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evinda

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MHB

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Hello! (Wave)

I want to prove that if the initial data of the initial value problem for the wave equation have compact support, then at each time the solution of the equation has also compact support.

Doesn't the fact that a function has compact support mean that the function is zero outside some bounded set? (Thinking)

The initial value problem for the wave equation is the following, right?

$$u_{tt}=c^2 u_{xx} \text{ in } \mathbb{R} \times (0,\infty) \\ u(\cdot, 0)=\phi \text{ in } \mathbb{R} \\ u_t(\cdot,0)=\psi \text{ in } \mathbb{R}$$

Is it meant with "the initial data of the initial value problem for the wave equation have compact support" that $\phi$ and $\psi$ are $0$ ?

If so, it follows directly that the solution is $0$, since it is equal to $u(x,t)=\frac{1}{2} \left[ \phi(x-ct)+\phi(x+ct)\right]+\frac{1}{2c} \int_{x-ct}^{x+ct} \psi(\tau) d{\tau}$, right?But if so, why is it stated as compact support? (Thinking)

I want to prove that if the initial data of the initial value problem for the wave equation have compact support, then at each time the solution of the equation has also compact support.

Doesn't the fact that a function has compact support mean that the function is zero outside some bounded set? (Thinking)

The initial value problem for the wave equation is the following, right?

$$u_{tt}=c^2 u_{xx} \text{ in } \mathbb{R} \times (0,\infty) \\ u(\cdot, 0)=\phi \text{ in } \mathbb{R} \\ u_t(\cdot,0)=\psi \text{ in } \mathbb{R}$$

Is it meant with "the initial data of the initial value problem for the wave equation have compact support" that $\phi$ and $\psi$ are $0$ ?

If so, it follows directly that the solution is $0$, since it is equal to $u(x,t)=\frac{1}{2} \left[ \phi(x-ct)+\phi(x+ct)\right]+\frac{1}{2c} \int_{x-ct}^{x+ct} \psi(\tau) d{\tau}$, right?But if so, why is it stated as compact support? (Thinking)

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