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vpde_lecture32 [2020/04/16 10:23] trinh |
vpde_lecture32 [2020/04/16 12:33] trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] |
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====== Lecture 33: Uniqueness of solutions I ====== | ====== Lecture 33: Uniqueness of solutions I ====== | ||
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In this lecture, we'll prove uniqueness for two variants of the heat equation. The different variants you should consider as as follows (all posed on $x\in[0, L]$): | In this lecture, we'll prove uniqueness for two variants of the heat equation. The different variants you should consider as as follows (all posed on $x\in[0, L]$): | ||
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so the energy is always decreasing. But note that $E'(0) = 0$ since $w(x, 0) = 0$. Finally, note that $E(t)$ is always $\geq 0$ by its form (the integral of a squared quantity). So the energy is always decreasing, begins from zero, and can never be negative. We have the three statements: | so the energy is always decreasing. But note that $E'(0) = 0$ since $w(x, 0) = 0$. Finally, note that $E(t)$ is always $\geq 0$ by its form (the integral of a squared quantity). So the energy is always decreasing, begins from zero, and can never be negative. We have the three statements: | ||
- | - $E'(t) \leq 0$ for all time | + | - $E'(t) \leq 0$ for all time, |
- | - $E(0) = 0$ | + | - $E(0) = 0$, |
- | - $E(t) \geq 0$ for all time | + | - $E(t) \geq 0$ for all time, |
- | Therefore | + | and you would conclude that it has to remain at its initial value, and therefore |
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E(t) \equiv 0 | E(t) \equiv 0 | ||
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- | for all time. Looking at the form of the integrand, you would conclude that | + | for all time. Looking at the form of the integrand, you would conclude that the only way this occurs is if the integrand is itself zero, or |
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w^2(x, t) = 0 | w^2(x, t) = 0 |