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2nd Quater
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Grading policy (1Q)
Each problem consists of 40 points. 60 points or above is needed for the credit.
Submission
Submission is through the submission forms placed on T2SCHOLA. Submit as PDF files.
Problem 1
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📝 Search in textbooks/web pages/etc and choose one particular PDE. Then
- explain the background (背景) of the equation;
- explain its derivation (導出). The derivation need not be mathematically rigorous.
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Comments:
- For example, if you choose the wave equation $u_t=u_{xx}$. (i) Explain what this model can describe: a vibrating string? Drumhead? Electromagnetic waves? What is the physical meaning of $u$, $x$, and $t$. In what areas are these equations used? (ii) How can one derive this equation? There may be several ways to do this.
- This problem is somewhat ambiguous. But this is a graduate course and the main aim is to encourage you to learn the subject and not to evaluate you. Be free to think and write what you can based on your motivation level and availability of time.
Problem 2
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📝 For Duffing’s equation
$$
\begin{dcases}
x'=y, \\
y'=-(kx+\alpha x^3)-\mu \textcolor{red}{y}, \\
(x,y)(0)=(x_0,y_0)\in \mathbb{R}^2
\end{dcases}
$$
with $k>0,\alpha \geq 0,\mu>0$,
- show that the “energy”
$$
E(t)=\frac{1}{2}y^2+\frac{\mu}{2}xy+\frac{\mu^2}{4}x^2+\frac{k}{2}x^2+\frac{\alpha}{4}x^4
$$
satisfy $dE(t)/dt+\nu E(t)\leq 0$ for some $\nu>0$. Hint: multiply Duffing’s equation by $\mu x/2$. Also use the identity for $E_0(t)=y^2/2+kx^2/2+\alpha x^4/4$ proved in the lecture;
- show that $y^2(t)\leq 4e^{-\nu t}E(0)$. In particular, $y(t)$ decays exponentially fast.
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Comments:
- The energy $E_0(t)=y^2/2+kx^2/2+\alpha x^4/4$ used in the lecture is natural since it is the sum of the kinetic and potential energy of the system. But how does one come up with the definition of $E(t)$ in this problem? Unfortunately, there is no general method to find good energy. This problem is intended not to give you the false impression that energies are always easily found
Problem 3
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📝 Prove the uniqueness part in Theorem 2.15. Hint Show that for two mild solutions $u$ and $v$, there exists $L(M)>0$ such that
$$
\| (u-v)(t) \|{\textcolor{red}{L^2}} \leq L(M)\int{0}^{t}\| (u-v)(s) \|_{\textcolor{red}{L^2}} \, ds
$$
holds. Here
$$
M=\max(\sup_{0\leq t\leq T}\| u(t) \|,\sup_{0\leq t\leq T}\| v(t) \|).
$$
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Comments (or Further Hints):