I have a reaction diffusion equation, $$ u_t = u_{xx}+f(u), ~x\in (-\infty,\infty) $$ with non-linearity $f$. Inserting the travelling wave ansatz $u(x,t)=v(x-ct)$, gives $$ u_t=v''+cv'+f(u) $$ and linearizing in $v$, $$ u_t=u''+cu'+f'(v)u $$ with the linear operator $L=\partial_x^2+c\partial_x+f'(v)$.
My question: Does this operator $L$ generate a $C_0$-semigroup?
It depends on which domain you are working on. But if you take the standard domains you can see that $L$ generates a $C_0$-semigroup by perturbation argument. In this case, $P=\partial_x^2 + c \partial_x$ is a $\partial_x^2$-bounded perturbation of $\partial_x^2$ and $\partial_x^2 + c \partial_x + f'(v)$ is a bounded perturbation of $P$.