Gronwall Inequality and Continuous Dependence

Let \(\phi _t\) and \(\psi _t\) be two flows on \(\mathbb {R}^2\setminus \{0\}\) induced by the autonomous linear ODEs \[ \frac {d}{dt}(x,y)=(x,y),\qquad \frac {d}{dt}(u,v)=(\alpha u,\beta v), \]

where \(\alpha , \beta > 0\) are constants.

1. Show that for any \(\alpha ,\beta >0\), the flows \(\phi _t\) and \(\psi _t\) are topologically conjugate on \(\mathbb {R}^2\setminus \{0\}\); i.e., there exists a homeomorphism \(h:\mathbb {R}^2\setminus \{0\}\to \mathbb {R}^2\setminus \{0\}\) such that \[ h(\phi _t(x,y))=\psi _t(h(x,y)) \] for all \((x,y)\ne (0,0)\) and all \(t\in \mathbb {R}\).
2. Find an explicit homeomorphism \(h\) realizing the conjugacy and prove it is a bijective continuous map with continuous inverse.
3. Discuss whether the conjugacy can be chosen differentiable (i.e., a \(C^1\) diffeomorphism). Under what conditions on \(\alpha ,\beta \) does a differentiable conjugacy exist?

Consider the following two second-order differential equations:

where \(f(t)\) is continuous and \(2\pi \)-periodic, i.e., \(f(t)=f(t+2\pi )\) for all \(t\in \mathbb {R}\). For both equations, we have the initial conditions \(x(0)=1\) and \(x'(0)=1\).

1. For equation (I), show the solution \(x(t)\) is unbounded.
2. For equation (II), show that \(x(t)\) is bounded.

Consider the family of ODE systems

with \(\|Df\|\le L\) and \(|g|\le M\). Show that

Hint: Look at the integral equation for \(z_\epsilon (t):=x_\epsilon (t)-x_0(t)\), estimate using Grönwall-type ideas.

Let \(f(t,x):\mathbb {R}\times \mathbb {R}^n\to \mathbb {R}^n\) be continuous such that for some

for all \(t\in \mathbb {R}\) and \(x,y\in \mathbb {R}^n\). Show that the initial value problem for \(x_0\in \mathbb {R}^n\) \[ \dot {x}=f(t,x)\quad \text {with }x(0)=x_0 \] has a unique solution that exists on \(t\in [0,\infty )\).

1. Let \(f:\mathbb {R}^d\to \mathbb {R}^d\) be a continuously differentiable function such that \(f(0)=0\). State the definition: \(z(t)=0\) is a Liapunov stable solution of \(\dot {u}=f(u)\) on \(0\le t<\infty \).
2. State conditions on the eigenvalues and eigenvectors of a real \(d \times d\) matrix \(A\) that hold if and only if \(z(t)=0\) is Liapunov stable solution for \(\dot {x}=Ax\) on \(0\le t<\infty \). Prove your conditions.
3. Let \(A(t)\) be a \(T>0\) periodic \(d\times d\) real continuous matrix function. State but don’t prove necessary and sufficient conditions on \(A(t)\) so that the solution \(z(t)\) \(= 0\) is Liapunov stable for \(\dot {x}=A(t)x\) on \(0\le t<\infty \).

[25] Let \(B(t)\) be a continuous real \(d\times d\) matrix function defined for \(t\in [0,\infty )\) such that \[ \int _0^\infty \|B(s)\|\,ds<\infty . \] Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\). Show that the initial value problem

(1)

has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem. Hint: Use the variation of parameters formula and Gronwall’s Inequality. Gronwall’s Inequality says that if \(c_0\ge 0\) and if \(f(t)\), \(\phi (t)\) are nonnegative continuous functions on the interval \([0,T]\) which satisfy \[ f(t)\le c_0+\int _0^t \phi (s)f(s)\,ds \] for all \(t\in [0,T]\), then \[ f(t)\le c_0\exp \left (\int _0^t \phi (s)\,ds\right ) \] for all \(t\in [0,T]\).]

Let \(f(t,x):[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\). Suppose that there are constants \(L,M<\infty \) such that

for all \(t\), \(y\), \(z\).

Show that the initial value problem

has a unique solution \(x(t)\in C^1([0,\infty ),\mathbb {R}^d)\). [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

1. Let \(A\) be a \(d\times d\) real matrix. State and prove a theorem that gives necessary and sufficient conditions on \(A\) so that every solution \(x(t)\) is bounded for \(t\in [0,\infty )\).

   \[ \dot {x}=Ax. \tag {1} \]

2. Let \(A(t)\) be a \(C^1\), \(T>0\) periodic, \(d\times d\) real continuous matrix function. State but don’t prove a theorem that gives necessary and sufficient conditions on \(A(t)\) so that every solution \(x(t)\) is bounded for \(t\in [0,\infty )\).

   \[ \dot {x}=A(t)x. \]

Let \(B(t)\) be a \(C^1\) real \(d\times d\) matrix function defined for \(t\in [0,\infty )\) such that

Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\). Show that there is \(\delta >0\) such that if \(|x_0|\le \delta \), then the initial value problem

(2)

has a solution such that \(x(t)\to 0\) as \(t\to \infty \). [Provide the complete proof. Do not just quote a theorem.]

[Hint: Use the variation of parameters formula and Gronwall’s Inequality, which says that for \(f(t)\), \(\phi (t)\) nonnegative continuous functions on the interval \([0,T]\) and for \(c_0\ge 0\), if \[ f(t)\le c_0+\int _0^t \phi (s)f(s)\,ds \] for all \(t\in [0,T]\), then \[ f(t)\le c_0\exp \left (\int _0^t \phi (s)\,ds\right ) \] for all \(t\in [0,T]\). You might first show that there is a constant \(k\) such that if \(\epsilon \) is small enough and \(|x_0|<\epsilon \) then the solution remains bounded \(|x(t)|\le k|x_0|\) for all \(t\in [0,T]\).]

1. What is the statement of Gronwall’s inequality? Where to start: Suppose \(\psi (x)\) satisfies the inequality \[ \psi (t)\le \alpha +\int _0^t \beta \psi (s)\,ds, \] then \(\cdots \).

2. Suppose \(f(t,x)\in C(U,\mathbb {R}^n)\) is Lipschitz continuous in its second argument, and let \(x(t)\) and \(y(t)\) be solutions of the same differential equation \(\frac {dx}{dt}=f(t,x)\) with different initial data \(x(0)=x_0\), \(y(0)=y_0\). Prove that

   \[ |x(t)-y(t)|\le |x_0-y_0|\exp (Lt), \]

   for some positive constant \(L\) and some positive time \(0 \le t < T\). What is an estimate for the constant \(L\)?

3. Why may the inequality fail to hold for all time? Give an example of where this fails.

Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\) and \(L\in \mathbb {R}\) be such that

Show that there is \(T>0\) such that the Picard iterates converge to a continuously differentiable solution of the initial value problem

on \([0,T]\). [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

Let \(A\) be a real matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\).

1. Show that there is \(\delta >0\) such that if \(|x_0|\le \delta \), then the initial value problem

   \[ \dot {x}=Ax+e^{-t}|x|x \]

   (1)

   \[ x(0)=x_0 \]

   has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem.]

2. Is the zero solution of (1) asymptotically stable? Explain. [Hint: let \(f(t)\), \(\phi (t)\) be nonnegative continuous functions on the interval \(J=(\alpha ,\beta )\) containing \(t_0\). Let \(c_0\ge 0\). Gronwall’s Lemma says that if \[ f(t)\le c_0+\left |\int _{t_0}^t \phi (s)f(s)\,ds\right | \] for all \(t\in J\), then \[ f(t)\le c_0\exp \left |\int _{t_0}^t \phi (s)\,ds\right | \] for all \(t\in J\).]

Let \(f \in C^1(U,\mathbb {R}^n)\) for \(U \subset \mathbb {R}^n\) and \(x_0 \in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\|x\|=\max _{0\le t\le T}|x(t)|\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\,ds \] for \(x \in X\). Define \(V=\{x\in X\mid \|x-x_0\|\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\).

1. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\), show that \[ |K(x(t))-K(y(t))|\le L_0t\|x-y\|. \] Hence, show that \[ \|K(x)-K(y)\|\le L_0T\|x-y\|,\quad x,y\in V. \]
2. Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the integral equation has a unique continuous solution \(x(t)\) for all \(t\in [0,T]\) and sufficiently small \(T\). Hence establish existence and uniqueness of the initial value problem \[ \frac {dx}{dt}=f(x),\quad x(0)=x_0. \]
3. Show that the solution of the equation \[ \dot {x}=x^2,\quad x(0)=x_0 \] blows up in finite time. Does this contradict the result of part (b)?

Consider the following linear equation for \(x\in \mathbb {R}^n\): \[ \dot {x}=Ax+B(t)x,\quad x(t_0)=x_0. \] where \(A\), \(B(t)\) are \(n\times n\) matrices. Suppose that all eigenvalues \(\lambda _j\), \(j=1,\ldots ,n\), of the matrix \(A\) satisfy \(\operatorname {Re}(\lambda _j)<0\), and let \(B(t)\) be continuous for \(0\le t<\infty \) with \[ \int _0^\infty \|B(t)\|\,dt<\infty . \]

1. Using the variation of constants formula show that there exist constants \(K,\sigma >0\) such that \[ |x(t)| \le K e^{-\sigma (t-t_0)}|x_0| +K\int _{t_0}^t e^{-\sigma (t-s)}|x(s)|\|B(s)\|\,ds. \]
2. Let \(u(t)=e^{\sigma t}|x(t)|\), \(v(t)=\|B(t)\|\) and \(c=Ke^{\sigma t_0}|x_0|\). Show that the inequality of part (a) can be rewritten as \[ u(t)\le c+\int _{t_0}^t v(s)u(s)\,ds. \]
3. From Gronwall’s inequality we have \[ u(t)\le c\exp \left (\int _{t_0}^t v(s)\,ds\right ). \] Use this to show that the zero solution of the initial value problem is asymptotically stable.

Suppose someone treats one of the variables in the Laplace equation as time and tries to solve the evolution problem \[ \begin {aligned} u_{tt}+u_{xx}&=0\quad (0<x<l,\ t>0),\\ u(0,t)&=u(l,t)=0,\qquad u(x,0)=\phi (x),\qquad u_t(x,0)=\psi (x), \end {aligned} \] similar to the wave equation. Show that this problem has no continuous dependence on data, even if the time \(t\) belongs to a finite interval \(0<t<T\) (\(T\) is a positive constant).

Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\). Show that there is \(T>0\) such that the initial value problem \[ \begin {aligned} \dot {x}&=f(t,x)\\ x(0)&=x_0 \end {aligned} \] has a solution \(x(t)\in C^1([0,T),\mathbb {R}^n)\). Is \(x(t)\) unique? [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

Let \(g:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuously differentiable function and \(p>1\) a constant such that \[ \abs {g(t,x)}\le \abs {x}^p\quad \text {for all }t\ge 0\text { and }x\in \mathbb {R}^d. \] Let \(A\) be a real matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\).

1. Show that there is \(\delta >0\) such that if \(\abs {x_0}\le \delta \), then the initial value problem \[ \begin {aligned} \dot {x}&=Ax+g(t,x)\\ x(0)&=x_0 \end {aligned} \tag {1} \] has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem.]

2. Is the zero solution of (1) asymptotically stable? Explain.

   [Hint: let \(f(t)\), \(\varphi (t)\) be nonnegative continuous functions on the interval \(J=(\alpha ,\beta )\) containing \(t_0\). Let \(c_0\ge 0\). Gronwall’s Lemma says that if \[ f(t)\le c_0+\left |\int _{t_0}^{t}\varphi (s)f(s)\dd s\right | \] for all \(t\in J\) then \[ f(t)\le c_0\exp \left |\int _{t_0}^{t}\varphi (s)\dd s\right | \] for all \(t\in J\).]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement \(\rho \) is the linear density of the material, and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1\), \(E_2\) and densities \(\rho _1\), \(\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave \(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

1. Let \(A(t)\) be a continuous, non-constant, real matrix-valued, \(T>0\) periodic function \(A(t+T)=A(t)\). For the system \[ \dot {x}=A(t)x \tag {2} \] define: Monodromy Matrix, Floquet Multipliers and Floquet Exponents.
2. If \(e^{T\gamma }=\rho \) is a Floquet multiplier, then there is a solution of (2) of the form \[ x(t)=e^{\gamma t}p(t) \] where \(p(t)=\pm p(t+T)\).
3. Let \(a,b\in \mathbb {R}\). Suppose that a Floquet multiplier is \(\mu _1=-1\). What can you say about the boundedness of all solutions on \([0,\infty )\)? \[ \begin {pmatrix}\dot {x}\\ \dot {y}\end {pmatrix} = \begin {pmatrix} -1 & a\\ b & \sin (\pi t) \end {pmatrix} \begin {pmatrix}x\\ y\end {pmatrix} \]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1\), \(E_2\) and densities \(\rho _1\), \(\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave \(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac {1}{2}[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\abs {x}<a\), \(u(x,0)=0\) if \(x>\abs {a}\); \(u_t(x,0)=0\)
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\abs {x}<a\), \(u_t(x,0)=0\) if \(x>\abs {a}\)

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 \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) 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. Find the first three successive approximations \(u_1(t)\), \(u_2(t)\), \(u_3(t)\) for the initial value problem (IVP) \[ \dot {x}=x^2,\qquad x(0)=1. \] Use mathematical induction to show that for all \(n\ge 1\), \(u_n(t)=1+t+\cdots +t^n+O(t^{n+1})\) as \(t\to 0\).
2. Solve the IVP in part (a) and show that the function \(x(t)=1/(1-t)\) is a solution to the IVP on the interval \((-\infty ,1)\). Compare with the approximation of part (a).
3. Let \(f\in C^1(U,\mathbb {R}^n)\) for \(U\subset \mathbb {R}^n\) and \(x_0\in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\norm {x}=\max _{0\le t\le T}\abs {x(t)}\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\dd s \] for \(x\in X\). Define \(V=\{x\in X\mid \norm {x-x_0}\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\). Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that \[ \abs {K(x(t))-K(y(t))}\le L_0t\norm {x-y}. \] Hence, show that \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y}\qquad x,y\in V. \]

Consider the following linear equation for \(x\in \mathbb {R}^n\): \[ \dot {x}=A+B(t)x,\qquad x(t_0)=x_0. \] where \(A,B(t)\) are \(n\times n\) matrices. Suppose that all eigenvalues \(\lambda _j\), \(j=1,\ldots ,n\), of the matrix \(A\) satisfy \(\Reop (\lambda _j)<0\), and let \(B(t)\) be continuous for \(0\le t<\infty \) with \(\int _0^\infty \norm {B(t)}\dd t<\infty \).

1. Using the variation of constants formula show that there exist constants \(K,\sigma >0\) such that \[ \abs {x(t)} \le Ke^{-\sigma (t-t_0)}\abs {x_0} +K\int _{t_0}^t e^{-\sigma (t-s)}\abs {x(s)}\norm {B(s)}\dd s. \]
2. Let \(u(t)=e^{\sigma t}\abs {x(t)}\), \(v(t)=\norm {B(t)}\) and \(c=Ke^{\sigma t_0}\abs {x_0}\). Show that the inequality of part (a) can be rewritten as \[ u(t)\le c+\int _{t_0}^t v(s)u(s)\dd s. \]
3. From Gronwall’s inequality we have \[ u(t)\le c\exp \left (\int _{t_0}^t v(s)\dd s\right ). \] Use this to show that the zero solution of the IVP is asymptotically stable.

Let \(f\in C^1(U,\mathbb {R}^n)\) for \(U\subset \mathbb {R}^n\) and \(x_0\in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\norm {x}=\max _{0\le t\le T}\abs {x(t)}\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\dd s \] for \(x\in X\). Define \(V=\{x\in X\mid \norm {x-x_0}\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\).

1. Give the definition of a locally Lipschitz function in an open set \(U\) of a normed vector space.
2. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that \[ \abs {K(x(t))-K(y(t))}\le L_0t\norm {x-y}. \] Hence, show that \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y}\qquad x,y\in V. \]
3. State the contraction mapping principle on a Banach space.
4. Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the integral equation has a unique continuous solution \(x(t)\) for all \(t\in [0,T]\) and sufficiently small \(T\). Hence establish existence and uniqueness of the initial value problem \[ \frac {dx}{dt}=f(x),\qquad x(0)=x_0. \]

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac {1}{2}[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\abs {x}<a\), \(u(x,0)=0\) if \(x>\abs {a}\); \(u_t(x,0)=0\)
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\abs {x}<a\), \(u_t(x,0)=0\) if \(x>\abs {a}\)

Let \(f\in C^1(U,\mathbb {R}^n)\) for \(U\subset \mathbb {R}^n\) and \(x_0\in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\norm {x}=\max _{0\le t\le T}\abs {x(t)}\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\dd s \] for \(x\in X\). Define \(V=\{x\in X\mid \norm {x-x_0}\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\).

1. Give the definition of a locally Lipschitz function in an open set \(U\) of a normed vector space.
2. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that \[ \abs {K(x(t))-K(y(t))}\le L_0t\norm {x-y}. \] Hence, show that \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y},\qquad x,y\in V. \]
3. State the contraction mapping principle on a Banach space.
4. Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the integral equation has a unique continuous solution \(x(t)\) for all \(t\in [0,T]\) and sufficiently small \(T\). Hence establish existence and uniqueness of the initial value problem \[ \frac {dx}{dt}=f(x),\qquad x(0)=x_0. \]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material, and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1,E_2\) and densities \(\rho _1,\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave \(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

A problem in the dynamics of the overhead power wire for an electric locomotive leads to the model \[ \frac {\partial ^2u}{\partial t^2} =\frac {\partial ^2u}{\partial x^2} \quad \text {for }x\ne X(t),\ t>0, \] where \(X\) is a prescribed smooth function with \(0<X'(t)<1\). Here \(X(t)\) is the locomotive position and \(u\) is the displacement of the wire. Across \(x=X\) there are prescribed discontinuities \[ \left [\frac {\partial u}{\partial x}\right ]_{X^-}^{X^+} =-V(X(t),t),\qquad [u]_{X^-}^{X^+}=0. \] Suppose that \(u=u_t=0\) at \(t=0\). By constructing a weak solution that takes into account the discontinuities across \(X\), show that \[ u(x,t)=\int _0^{\tau _0}V(X(\tau ),\tau )\left (1-X'(\tau )^2\right )d\tau , \] where the range of \(\tau \) is taken so that \(y=X(\tau )\) lies within the range \[ \tau -t<y-x<t-\tau ,\qquad \tau >0. \] [Hint: you will need to derive a relationship between discontinuities in \(\partial _tu\) and in \(\partial _xu\) along \(X(t)\). You will also need to consider the domain of dependence in the \((x,t)\)-plane].

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1,E_2\) and densities \(\rho _1,\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=R e^{i(t+x/c_1)}\) and a transmitted wave \(u_T=T e^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\norm {x}<a\), \(u(x,0)=0\) if \(x>\norm {a}\); \(u_t(x,0)=0\).
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\norm {x}<a\), \(u_t(x,0)=0\) if \(x>\norm {a}\).

Let \(f(x,t)\) be continuous for all \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Suppose that there is \(M\in \mathbb {R}\) such that \[ \norm {f(x,t)-f(y,t)}\le M\norm {x-y}, \qquad \text {for all }(x,t),(y,t)\in \mathbb {R}^n\times \mathbb {R}. \] Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1,E_2\) and densities \(\rho _1,\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=R e^{i(t+x/c_1)}\) and a transmitted wave \(u_T=T e^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\norm {x}<a\), \(u(x,0)=0\) if \(x>\norm {a}\); \(u_t(x,0)=0\).
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\norm {x}<a\), \(u_t(x,0)=0\) if \(x>\norm {a}\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Briefly explain the concepts of domain of dependence, and finite propogation speed, and prove that these properties follow d’Alembert’s Formula.
3. Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)=0\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s Formula as a starting point, find a closed form solution for \(u(x,t)\) with \(x,t>0\).

Let \(f(t,x)\) be continuous function on \(\mathbb {R}\times \mathbb {R}^n\) that satisfies \[ \norm {f(t,x_1)-f(t,x_2)}\le \norm {x_2-x_1} \] for all \(t\in \mathbb {R}\) and \(x_1,x_2\in \mathbb {R}^n\). Let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Let \(f(x,t)\) be continuous for \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Suppose \(f(x,t)\) is continuous for \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\) be a point. Assume that there is a constant \(L<\infty \) so that \[ \lvert f(x_1,t)-f(x_2,t)\rvert \le L\lvert x_1-x_2\rvert \] for all \(x_1,x_2,t\). Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \]

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\quad x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)]+\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Briefly explain the concepts of domain of dependence, and finite propogation speed, and prove that these properties follow d’Alembert’s Formula.
3. Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)=0\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s Formula as a starting point, find a closed form solution for \(u(x,t)\) with \(x,t>0\).

Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\Re \lambda _i<0\) for all \(i=1,\ldots ,n\). Let \(g(x,t)\in C^\infty (\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) such that \(\lvert g(x,t)\rvert \le \lvert x\rvert ^2\) for all \(x\in \mathbb {R}^n\). Consider the ODE \[ y'=Ay+g(y,t). \]

1. Show that if the initial value \(\lVert x(0)\rVert \) is small enough, then the solution exists for all time and \(\lim _{t\to \infty }x(t)=0\). [Hint: Gronwall’s Inequality for nonnegative functions \(u(t)\) and \(v(t)\) that satisfy \(v(t)\le c_0+\int _0^t u(s)v(s)\dd s\) is \(v(t)\le c_0\exp \left (\int _0^t u(s)\dd s\right )\).]
2. Let \(z(t)=0\) for all \(t\) be the zero solution. Define what it means for \(z\) to be Stable (often called Liapunov Stable.) Discuss the stability of \(z\) for this equation.

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\quad x\in \mathbb {R}. \] along with the intial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s formula \[ u(x,t)=\frac 12\bigl (g(x+ct)+g(x-ct)\bigr ) +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Briefly explain the concepts of domain of dependence, and finite propagation speed, and prove that these properties follow from d’Alembert’s formula.
3. Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s formula as a starting point, find a closed-form solution for \(u(x,t)\), with \(x,t>0\).

Consider the linear non–autonomous first order system \[ \dot {x}=Ax+B(t)x,\qquad x\in \mathbb {R}^n \] with \(A\) non-singular and \(B(t)\) continuous for \(t\ge 0\). Further, assume that

• the eigenvalues of \(A\) have non–positive real parts and those with zero real part are non-degenerate
• \(\displaystyle \int _0^\infty \lVert B(t)\rVert \dd t=c_1\) with \(c_1\) a positive constant.

1. Let \(\Phi (t)\) be the fundamental matrix of the equation \(\dot {x}=Ax\) with \(\Phi (0)=I\). Derive the variation of constants formula \[ x(t)=\Phi (t)x(0)+\int _0^t \Phi (t-s)B(s)x(s)\dd s \]
2. Prove that the solution \(x(t)\) is bounded for all times \(t>0\). [Hint: Use part (a) and Gronwall’s lemma in the following form: \(v(t)\le v_0+\int _0^t u(s)v(s)\dd s\) implies that \(v(t)\le v_0\exp (\int _0^t u(s)\dd s)\) for \(t>0\)]. What does this imply about the stability of the origin?

Use an indirect argument (and basic existence uniqueness principles) to prove the following:

If \(u:\mathbb {R}\to \mathbb {R}\) belonging to \(C^2(-\infty ,\infty )\) is a one-signed solution (i.e. it either is everywhere nonnegative or everywhere nonpositve) of the equation \[ -u''+V(t)u=u^4, \] where \(V:\mathbb {R}\to \mathbb {R}\) is continuous, then \(u\) is either identically zero or never zero.

Limit Sets. Let \(\varphi _t(x_0)\) be the flow of the differential equation \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\in C^1(\mathbb {R}^n)\) satisfies a global Lipschitz condition, and \(x_0\in \mathbb {R}^n\).

1. Given a trajectory \(\Gamma \) of \(\varphi _t\), define the \(\omega \)-limit set \(\omega (\Gamma )\).
2. Prove that \(\omega (\Gamma )\) is closed and invariant with respect to the flow \(\varphi _t\).

Dependence on Initial Conditions. Consider the scalar initial value problem \[ \begin {aligned} y'' &= y-y^3,\\ y(0) &= a,\\ y'(0) &= b. \end {aligned} \] Denote the dependence of \(y(t)\) on its initial conditions by \(y(t;a,b)\). Without explicitly solving for \(y(t;a,b)\), find \(\frac {\partial y}{\partial a}(t;0,0)\) and \(\frac {\partial y}{\partial b}(t;0,0)\), then write down a first-order approximation for \(y(t;a,b)\) which is valid for small \(\lvert a\rvert ,\lvert b\rvert \).

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be open. Suppose \(u\in C(\Omega )\) satisfies the mean value property: for every ball \(B_r(\xi )=\{x\in \mathbb {R}^n:\lvert x-\xi \rvert <r\}\) with \(B_r(\xi )\subset \Omega \), \[ u(\xi )=\frac {1}{\omega _n}\int _{\lvert x\rvert =1}u(\xi +rx)\,dS_x, \] where \(\omega _n\) is the measure of the \((n-1)\)-dimensional unit sphere in \(\mathbb {R}^n\). Prove that \(\Delta u=0\) in \(\Omega \).

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. 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.

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be a domain. Let \(\Omega _1\) be a bounded subdomain with \(\overline {\Omega }_1\subset \Omega \).

1. State the Harnack Inequality for harmonic functions.
2. Let \(\{u_n\}\) be a monotone increasing sequence of nonnegative harmonic functions on \(\Omega \). Suppose that for some point \(\xi \in \Omega _1\), the sequence of real numbers \(\{u_n(\xi )\}\) is bounded. Prove that \(\{u_n\}\) converges uniformly on \(\Omega _1\) to a harmonic function. You may use standard theorems about convergence of sequences of harmonic functions without proof.

Distributions. Suppose \(f(\xi ),g(\xi )\in C^2_0(\mathbb {R})\) are compactly supported twice continuously differentiable functions. Then \[ u(x,t)=f(x+t)+g(x-t) \tag {2} \] is a classical solution of the wave equation \[ u_{tt}-u_{xx}=0\qquad \text {for }-\infty <x,t<\infty . \tag {3} \] Show that if \(f(\xi ),g(\xi )\in C_0(\mathbb {R})\) are merely compactly supported continuous functions, then (2) still defines a solution to (3) in the sense of distributions.

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.

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?

Let \(E\) be an open subset of \(\mathbb {R}^n\) containing the origin and \(x_0\). Assume that \(\mathbf {f}\) is a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\). Suppose \(\mathbf {f}(0)=0\) and \(A=D\mathbf {f}(0)\). Let \(U\) and \(V\) be subsets of \(E\) containing the origin.

1. Define the maximal interval of existence \(I(x_0)\).
2. Define the flow \(\phi _t(x_0)\).
3. Define a homeomorphism \(H\) of \(U\) onto \(V\).
4. Suppose \(x_0\in U\). State the Hartman-Grobman theorem in terms of \(\phi _t\), \(A\), \(H\) and \(x_0\). Be sure to include all necessary conditions.
5. Consider \(x=(x_1,x_2)\) with \[ \begin {aligned} \dot {x}_1&=-x_1,\\ \dot {x}_2&=x_2-x_1^2, \end {aligned} \] and \(x(0)=x_0\). Write down the explicit solution to this equation. Give the linearized system \(\dot {y}=Ay\), \(y(0)=y_0\) and its explicit solution. Give the homeomorphism \(H\) which relates the two solutions.

Use the above and a Gronwall inequality argument (note: \(\lvert f(t,x)\rvert \le \lvert f(t,x)-f(t,0)\rvert +\lvert f(t,0)\rvert \)) to show that solutions of (2.2) are defined on all of \(\mathbb {R}\).

Let \(G\) be an open subset of \(\mathbb {R}^N\) and let \(f\in L^2(G)\). Show that \(f\) defines a continuous linear functional on \(H^1(G)\).

Consider the problem \[ \begin {aligned} \frac {dx}{dt} &= f(x,t),\\ x(t_0) &= x_0 \end {aligned} \tag {1.1} \] where \(f\in C(D,\mathbb {R}^n)\) and \(D\subset \mathbb {R}^n\times \mathbb {R}\) is open. Suppose that \(f\) is Lipschitz continuous in \(x\) for \((x,t)\in D\).

1. Prove the existence of a local (in \(t\)) solution to (1.1) using successive approximations.
2. Prove the uniqueness of this solution and show that it can be extended until it meets \(\partial D\).
3. Prove that the solution is continuous in \((x_0,t_0)\).

Show that \(\dfrac {\partial \varphi }{\partial \xi }(t;\xi )\) exists for all \(t\ge 0\), \(\xi \in \mathbb {R}\).

Let \(f(t,y)=\dfrac {y-3}{1+y^2+t^2}\). Find \(\dfrac {\partial \varphi }{\partial \xi }(t;3)\).

Assume that \(A\) is a real matrix whose eigenvalues have negative real part \((\Reals \,\sigma (A)<0)\) and \(B(t)\) is continuous for all \(t\in \mathbb {R}\). Show that there is an \(\epsilon >0\) depending on \(A\) so that if \(\norm {B}\le \epsilon \) then the zero solution of (2) with \(f(x,t)=Ax+B(t)x\) is uniformly Liapunov stable.

Show that there is a sequence of functions \(u_i\in S_\varphi \) so that \(u_i\to w_\varphi \) uniformly in \(\overline {B_r(\xi )}\).

Show that \(w_\varphi \) is harmonic in \(B_r(\xi )\).

Let \(\zeta \in C_0(\mathbb {R})\) be continuous with compact support such that the integral \[ \int _{-\infty }^{\infty }\zeta (x)\,dx=1. \] Show that as distributions, the convolutions \(\zeta _\epsilon *F\to F\) as \(\epsilon \to 0\), where \[ \zeta _\epsilon (x):=\frac {1}{\epsilon }\zeta \left (\frac {x}{\epsilon }\right ). \]

State the Poincaré inequality for functions \(u\in H^1_0(\Omega )\).

Let \(f\) have the special form \[ f(t,x)=Ax+h(t), \tag {2.10} \] where \(h:\mathbb {R}\to \mathbb {R}^N\) is a continuous function of period \(T\) and \(A\) is a constant \(N\times N\) matrix which is positive (negative) definite (i.e. \(Ax\cdot x\ge \mu \lvert x\rvert ^2\), resp. \(Ax\cdot x\le -\mu \lvert x\rvert ^2\), \(\mu >0\)). Use the result proved above to show that the equation \[ x'=Ax+h(t) \tag {2.11} \] has a unique \(T\)-periodic solution.

Use Gronwall’s inequality (or a similar argument) to establish the existence of a priori bounds on \(\lvert x(t)\rvert ^2\) for a solution \(x\) of (2.13).

Apply this theorem to the system \[ \begin {aligned} u'&=-v\\ v'&=-v+g(u)\\ x&=\begin {pmatrix}u\\v\end {pmatrix}, \end {aligned} \] where \(g:\mathbb {R}\to \mathbb {R}\) is a \(C^1\) function such that \[ ug(u)>0,\quad u\ne 0, \] using \[ \phi (u,v)=\frac {v^2}{2}+\int _0^u g(\xi )\,d\xi . \] What is your conclusion?

Suppose \(D\) is the rectangle defined by \(\lvert x-x_0\rvert \le a\), \(\lvert y-y_0\rvert \le b\) in the plane \(\mathbb {R}^2\). Suppose that \(f\) is a real valued function defined on \(D\) which is continuous, hence bounded by some nonnegative number \(M\) on \(D\).

1. Show that if \(\alpha =\min \{a,b/M\}\), then there exists a function \(y(x)\), defined on \(\lvert x-x_0\rvert \le \alpha \), such that \[ y'(x)=f(x,y(x))\quad \text {and}\quad y(x_0)=y_0. \]
2. Show that the solutions \(y\) guaranteed in part (i) may not be unique; that is, give an example in which (i) holds and more than one solution \(y(x)\) exists through the point \((x_0,y_0)\) in your example.

Consider the initial value problem \[ x'=f(t,x),\qquad x(t_0)=x_0 \] on the set \(X:\lvert t-t_0\rvert \le a\), \(\lvert x-x_0\rvert \le b\). Let \(M=\max \lvert f(t,x)\rvert \) on \(X\), \(h=\min (a,b/M)\). Assume that \(f\) is continuous on \(X\), \(L\ge 0\) and \(f\) satisfies the Lipschitz condition \(\lvert f(t,x)-f(t,y)\rvert \le L\lvert x-y\rvert \) for \((t,x)\in X\), \((t,y)\in X\).

Prove the Picard–Lindelöf existence and uniqueness theorem: There exists a unique solution \(x(t)\) defined on the interval \(\lvert t-t_0\rvert \le h\).

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\). Complete the following:

1. Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\xi =re^{i\theta }\) and \(\sqrt {\xi }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \). \[ G(x,y,\xi )= \begin {cases} e^{i\sqrt {\xi }y}\sin (\sqrt {\xi }x)/\sqrt {\xi } & \text {for }0\le x\le y<\infty ,\\ e^{i\sqrt {\xi }x}\sin (\sqrt {\xi }y)/\sqrt {\xi } & \text {for }0\le y\le x<\infty . \end {cases} \]
2. Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.

Assume the conditions of d) and let \(B(t)\) be a continuous matrix function with \[ \int _0^\infty \lVert A(t)-B(t)\rVert \,dt <+\infty . \] Let \(\Psi \) be a solution of \[ \Psi '=B(t)\Psi . \] Show that \(\Psi \) must be uniformly bounded on \([0,\infty )\).

Hint: \(\Psi \) also solves \[ \Psi '=A(t)\Psi +(B(t)-A(t))\Psi . \] Use variation of constants.

Let \(G\) be an open set in \(\mathbb {R}^n\) with \[ \sup \{\lvert x_1\rvert :(x_1,x_2,\ldots ,x_n)\in G\}=K<+\infty . \] Then \[ \lVert \varphi \rVert _{L^2(G)}\le 2K\lVert \partial _1\varphi \rVert _{L^2(G)}, \quad \forall \varphi \in H_0^1(G). \tag {6} \] \[ \left (\text {Hint: For }\varphi \in C_0^\infty (G)\text { compute } \frac {\partial }{\partial x_1}(x_1\lvert \varphi (x)\rvert ^2)\right . \] \[ \left .\text {and use the divergence theorem.}\right ) \] Use (6) to obtain Poincaré’s inequality for functions \(\varphi \in H_0^1(G)\), whenever \(G\) is a bounded domain.

Define the concepts “strongly continuous semigroup on \(H\)” and “infinitesimal generator” of the semigroup.

Let \(\{T(t):t\ge 0\}\) be a strongly continuous semigroup on \(H\) with infinitesimal generator \(A\). Show that \[ T(t):D(A)\longrightarrow D(A) \] and that the map \[ t\longmapsto T(t)x,\qquad t\in [0,\infty ), \] is differentiable and \[ \frac {d}{dt}T(t)x=AT(t)x=T(t)Ax,\quad x\in D(A). \]

State necessary and sufficient conditions in order that a linear operator be the generator of a strongly continuous contraction semigroup on \(H\).

1. What does Floquet’s Theorem say about solutions of linear \(n\times n\) systems \(X'=A(t)X\) (\(A\) is a continuous \(n\times n\) matrix)?
2. Show that, given any \(n\times n\) matrices \(B\) (constant) and \(A(t)\) (continuous), there is a change of variables \(Y=P(t)X\) which reduces the system \(X'=A(t)X\) to the system \(Y'=BY\).
3. Show that there is a particular choice of \(P(t)\) and \(B\) that is related to Floquet’s Theorem. Why might you be interested in that choice?

Consider the Dirichlet problem in a bounded plane domain \(\Omega \): \[ u_{xx}+u_{yy}=0\quad \text {in }\Omega , \] \[ u=f\quad \text {on }\partial \Omega , \] where \(f\) is continuous on \(\partial \Omega \).

1. Prove that this Dirichlet problem has at most one solution. Define clearly what you mean by a solution.

2. Prove that if \(\Omega \) is a bounded domain for which the Dirichlet problem is solvable for all \(f\in C(\partial \Omega )\) then the mapping \(f\in C(\partial \Omega )\longrightarrow u\in C(\overline {\Omega })\) is continuous in the uniform norms. (\(\overline {\Omega }\) denotes the closure of \(\Omega \).)

DE Qualifying Examination 1992

Consider the integral \[ u(x,t)=\frac {1}{2\sqrt {\pi t}}\int _{-\infty }^{\infty } f(\xi )e^{-(x-\xi )^2/4t}\,d\xi . \]

1. Prove that if \(f(x)\) is continuous and bounded for all \(x\) then \(u\) is a solution of the initial value problem \[ u_t=u_{xx}\quad \text {for all }-\infty <x<\infty ,\ t>0. \] \[ u(x,0)=f(x)\quad \text {for all }-\infty <x<\infty . \] Formulate clearly what you mean by a solution.
2. Show how the results of part a) can be used to solve the initial-boundary value problem \[ u_t=u_{xx}\quad \text {for all }0<x<1,\ t>0, \] \[ u(x,0)=g(x)\quad \text {for }0\le x\le 1, \] \[ u(0,t)=u(1,t)=0\quad \text {for }t\ge 0. \] Define what you mean by a solution and state what conditions on \(g\) guarantee that such a solution exists.

DE Qualifying Examination 1992