# How to prove that a linear differential operator generates a semigroup?

I am working on a nonlinear PDE and I should now prove that the linear operators $$L:=\partial_x^3+\partial_x^2$$, $$S(u):=u L$$ and $$T(u):=L+S(u)$$ are generators of a $$C_0$$ semigroup such that $$\|e^{-s A}\|_{B(X)}\leq e^{\beta s}$$, for a certain constant $$\beta$$. Above, $$A$$ denotes any of the operator and $$B(X)$$ is the set of bounded operators in $$X$$. In my the concrete case, $$X=H^s(\mathbb{R})$$ (Sobolev space), with $$s>1/2$$.

In principle I believe that I shoud restrict these operators to the domain $$H^{s+3}$$ and them prove that $$\langle A\phi,\phi\rangle_X\geq -\beta\|\phi\|^2_X$$ and also that there exists a number $$\lambda>\beta$$ such that $$(A+\lambda)$$ is an onto operator (again, $$A$$ means any of the operators).

The situation for $$L$$ is rather simple than for the other two. However, I am facing dificulties to deal with $$S(u)$$ and the sum of the first two operators given by $$T(u)$$.