Equation of the type evolution is one important topic in mathematics. Today, I read some very theoretical basis of this subject.
$$
\partial_t f = \Delta f(x) + a(x) \cdot \nabla f(x) + c(x) f(x) + \int_{\mathbb{R}^d} b(y,x) f(y) dy
$$
with the condition that $a \in W^{1,\infty}, c \in L^{\infty}, b \in L^2(\mathbb{R^d \times R^d})$.
Although there are many equations (maybe of types degenerate), but this model describes the equation of diffusion, of branching, of transport and also of mean-field, so it is already very large. Of course, we will talk about the existence, uniqueness of this equation.
The main frame to talk about the evolution equation is to design a Hilbert space $H$ and a Banach subspace $V \subset H$, the advantage is to use the duality of Hilbert space to get
$$
V \subset H = H' \subset V' .
$$
so we can talk about the weak solution of the equation and we use $|\cdot|, \|\cdot\|$ to represent respectively the norm in $H$ and $V$.
One propose a good condition, similar to that of Lax-Milgram. If we denote by $L$ the generator, then we requires that
(1)$L : V \rightarrow V'$ is bounded.
(2)$L$ is coercive + dissipative that $\langle Lg, g \rangle \leq - \alpha \|g\|^2 + b|g|^2$ for every $g \in V$.
This condition gives an inequality
$$
|g(T)|^2_H + 2 \alpha \int_0^T \|g(s)\|_V^2 ds \leq e^{2bT}|g_0|^2
$$
at first correct for a $g$ more regular and then we can pass to general function by approximation. This is the energy inequality, it says a lot of things and the most important the uniqueness and make sure that the solution stays always in the function space $g \in L^{\infty}(0, T; H) \cap L^2(0,T;V) \cap H^1(0, T; V')$.
(One remark : the dissipative part just says the Cauchy-Liptchitz condition. )
Later, we use a theorem : $ L^2(0,T;V) \cap H^1(0, T; V') \subset C([0,T];H)$. This makes sense when we do formally $g'$ and then we can manipulate as if it is regular.
The existence theory is a variant theorem of Lax-Milgram theory. When we discretize the problem, it becomes a variant elliptic equation in small discrete steps. Then, we do a sub-sequence of weak limit to get the solution of the equation. .
