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Suppose one is given a 1-dimensional heat equation $u_t=\Delta u$ with continuous and bounded initial condition $u(x,0)=g$, and $g$ satisfies $g=0$ for $|x|\leq100$, then would $u(0,t)$ be zero for $t$ sufficiently small? Why? Also if one considers the 2-dimensional wave equation with initial conditions $u(x,0)=g, u_t(x,0)=h$, and $g,h$ also satisfy the condition above, then would the same conclusion applies? Why?

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  1. No. The heat equation exhibits infinite propagation speed. Look at the integral formula for its solution: it has a kernel that is positive on all $\mathbb R$, no matter how small $t$ is.

  2. Yes. The wave equation exhibits finite propagation speed. Look at the integral formula for its solution: it has a kernel that is zero when $|x-y|>ct$.

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