. Show that if \(S_{nT}\) has a unique fixed point, then so does \(S_T\).
4. Apply this to show that if \(h\) is a continuous periodic function of period \(T\), then (3) has a unique \(T\)-periodic solution.

Let \[ f:\mathbb {R}^N\to \mathbb {R}^N \] be a \(C^1\) function and consider the system of differential equations \[ u'=f(u). \tag {4} \] Let there exist, for a given \(v\in \mathbb {R}^N\), a compact subset \(K\) of \(\mathbb {R}^N\), such that \[ u(t,v)\in K,\qquad t\in I_v, \] where \(I_v\) is the maximal interval of existence of the solution \(u(t,v)\) of (4) which satisfies \[ u(0,v)=v. \] Sketch proofs of the following:

1. \[ I_v=(-\infty ,\infty ). \]
2. The \(\omega \) limit set \(\Gamma (v)\) (define this term) is a nonempty, compact, connected, and invariant set.

What more can you say about the set \(\Gamma (v)\) in case of dimension \(N=2\)? Are there similar statements in higher dimensions?

Consider the system in the plane \[ \begin {aligned} u'&=-v+uw,\\ v'&=u+vw, \end {aligned} \tag {5} \] where \[ r^2=u^2+v^2,\qquad w=r^4-6r^2+8. \] Show that (5) has precisely two nontrivial periodic orbits and provide a “complete” phase plane analysis of the system.

Use the implicit function theorem for mappings between Banach spaces to discuss the unique solvability of the scalar boundary value problem \[ u''+2u+u^2=h(t),\qquad u(0)=u(\pi ), \tag {6} \] for continuous functions \(h\) of small maximum norm. Results needed from Sturm-Liouville theory may be assumed but should be quoted.

Let \(G\subset \mathbb {R}^N\) be an open set. Define what is meant by a distribution on \(G\). Also define the concept of distributional derivatives of any order \(\alpha \), where \(\alpha \) is a multiindex.

Give a definition of the Sobolev spaces \(H^m(G)\), \(H^m_0(G)\), where \(m\) is a nonnegative integer. Use properties of distributions to prove the generalized integration by parts formula \[ \inner {\partial ^\alpha f}{g} = (-1)^{\lvert \alpha \rvert }\inner {f}{\partial ^\alpha g}, \qquad f\in H^m(G),\ g\in H^m_0(G),\ \lvert \alpha \rvert \le m, \] and \(\inner {\cdot }{\cdot }\) is the \(L^2(G)\) inner product.

Consider the space of real-valued functions \(H^2(0,1)\) and define the quadratic form \[ a(u,v)=\int _0^1 u''v''\,dx+b\int _0^1 u'v'\,dx+c\int _0^1 uv\,dx, \] where \(b\) and \(c\) are positive constants. Show that \(a\) is a form to which the Lax-Milgram theorem may be applied and state the result thus obtained. Let \(f\in L^2(0,1)\), and restrict the form \(a\) to the space \(H^2_0(0,1)\). Show that \(f\) may be thought of as a continuous linear functional on \(H^2_0(0,1)\), and apply the Lax-Milgram theorem to the pair \(a,f\). Determine a boundary value problem which is solved in a distributional sense by the Lax-Milgram solution and show that this solution is at least three times continuously differentiable.

Let \(G\) be a bounded open set in \(\mathbb {R}^N\) with smooth boundary. State and indicate a proof of Poincaré’s inequality for functions \(u\in H^1_0(G)\). Using the inequality show that \[ \norm {u}_0^2=\int _G \lvert \nabla u\rvert ^2\,dx \] defines a norm equivalent to the \(H^1\) norm. Consider the form \[ a(u,v)=\int _G \nabla u\cdot \overline {\nabla v} +\int _G c(x)\cdot \nabla u\,\overline {v}\,dx, \tag {7} \] where the components of the vector function \(c\) belong to \(L^\infty (G)\).

Provide a condition on the function \(c\) in order that \(a\) is a sesquilinear form satisfying the conditions of the Lax-Milgram theorem. Determine a boundary value problem solved by the Lax-Milgram solution.

1. Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0,\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

2. Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_x&=0,\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \] Solve this problem explicitly.

   Also state a result about the initial value problem \[ \begin {aligned} u_t-u_x&=f(t,x),\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \]

1. State the theorem of Rellich-Kondrachov for bounded open sets \(G\subset \mathbb {R}^N\).
2. Let \[ f:\overline {G}\times \mathbb {R}\to \mathbb {R} \] be a continuous bounded function such that \[ \lvert f(x,u)\rvert \le M,\qquad (x,u)\in \overline {G}\times \mathbb {R} \] where \(M\) is a constant. Show that the boundary value problem: \[ -\Delta u=f(x,u),\qquad u\in H^1_0(G), \tag {8} \] has a solution by converting (8) into an equivalent fixed point problem in \(L^2(G)\) and by applying Schauder’s fixed point theorem (give reasons why its hypotheses hold!).