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vpde_lecture32 [2020/04/16 12:15] trinh |
vpde_lecture32 [2020/04/16 12:33] trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] |
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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 | ||