GATE 2020 MA – Question 24
The heat-equation IVP $u_t=u_{xx}$, u(x,0) = f(x), x∈R, t > 0, has solution $u(x,t)=\int_{\mathbb R}K(x-y,t)f(y)dy$ for every bounded continuous f. Find $\int_{\mathbb R}K(x,t)dx$.
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Correct answer: 1 to 1
Explanation
Take f identically 1. The constant function u(x,t) = 1 solves the heat equation with that initial value. The given representation then reads $1=\int_{\mathbb R}K(x-y,t)dy$. A translation of the integration variable gives $\int K(z,t)dz=1$. This is conservation of total mass of the heat kernel.