Heat operator ($L={\partial }_t-\Delta$)

For the sign convention $\Delta =\mathrm{div}\nabla$, the heat operator is the differential operator $L={\partial }_t-\Delta$. The heat equation is $Lu=0$. If $P=-\Delta$ is the positive Laplace-Beltrami operator, the same operator is ${\partial }_t+P$. A Riemannian heat kernel is a fundamental solution with the initial Dirac delta distribution.