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I am currently interested in the system of equations of the form, \begin{equation} \begin{cases} \dfrac{du}{dt} + Au\ni f(u)\\[0.2cm] u(0)=u_0 \end{cases} \end{equation}

where the operator $A:D(A)\subset X\to X^*$ is nonlinear, $X$ is a Banach space and the functional $f=f(u)$ is linear.

I know some of the wellposedness results for the nonhomogeneous problems where the RHS term $f=f(t)$. I am also aware of some results for the evolution equations (as above) for semilinear equations where $A$ is linear and $f=f(u)$ is nonlinear. Can somebody point out some references / articles / books which discusses the type of evolution equation which I mentioned. Thanks in advance.

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You can check the following recent paper and the references therein

COMPACT ALMOST AUTOMORPHIC WEAK SOLUTIONS FOR SOME MONOTONE DIFFERENTIAL INCLUSIONS: APPLICATIONS TO PARABOLIC AND HYPERBOLIC EQUATIONS, B. Es-sebbar et al. (2019).

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    $\begingroup$ Thank you. I will have a look. $\endgroup$ – Hirak Dec 31 '19 at 9:53

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