HW5 (MTH847 Fall 2017)

General Info

Remember: for every question you must indicate all the people with whom you collaborated. If you didn't collaborate with anyone else on a given problem, write "Collaborators: none."

Please write legibly, or type-up your solutions in LaTeX. The grader will greatly appreciate it.

This problem set will be due in class, Wednesday November 15.

Problems

1. The conclusions of question 9 of HW4 is true for all odd dimensions $\geq 3$. The first part, however, fails when $d = 1$:
   Suppose $u: \mathbb{R}\times\mathbb{R}\to \mathbb{R}$ solves $$- \partial^2_{tt} u + \partial^2_{xx} u = f$$ with data $$u(0,x) = g,\qquad \partial_t u(0,x) = h$$ where $f \in C^\infty_c(\mathbb{R}\times \mathbb{R})$ and $g \in C^\infty_c(\mathbb{R})$ and $h\in C^\infty_c(\mathbb{R})$. Prove that, if $h\geq 0$ and $f \leq 0$ are such that they are not both identically vanishing, then for every $x_0\in \mathbb{R}$ the function $t \mapsto u(t,x_0)$ has noncompact support.
2. Consider the nonlinear wave equation on $\mathbb{R}\times\mathbb{R}$: $$\begin{gather} \Box u = \lambda u^p \newline u(0,x) = g(x)\newline \partial_t u(0,x) = h(x) \end{gather}$$ where $\lambda\in \mathbb{R}$ and $p\geq 2$ is an integer. We can rewrite this using Duhamel's formula applied to D'Alembert's solution as the integral equation $$u(t,x) = \frac12 [ g(x+t) + g(x-t) + \int_{x-t}^{x+t} h(y) ~\mathrm{d}y ] - \frac12 \int_0^t \int_{x - (t-s)}^{x + (t-s)} \lambda u(s,y)^p ~\mathrm{d}y ~\mathrm{d}s.$$ Prove that for every $g, h\in C^0_c(\mathbb{R})$, there exists a time $T$ and a function $u\in C^0([0,T]\times \mathbb{R})$ such that $u$ verifies the integral equation given above.
   (Hint: emulate our proof of local existence for the nonlinear heat equation. Construct a sequence of functions by setting $$u_{i+1}(t,x) = \frac12 [ g(x+t) + g(x-t) + \int_{x-t}^{x+t} h(y) ~\mathrm{d}y ] - \frac12 \int_0^t \int_{x - (t-s)}^{x + (t-s)} \lambda u_i(s,y)^p ~\mathrm{d}y ~\mathrm{d}s$$ and showing that the sequence converges uniformly on $[0,T]\times\mathbb{R}$ if $T$ is chosen small enough.)
   Solution to this question is now available
3. (This question uses the same notation as HW4 question 10). Recall from class that if $\vec{v} = (v_0, v_1, \ldots, v_d)$ is a vector field, we define its deformation tensor to be $${}^{(\vec{v})}\pi_{jk} = \sum_{i = 0}^d m_{ik} \partial_{x_i} v_j + m_{ij} \partial_{x_i} v_k.$$ Suppose $\vec{v}$ is such that its deformation tensor vanishes identically.
   • Prove that the components of $\vec{v}$ satisfies $\Box v_i = 0$.
   • Prove that whenever $\phi$ solves the wave equation, so does the quantity $\psi = \sum_{i = 0}^d v_i \partial_{x_i} \phi$.
     
4. Let $\vec{v},\vec{w}$ be two time-like vectors. Prove that
   • $\vec{v}\cdot\vec{w} \neq 0$
   • the sign of $\vec{v} \cdot \vec{w}$ is determined by whether the $v_0$ and $w_0$ components have the same or opposite signs.
5. Let $\vec{v}, \vec{w}$ be any two time-like vectors, such that $\vec{v} \cdot \vec{w} > 0$. Prove that ${}^{(\vec{v})}J \cdot \vec{w} \leq 0$ with equality if and only if $\nabla \phi = 0$.
6. Suppose $\phi,\psi\in C^2(\mathbb{R}\times\mathbb{R}^d)$ both solve the wave equation. Suppose further that on the set $$\{ t^2 + |x|^2 \leq 1 \}$$ the two solutions agree. What is the largest space-time subset on which you can guarantee that $\phi = \psi$?
7. Let $\phi\in C^2(\mathbb{R}\times\mathbb{R}^d)$ solve the homogeneous wave equation. Suppose we know that along the set $${ (-1,x) : x_1 < \lambda } \cup { (1,x) : x_1 > - \lambda }$$ both $\phi$ and $\partial_t\phi$ vanish.
   • Prove that if $\lambda \geq 1$ then $\phi \equiv 0$.
   • Prove that if $\lambda < 1$ there exists a $\phi$, not identically vanishing, satisfying the conditions set out above.
8. Suppose $\phi\in C^4(\mathbb{R}\times\mathbb{R}^d)$ solves the homogeneous wave equation, with boundary conditions $$\phi(0,x) = \partial^3_{ttt} \phi(0,x) = 0.$$ Suppose further that $\partial_t \phi(0,x)$ is a bounded function on $\mathbb{R}^d$. Prove that there exists a constant $c$ such that $\phi(t,x) = c t$.
   (Hint: first study the function $\psi(t,x) = \partial^2_{tt} \phi(t,x)$.)
9. Let $\Omega\subset \mathbb{R}^d$ be a bounded open set with $C^1$ boundary. Consider the system $$\Box \phi = 0$$ on $(0,T) \times \Omega$ with initial-boundary conditions $$\phi(0,x) = \partial_t \phi(0,x) = 0, \quad x\in \Omega$$ and $$\partial_t \phi + \partial_n \phi = 0 \quad \text{ on } [0,T]\times\partial\Omega,$$ where $\partial_n \phi$ denotes the outward normal derivative on $\partial\Omega$. Prove, using the energy method, that if $\phi \in C^2([0,T]\times\overline{\Omega})$ solves the above system, then $\phi \equiv 0$ on $[0,T]\times\overline{\Omega}$.
   (Remark: the boundary condition $\partial_t \phi + \partial_n \phi = 0$ is known as the "no incoming radiation" boundary condition.)
10. Prove that the "time-reversed" version of the previous question is false in general. More precisely, find $\phi \in C^2(\mathbb{R}\times\mathbb{R})$ such that $\Box \phi = 0$; $\phi(1,x) = \partial_t \phi(1,x) = 0$ for every $x\in [-1,1]$; $\partial_t\phi + \partial_n \phi = 0$ along $[0,1]\times {-1,1}$; but $\phi(0,x) \not\equiv 0$ on $[-1,1]$.
    (Remark: as the name indicates, the "no incoming radiation" boundary condition prevents waves flowing into the domain from outside the domain, but it does not prevent waves flowing out of the domain from the inside.)