Heat kernel with constant absorption ($K_c(x,t)=e^{-ct}(4\pi t{)}^{-1/2}{e}^{-x^2/(4t)}$)

The displayed fundamental solution solves $u_t=u_{xx}-cu$ on the real line with a Dirac delta at the initial time. It is the ordinary heat kernel multiplied by $e^{-ct}$, because that factor removes the constant reaction term. Its integral is $e^{-ct}$ and it converges to the Dirac delta as $t\downarrow 0$. Convolution with initial data gives the corresponding solution. For $c>0$ the total mass decays by absorption; for $c<0$ it grows instead.