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Let $u$ be the unique solution of

$$\left.\begin{array}{lr}\dfrac{\partial u}{\partial t}=\dfrac{\partial^{2} u}{\partial x^{2}} \text { where }(x, t) \in(0,1) \times(0, \infty) \\ u(x, 0)=\sin \pi x, & x \in(0,1) \\ u(0, t)=u(1, t)=0, & t \in(0, \infty)\end{array}\right\}$$

Then which of the following is true?

1. There exists $(x, t) \in(0,1) \times(0, \infty)$ such that $u(x, t)=0$

2. There exists $(x, t) \in(0,1) \times(0, \infty)$ such that $\frac{\partial u}{\partial t}(x, t)=0$

3. The function $e^{t} u(x, t)$ is bounded for $(x, t) \in(0,1) \times(0, \infty)$

4. There exists $(x, t) \in(0,1) \times(0, \infty)$ such that $u(x, t)>1$
in Differential Equation by Expert | 485 views

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