I am currently reading the following paper:
Chen, X. The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Rational Mech. Anal. 123, 117–151 (1993). https://doi.org/10.1007/BF00695274
But I am stuck with the following step:
We want to prove existence and uniqueness for this PDE \begin{equation} f_{t} + \varepsilon J^{-4}(t,x) f_{xxxx} = F(t,x) \end{equation} Chen then approximates $ \frac{1}{2} \leqq J \leqq 2$ and $\vert F(t,x) \vert_{L^{2} (S^1 \times (0,T))}^2 \leqq C \varepsilon^2 (E_0 T + M)$, where $0<E_0 < \infty $ and $T,M>0$ can be made as small as needed. From these inequalities he then concludes that the PDE admits a unique solution in $L^2[0,T;H^4(S^1)] \cap H^1[0,T;L^2(S^1)]$ by classical semigroup.
I am not so familiar with semigroup theory, but I know that in the homogenous case the statement is true by virtue of the Hille-Yosida Theorem for unbounded operators. However I am not familiar with a general proof for inhomogeneous PDEs.
My question, therefore, is how one can prove the existence in this case. Is there a theorem for this? What is the technique to use?
Thank you in advance.