Consider the problem: $$u_t -div(A(x) \nabla u) +a(x) u = f $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) $$
and its variational formulation
$$ \lt \dot{u(t)},v \gt_* + B(u(t),v;t) =(f,v)_{L^2} \quad \forall v \in H^1_0(\Omega) \quad a.a. t\in(0,T) $$
if $$B(u,v;t)=\int_{\Omega}A(x)\nabla u \nabla v \ + \int_{\Omega}a(x)uv$$
is weakly cohercive,
than the equation satisfies a "Maximum" Principle:
$$ f \geq 0, \ g \geq 0 \Rightarrow u\geq0 $$
I was wandering, is there any generalization of the problem?
For example the equation could be:
$$u_t -div(A(x) \nabla u) + div(\vec{b}u) + \vec{c} \nabla u +a(x) u = f $$
and the boundary conditions different, but I couldn't find it anywhere