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 $T(u)$.


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