Let the Heat equation \begin{align} u_t&=\Delta{u}\text{ in }\mathbb{R}_{+}\times \mathbb{R}^n\\ u(0)&=u_0 \end{align}
The heat kernel is $p_t(x)=\frac{1}{(4\pi t)^{n/2}}\exp\left( -\frac{|x|^2}{4t}\right)$ and this kernel satisfies $\int p_t(x)\,dx=1$, $p_t$ is positive and $p_t$ is smooth $\mathbb{R}_{+}\times \mathbb{R}^n$ and satisfies the heat equation.
From what I have studied, another way to represent the kernel of the heat equation is by using the Fourier transform, that is, $p_t(x)=\int_{\mathbb{R}^n}\mathrm{e}^{ix\cdot \xi}\mathrm{e}^{-t|\xi|^2}\,d\xi$ then $p_t$ is a Schwartz function because $\mathrm{e}^{-t|\xi|^2}$ is a Schwartz function and how the transform is an automorphism on the Schwartz function, then $p_t$ is a Schwartz function etc.
In general, if \begin{align} u_t&=Pu,\quad t>0\\ u(0)&=u_0 \end{align} with $P$ an differential operator.
What is the definition of an kernel for the operator $P$?
This kernel must always satisfy that $P_t$ is smooth, $P_t$ positive and $\int P_t=1$? or does this change according to each differential equation?