Existence and Uniqueness Theorems

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 \(t\in [0,\infty )\).

[25] Let \((t_0,x_0)\in \Omega \) be a point in an open set \(\Omega \subset \mathbb {R}\times \mathbb {R}^d\) and \(f:\Omega \to \mathbb {R}^d\) be a continuously differentiable function. Show that for some \(\epsilon > 0\) the initial value problem

has a unique continuously differentiable solution \(u(t):[0,\epsilon ]\to \mathbb {R}^d\) such that \((t,u(t))\in \Omega \) for all \(0 \le t \le \epsilon \). [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

[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 \(t\in [0,T]\).]

1. For the system \(\dot {u}=f(u)\) where \(f\) is a continuously differentiable function on \(\mathbb {R}^d\), give the definition of an invariant set, attracting set and \(\omega \)-limit set.

2. Determine the \(\omega \)-limit sets for this system and prove their existence. Sketch the phase portrait.

   \[ \dot {x}=x-y-x^3 \]
   \[ \dot {y}=-x+y-y^3. \]

Let \(A\) be a real \(d\times d\) matrix. Consider the initial value problem

(3)

where \(h(t)\in \mathbb {R}^d\) is a real \(T\)-periodic continuous function.

1. Define the Poincare Map in this context and give an explicit formula for the Poincare Map in terms of \(A\), \(h\) and \(x_0\).
2. Under what conditions on \(A\) and \(h\) is there is a unique \(T\)-periodic solution of (3)?
3. Compute the derivative of the Poincare map at the periodic solution. What stability property do you conclude for the periodic solution?

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.]

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 \(t\in [0,T]\).]

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

where \(h(t)\in \mathbb {R}^d\) is a \(T\)-periodic continuous function.

1. Define the Poincare Map in this context and give an explicit formula for the Poincare Map in terms of \(A\) and \(h\)
2. Show that there is a unique \(T\)-periodic solution of (3). [Hint: there is again an explicit equation in terms of \(A\) and \(h\).]
3. What stability property does the periodic orbit have? Provide a justification of your answer with explicit calculation/proof.
4. How could the requirements stipulated on the eigenvalues of \(A\) be generalized to still maintain the existence and uniqueness results for the periodic orbit from (b)?

1. Suppose the function \(f(t,x)\) is periodic in \(t\), \(f(t+1,x)=f(t,x)\). Define the Poincare map for the differential equation \(\dot {x}=f(t,x)\).

2. Consider the differential equation

   \[ \dot {x}=x(1-x)-h(t)x, \]

   where \(h(t)\ge 0\) is periodic, \(h(t+1)=h(t)\). Under what conditions on \(h(t)\) does this equation have a strictly positive periodic solution?

3. Suppose \(h(t)=a\sin ^2(\pi x)\). What condition on \(a\) guarantees the existence of a positive periodic solution?

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 \(t\in J\).]

Consider the differential equation \[ \frac {dx}{dt}=f(x,t),\quad x\in \mathbb {R}^n, \] with initial condition \(x(0)=x_0\).

1. State a theorem, with conditions on \(f(x,t)\), guaranteeing the existence and uniqueness of solutions of the differential equation on some interval \(0<t<T_0\).
2. Outline the proof of your theorem, indicating the main steps of the proof (without all the details), indicating where your stated assumptions on \(f(x,t)\) are invoked. What are Picard iterates and how do they play a role in your proof?

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.

The goal of this problem is to use Picard iterates to find the solution of the initial value problem \[ \frac {dx}{dt}=x^2,\qquad x(0)=1 \tag {1} \]

1. State a theorem, without proof, with necessary conditions on \(f(x,t)\), guaranteeing the existence of a unique solution of the differential equation \(\frac {dx}{dt}=f(x,t)\), \(x(0)=x_0\).
2. Verify that the function \(f=x^2\) satisfies these necessary conditions.
3. Define the Picard iterates for the equation (1) and calculate the first three functions \(x_0(t)\), \(x_1(t)\), and \(x_2(t)\).
4. Prove by induction that \[ x_n(t)=1+t+t^2+\cdots +t^n+O(t^{n+1}) =\sum _{j=0}^{n}t^j+O(t^{n+1}). \tag {2} \] Hint: You will need to show that \[ \sum _{j=0}^{n}t^j\sum _{k=0}^{n}t^k =\sum _{j=0}^{n}jt^j+O(t^{n+1}). \]
5. Find the exact solution of (1). How does it compare to the approximate solution found in the previous part?

1. Show that the wave equation \(u_{tt}=c^2u_{xx}\) \((-\infty <x<\infty ,\ t>0)\) conserves energy \[ E=\int _{-\infty }^{\infty }\left [(u_t)^2+c^2(u_x)^2\right ]d x. \]
2. Show the uniqueness: If two solutions of the wave equation satisfy the same initial conditions, then the solutions are identical.

1. Show the uniqueness of Robin’s problem \[ \begin {aligned} \Delta u&=f(x)\ [x\in D],\\ \nabla u\cdot \mathbf {n}+a(x)u&=h(x)\ [x\in \partial D],\qquad [D\text { is a 3D domain, }a(x)>0]. \end {aligned} \]
2. When \(a\equiv 0\), we have the Neumann problem. Show that it has no solution, unless \[ \int _D fd x=\oint _{\partial D}hd A. \] Describe physical meaning of this fact in the case of steady heat conduction.
3. Prove the uniqueness of the Neumann problem up to an additive constant.

1. Let \(f(t,x):\mathbb {R}\times \mathbb {R}^n\to \mathbb {R}^n\) be a continuously differentiable function. Show that there is an \(\epsilon >0\) and a unique continuously differentiable function \(y(t)\in \mathbb {R}^n\) for \(t\in [0,\epsilon ]\) that satisfies the initial value problem \[ \frac {dx}{dt}=f(t,x),\qquad x(0)=x_0. \]
2. Construct the first few Picard iterates for the differential equation \[ \dot {x}=2t-\sqrt {\max (0,x)},\qquad x(0)=0. \tag {1} \] Show that the Picard iterates do not converge. Why do they not converge?

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 \(t\in J\).]

Consider the torqued pendulum with constant drag \(\beta >0\) and torque \(\mu >1\) large enough to overcome the pull of gravity, no matter what the angle \(x\) of the pendulum may be. \[ \begin {aligned} \dot {x}&=y\\ \dot {y}&=\mu -\beta y-\sin x \end {aligned} \]

1. Show that the solution of the initial value problem with \(x(0)=0\) and \(y(0)=y_0\) exists for all time.
2. Show that there is a solution such that the pendulum continues to rotate indefinitely in a \(T\)-periodic fashion. It would have \(x^*(T+t)=x^*(t)+2\pi \) and \(y^*(T+t)=y^*(t)\).
3. Show that the periodic solution is unique.
4. Show that the periodic solution is orbitally stable.

Suppose that the initial value problem for the wave equation \[ \begin {aligned} \frac {\partial ^2u}{\partial t^2}-c^2\nabla ^2u&=0, &&x\in \mathbb {R}^n,\quad t>0,\\ u(0,x)&=g(x),\\ \frac {\partial u}{\partial t}(0,x)&=0, &&x\in \mathbb {R}^n \end {aligned} \] has a solution \(u_g\). Find (and verify) the solution \(u_h\) of the same equation subject to the initial conditions \[ \begin {aligned} u(0,x)&=g(x),\\ \frac {\partial u}{\partial t}(0,x)&=h(x), &&x\in \mathbb {R}^n \end {aligned} \] in terms of \(u_g\), \(g\) and \(h\).

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.

Let \(p(t)\), \(q(t)\) and \(r(t)\) be continuous functions defined for \(t\in \mathbb {R}\) and \(t_0,x_0,x_1\in \mathbb {R}\). Consider the initial value problem \[ \begin {cases} \ddot {x}+p(t)\dot {x}+q(t)x=r(t)\\ x(t_0)=x_0,\qquad \dot {x}(t_0)=x_1. \end {cases} \tag {1} \]

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. Carefully reformulate (1) as a fixed point problem. Prove that the fixed point problem is equivalent to the local existence problem for (1).
3. Using the Contraction Mapping Theorem, show that there is a \(\delta >0\) and a unique function \(x(t)\in C^2([t_0,t_0+\delta ])\) that satisfies your fixed point problem, and hence (1), for \(t_0\le t\le t_0+\delta \).

Assuming the local existence result in Problem A1 for equation (1), show that in fact a unique solution exists globally for all \(t\in \mathbb {R}\).

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.

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 \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y}\qquad x,y\in V. \]

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 \[ \frac {dx}{dt}=f(x),\qquad x(0)=x_0. \]

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.

1. Consider a string clamped at the end points \(a,b\) with \(u(a,t)=u_a\), \(u(b,t)=u_b\), where \(u(x,t)\) is the string’s deviation from the horizontal rest position. The kinetic energy of the string is \[ E_k=\frac {1}{2}\int _a^b \rho u_t^2\dd s,\qquad \dd s=\sqrt {1+u_x^2}\dd x, \] where \(\rho (x,t)\) is the mass density. The potential energy due to stretching of the string is \[ E_p=\int _a^b k\left (\sqrt {1+u_x^2}-1\right )\dd x, \] where \(k(x,t)\) is the elastic coefficient.

   1. Find the first variation of the action \[ A=\int _0^T(E_k-E_p)\dd t. \]
   2. Integrating by parts, imposing the boundary conditions, and equating the first variation to zero, derive the PDE \[ \left (\rho u_t\sqrt {1+u_x^2}\right )_t = \left (k(1+u_x^2)^{-1/2}u_x\right )_x. \] Assuming \(\rho ,k\) are constants and using the approximation \(\abs {u_x}\ll 1\) derive the classical one-dimensional wave equation.
2. Consider the functional \[ Y[u]=\int _D\left (\frac {1}{2}\abs {\nabla u}^2+\frac {1}{2}u^2+fu\right )\dd x, \] where \(D\) is a bounded domain in \(\mathbb {R}^n\) and \(f\) is a bounded, continuous function in \(D\). Suppose that there exists a minimizer \(u^*\) of \(Y[u]\) in the Sobolev space \(H_1(D)\). Use this to prove existence and uniqueness of the weak formulation of the PDE (for which \(u\in C^1(D)\)) \[ -\nabla ^2u+u=-f,\qquad x\in D\subset \mathbb {R}^n,\qquad \partial _nu=0,\qquad x\in \partial D. \] [Hint: show that the weak formulation can be expressed as \((u,\psi )_{H_1(D)}=0\) for all \(\psi \in H_1(D)\).]

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

Let \(A\) be an \(n\times n\) real matrix and let \(\varphi (t,x_0)\) denote the solution of the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Suppose that \(k\) not necessarily distinct eigenvalues of \(A\) have non-positive real part and the rest have positive real part \[ \Re \lambda _1\le \cdots \le \Re \lambda _k\le 0 <\Re \lambda _{k+1}\le \cdots \le \Re \lambda _n. \] Show that there is a \(k\)-dimensional subspace \(E\subset \mathbb {R}^n\) such that

1. \(E\) is forward invariant: for every \(x_0\in E\) we have \(\varphi (t,x_0)\in E\) for all \(t\ge 0\);
2. If \(x_0\notin E\) then the distance from \(\varphi (t,x_0)\) to \(E\) tends to infinity as \(t\to \infty \).

Let \(f(x)\) be continuously differentiable for all \(X\in \mathbb {R}^n\) and let \(x_0\in \mathbb {R}^n\times \mathbb {R}\). 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 \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \]

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.

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 \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Let \(A\) be an \(n\times n\) real matrix. Consider the linear initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Give an algebraic condition on the matrix \(A\) which holds if and only if all solutions of the initial value problem are bounded for all \(t\in \mathbb {R}\). Prove the equivalence.

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.

Let \(f(x)\) be continuously differentiable for all \(x\in \mathbb {R}^n\) 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 \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \]

Let \(u\) solve the initial value problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0, && \text {in }\mathbb {R}\times (0,\infty ),\\ u=g,\quad u_t=h, &&& \text {on }\mathbb {R}\times \{0\}. \end {aligned} \] Suppose \(g,h\) have compact support. Define the kinetic energy \(k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\dd x\) and the potential energy \(p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\dd x\). Prove

1. The total energy \(E=k(t)+p(t)\) is constant in \(t\).
2. Assume that there are two distinct solutions \(u,v\) to the initial value problem. By considering the energy of the difference solution \(U=u-v\), prove that \(u=v\), that is, the solution is unique.

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 \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Assume that \(u\in C^2(\mathbb {R}^n\times [0,\infty ])\) satisfies the initial value problem for the wave equation \[ \begin {cases} u_{tt}-\Delta u=0, & \text {in }\mathbb {R}^n\times (0,\infty ),\\ u=g,\ u_t=h, & \text {on }\mathbb {R}^n\times \{t=0\}. \end {cases} \] Recall that when \(n=3\), using spherical means and the Euler-Poisson-Darboux equation, one can derive Kirchhoff’s formula \[ \begin {aligned} u(x,t) &= \frac {\partial }{\partial t}\left (t\fint _{\partial B(x,t)}g\dd S\right ) +t\fint _{\partial B(x,t)}h\dd S\\ &= \fint _{\partial B(x,t)} \bigl (th(y)+g(y)+Dg(y)\cdot (y-x)\bigr )\dd S(y),\qquad t>0. \end {aligned} \]

1. Outline the procedure (the “method of descent”) for deriving a similar formula (Poisson’s formula) for the initial value problem in the case \(n=2\).
2. Explain how a disturbance originating at a point \(x\in \mathbb {R}^2\) at time \(t=0\) propagates in relation to the cone \(C=\{(y,t):t>0,\ \lvert x-y\rvert <t\}\). In particular, using the Kirchhoff and Poisson formulas, show that there is a qualitative difference in propagation behavior between the cases \(n=2\) and \(n=3\).

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

Let \(u\) solve the initial value problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0, && \text {in }\mathbb {R}\times (0,\infty ),\\ u=g,\quad u_t=h, &&& \text {on }\mathbb {R}\times \{0\}. \end {aligned} \] Suppose \(g,h\) have compact support. Define the kinetic energy \(k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\dd x\) and the potential energy \(p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\dd x\). Prove

1. \(k(t)+p(t)\) is constant in \(t\).
2. \(k(t)=p(t)\) for all \(t\) sufficiently large.

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 \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Consider the problem \[ \begin {aligned} u_t &= u_{xx}, && t\in (0,1),\ x\in (0,1),\\ u(0,t)=u(1,t) &= 0, && t\in (0,1),\\ u(x,1) &= f, && x\in (0,1). \end {aligned} \]

1. What are the least restrictive condition you can find on \(f\) which guarantee that the problem admits a solution \(u\), bounded in the norm \[ \norm {u}_B=\sup _{t\in [0,1]}\left (\int _0^1 u(x,t)^2d x\right )^{1/2}\ ? \]
2. Under what conditions on \(f\) does the stability estimate \[ \norm {u}_B\le C\norm {f}_{L^2(0,1)} \] hold?
3. Assuming \(u\in C^2(\overline {U_1})\), where \(U_1=(0,1)\times (0,1]\), is uniqueness of solutions guaranteed?

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 \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \]

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.

The response of a certain biological oscillator, \((x,y)\), \(x\ge 0\), \(y\ge 0\), to a stimulus of size \(b\) satisfies the differential equation \[ \begin {aligned} \dot {x}&=x-ay+b,\quad &\dot {y}&=x-cy && \text {for } x\ge 0,\ y\ge 0;\\ &&\dot {y}&=-cy && \text {for } x=0, \end {aligned} \] with \(a,b,c>0\).

1. Using phase-plane analysis show that there exists a limit cycle, part of which lies on the \(y\)-axis, when \(c<1\) and \(4a>(1+c)^2\). [Hint: existence of limit cycle depends on existence of an unstable spiral].
2. Show that the period of the orbit is independent of \(b\).

Let \(U\) be an open subset of \(\mathbb {R}^2\). Prove that if \(u\in C^2(U)\) satisfies \[ \Delta u-u=0 \quad \text {in } U, \] then \[ u(x)=\frac {1}{I(r)}\int _{\partial B(x,r)} u\,dS_y \] for every ball \(B(x,r)\subset U\), where \[ I(r)=\frac {1}{\pi }\int _0^\pi e^{r\cos t}\,dt. \] Assume without proof that \(I(r)\) is the unique solution to the modified Bessel equation \[ r\frac {d^2\varphi }{dr^2}+\frac {d\varphi }{dr}-r\varphi =0,\quad r>0, \] which satisfies \(\varphi (0)=1\).

Let \(U=\{x\in \mathbb {R}^2:\frac {1}{2}<|x|<1\}\), and denote \(\partial U=\Gamma _1\cup \Gamma _2\) where \[ \Gamma _1=\{|x|=\tfrac {1}{2}\},\quad \Gamma _2=\{|x|=1\}. \] Consider \[ \begin {aligned} -\Delta u &= f, && \text {in } U,\\ \alpha u+\frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _1,\\ \frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _2, \end {aligned} \] where \(\alpha >0\), \(\eta \) is the unit outward normal on \(\partial U\), and \(f\in L^2(U)\).

Prove that there exists a constant \(C>0\) such that \[ C\int _U u^2\,dx \le \int _U |Du|^2\,dx+\int _{\Gamma _1} u^2\,dS \] for all \(u\in C^1(\overline {U})\). Polar coordinates may be helpful. Explain how this estimate could be used in a proof of existence and uniqueness for solutions of the problem above.

Contraction Mapping Theorem. Show that the problem \[ \begin {aligned} \dot {x}(t) &= f(x(t))+\frac {1}{a}\int _0^a x(s)\,ds,\quad t\in (0,a),\\ x(0) &= 0, \end {aligned} \] admits a unique solution \(x\in C([0,a];\mathbb {R}^n)\) for \(a>0\) sufficiently small. Assume \(f:\mathbb {R}^n\to \mathbb {R}^n\) satisfies \(\lvert f(x)-f(y)\rvert \le M\lvert x-y\rvert \) for all \(x,y\in \mathbb {R}^n\).

Linear Systems. Consider the linear initial value problem \[ \begin {aligned} \dot {x} &= Ax,\\ x(0) &= x_0, \end {aligned} \] where \(A\in \mathbb {R}^{n\times n}\) is constant, and \(x_0\in \mathbb {R}^n\) is given.

1. Prove that the matrix exponential \(e^{At}\) exists and satisfies \(\lVert e^{At}\rVert \le e^{\lVert A\rVert t}\), where \(\lVert \cdot \rVert \) is the operator norm.
2. Prove directly that \(x(t)=e^{At}x_0\) is the unique solution to the initial value problem above.

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 \(\lvert a\rvert ,\lvert b\rvert \).

Variational Principles. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary \(\partial \Omega \) and unit outward normal \(\eta \). Consider the problem \[ \begin {aligned} \Delta u-\epsilon u &= f,\quad \text {in }\Omega ,\\ \frac {\partial u}{\partial \eta } &= 0,\quad \text {on }\partial \Omega , \end {aligned} \] where \(f\in L^2(\Omega )\) satisfies \(\int _\Omega f=0\).

1. Prove that for each real constant \(\epsilon >0\), there exists a unique weak solution \(u_\epsilon \in H^1(\Omega )\).
2. In the case \(\epsilon =0\), find an additional condition on \(u\) which guarantees a unique weak solution \(u_0\), such that the solutions \(u_\epsilon \) from part (a) converge to \(u_0\) as \(\epsilon \to 0^+\). Prove existence and uniqueness of \(u_0\) (with the extra condition), and prove \(u_\epsilon \to u_0\) in \(H^1(\Omega )\) as \(\epsilon \to 0^+\).

Heat Equation. Given the constant \(\beta \in \mathbb {R}\), consider the initial value problem \[ \begin {aligned} u_t &= \beta \Delta u,\quad x\in \mathbb {R}^n,\quad t>0,\\ u(x,0) &= g(x),\quad x\in \mathbb {R}^n. \end {aligned} \]

1. Assume \(\beta >0\) and let \(g\in C(\mathbb {R}^n)\) be bounded. Prove that there exists a bounded solution \(u(x,t)\) which is \(C^\infty \) for all \(t>0\). Under what condition are solutions unique?
2. Assume \(\beta <0\) and let \(g\in C^\infty (\mathbb {R}^n)\) be bounded. Are solutions \(u(x,t)\) necessarily bounded and continuous for \(t>0\)? Prove, or provide a counterexample.

Conservation Laws. Consider the first-order equation \[ (G(u))_x+u_y=0, \] where \(G\in C^\infty \), with initial condition \(u(x,0)=h(x)\).

1. Using the method of characteristics, derive the general solution of the initial value problem. Assuming \(h\in C^\infty \), find a condition on \(G'(h(x))\) which implies the solution \(u(x,y)\) will develop a singularity in finite time \(y>0\).
2. Set \(G(u)=\frac {1}{2}u^2\). Given \(u_0\in \mathbb {R}\), find explicit weak solution(s) for \[ h(x)= \begin {cases} u_0, & x<0,\\ 0, & x>0. \end {cases} \] Show that your solutions are weak solutions. Which properties do your solutions exhibit: shocks, rarefaction, nonuniqueness?

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. 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.

Contraction Mapping. Consider the initial value problem \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\) satisfies for all \(x,y\in \mathbb {R}^n\) \[ \lvert f(x)-f(y)\rvert \le (1+\lvert x\rvert +\lvert y\rvert )\lvert x-y\rvert . \] Use the contraction mapping principle to find a condition on \(\lvert x_0\rvert \) which guarantees local existence and uniqueness of solutions. How does the local time interval upon which existence is obtained depend on \(\lvert x_0\rvert \)?

Global Existence. 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 \(f\in C^1(E)\), \(E\subset \mathbb {R}^n\) is open, and \(x_0\in E\). Prove that if the system has a Liapunov function \(V\) with \(\dot {V}(x)=\left .\frac {d}{dt}V(\varphi _t(x))\right |_{t=0}\le 0\) for all \(x\in E\), and the set \(S=\{x\in E:V(x)\le 1\}\) is compact, then solutions \(\varphi _t(x_0)\) with \(x_0\in S\) exist for all positive times \(t>0\).

Linear Systems. Suppose \(A\) is a real \(n\times n\) matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i\le 0\). Show that there are constants \(C<\infty \) and \(k\le n-1\) so that any solution \(x(t)\in \mathbb {R}^n\) of the initial value problem \[ \begin {aligned} \frac {dx}{dt} &= Ax,\\ x(0) &= x_0, \end {aligned} \] satisfies \[ \lvert x(t)\rvert \le C(1+\lvert t\rvert ^k)\lvert x_0\rvert \] for all \(t\in \mathbb {R}\) and \(x_0\in \mathbb {R}^n\).

Wave Propagation. Consider the equation \[ u_{tt}-u_{xx}+\lambda u=0,\quad x\in \mathbb {R},\quad t>0, \] where \(\lambda >0\) is fixed.

1. Explain what is meant by a dispersive solution, and find a dispersion relation for the equation.
2. Define an “energy” functional \(E(t)\) for this equation and show that each solution \(u(x,t)\) with finite initial energy conserves energy. State conditions under which uniqueness of solutions holds.

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?

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. \[ \begin {cases} \dfrac {dx}{dt}=Ax+e^{-t}(x+v_0),\\ x(0)=x_0. \end {cases} \tag {2.} \]

Peano Existence Theorem. Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Consider the system of ODE’s \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \qquad (\spadesuit ) \]

1. Find an example that shows that the solution of \((\spadesuit )\) may not be unique. Give another example that shows that the solution may not exist for all time.
2. Prove the Peano Existence theorem: Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R})\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Then there is an \(\epsilon >0\) and a function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) which solves \((\spadesuit )\).

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?

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 \[ \begin {aligned} u_t-u_x&=f(t,x),\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \]

Let \(E\) be an open subset of \(\mathbb {R}^n\) containing \(x_0\) and assume that \(\mathbf {f}\) is a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\).

1. Use the method of successive approximations to prove that the initial value problem \(x'=\mathbf {f}(x)\), \(x(0)=x_0\), has a unique solution on some interval \([-a,a]\), and give an estimate for \(a\).
2. Indicate how the result in (a) can be used to prove a similar result for the nonautonomous case.

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.

Consider the equation \[ a(x,y)u_x+b(x,y)u_y=c(x,y,u) \] where \(a\), \(b\) and \(c\) are suitably smooth.

1. Define what is meant by an integral surface for the above equation.
2. Write down the characteristic equations and show how they can be used to construct an integral surface that contains the curve \(x=\phi (s)\), \(y=\psi (s)\), \(u=\eta (s)\), where \(\phi \), \(\psi \) and \(\eta \) are \(C^1\).
3. What additional conditions, if any, must be placed on the data to guarantee a locally unique solution? Explain carefully.
4. Demonstrate the above method by solving \[ \begin {aligned} u_x+2u_y&=-u,\\ u(0,y)&=f(y) \end {aligned} \] where \(f\) is of class \(C^1\).

State the fundamental existence uniqueness theorem for solutions of (2.2) and sketch a proof.

State a general principle about the existence intervals for solutions of (2.2) and the behavior of solutions at the endpoints of these intervals.

Use general existence uniqueness principles to give the form of all fundamental matrix solutions of (2.3).

Show that in the sense of distributions there exists a unique solution \(u\in H^1_0(G)\) of the equation \[ -\Delta u+u=f. \]

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 \[ \begin {aligned} u_t-u_x&=f(t,x),\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \]

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 \((x_0,t_0)\).

Consider the equation \[ a(x,y)u_x+b(x,y)u_y=c(x,y,u) \tag {8.1} \] where \(a\), \(b\) and \(c\) are suitably smooth.

1. Define what is meant by an integral surface for (8.1).
2. Write down the characteristic equations for (8.1) and show how they can be used to construct an integral surface of (8.1) that contains the curve \[ x=\varphi (s)\qquad y=\psi (s)\qquad u=\eta (s), \] where \(\varphi ,\psi \) and \(\eta \) are \(C^1\).
3. What additional conditions, if any, must be placed on the data to guarantee a locally unique solution? Explain carefully.
4. Consider the equation \[ u_t+uu_x=0 \tag {8.2} \] with initial data given by \[ u(x,0)=u_0(x)= \begin {cases} 1, & x\le 0,\\ 1-x, & x\in (0,1),\\ 0, & x\ge 1. \end {cases} \tag {8.3} \] Does (8.2)–(8.3) have a unique classical solution for \(t\ge 0\)? A unique weak solution for \(t\ge 0\)? Explain.

Using the method of characteristics, solve the initial value problem (3) assuming \(f(x,t)=x\) and \(\varphi (s)=s\).

Find a condition relating \(\Omega \) and \(V\) which implies there exists a unique weak solution \(u\in H^1_0(\Omega )\) for problem (6). Prove the existence and uniqueness under your condition.

Let \[ E_0=E\cap \{u:u(0)=u(\pi )=0\}, \] and restrict \(f\) to \(E_0\). Show that, except for a countable set of values of \(\lambda \) (describe this set !), the following holds: There exists \(\delta =\delta (\lambda )\) such that for all \(h\in X\), \(\norm {h}_X<\delta \), there exists a unique \(u\in E_0\) such that \[ f(u)=h, \tag {2.2} \] i.e. the problem \[ \begin {cases} u''+\lambda u+u^3=h,\\ u(0)=0=u(\pi ) \end {cases} \tag {2.3} \] has a unique 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).

Show that in the sense of distributions there exists a unique solution \(u\in W_0^{1,2}(G)\) of the equation \[ -\Delta u+u=f. \]

Let \(G\) be bounded. Show that in the sense of distributions there exists a unique solution \(u\in W_0^{1,2}(G)\) of the equation \[ -\Delta u=f. \]

1. Suppose \(A\) is an \(n\times n\) matrix whose elements \(a_{ij}(t)\) are each of period \(T\). You may assume existence and uniqueness of a fundamental matrix \(\Phi (t)\) for the equation \[ \dot {x}(t)=A(t)x(t). \] Show that \(\Phi (t+T)\) is also a fundamental matrix for the equation, hence \[ \Phi (t+T)=\Phi (t)\cdot C \] for some non-singular matrix \(C\).

2. You may assume that a non-singular matrix \(C\) has a logarithm, i.e., \(C=e^B\) for some \(n\times n\) matrix \(B\).

   Show that there exists a non-singular matrix \(P(t)\) of period \(T\) such that \[ \Phi (t)=P(t)e^{Bt} \qquad \text {Floquet's Theorem.} \]

   Hint: Solve for \(P(t)\) and check.

3. Make the change of variables \(y=P(t)u(t)\) and change the periodic system \(y'=A(t)y\) into the system \[ u'=Bu. \]

Consider the Sturm-Liouville operator \[ \mathcal {L} u=\frac {1}{w(x)}\{-(p(x)u')'+q(x)u\} \] where \(p(x)>0\), \(w(x)>0\) and \(1/p(x)\), \(q(x)\) and \(w(x)\) are locally Lebesgue integrable on an arbitrary interval \(I\) (open or closed, bounded or unbounded).

1. Define what is meant by a solution of \(\mathcal {L} u=\lambda u\) on \(I\).
2. Show that the initial value problem for the differential equation \(\mathcal {L} u=\lambda u\) on \(I\) with \(u(x_0)=c_1\), \((pu')(x_0)=c_2\) at \(x_0\in I\) is equivalent to an integral equation for the vector \((u_1,u_2)=(u,pu')\). Prove the equivalence.

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 \(\lvert t-t_0\rvert \le h\).

Show that solutions of initial value problems for (2) are defined on \((-\infty ,\infty )\).

Use the above to establish the existence and uniqueness of solutions to the initial value problem \[ U_t=U_{xx}+f(t,x),\quad t>0,\quad x\in (0,1) \] \[ U(0,x)=\varphi (x) \] \[ U(t,0)=U(t,1)=0; \] be sure to provide the appropriate function spaces for \(U\), \(f\), \(\varphi \).

Let \(E\) be an open subset of \(\mathbb {R}^n\). Let \(f\) be a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\), and let \(x_0\) belong to \(E\).

1. Use the method of successive approximations to prove that the initial value problem \(x'=f(x)\), \(x(0)=x_0\) has a unique solution on some interval \([-a,a]\), and give an estimate for \(a\).
2. Suggest how the result in a) can be used to prove a similar result for the nonautonomous case.

Let \(A\) be a constant \(n\times n\) matrix, and define the matrix exponential by \[ e^A=\sum _{k=0}^{\infty }A^k/k!. \]

1. Prove that \(e^{At}\) is differentiable (you may use known theorems about scalar power series), and that it satisfies the initial value problem \(X'=AX\), \(X(0)=I\) (the identity matrix). Justify your steps.
2. Let \(A\) have distinct eigenvalues. Prove that for the origin to be a globally asymptotically stable point for \(X'=AX\) it is sufficient that all of the eigenvalues of \(A\) have negative real parts.
3. Is the condition in b) that \(A\) have distinct eigenvalues required in order to prove the result? (no proof necessary)
4. Is the sufficient condition in b) also necessary? Justify your answer.

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