Infinitisemal generator of a semigroup with parameter

Question

When we talk about evolution equation the first idea who comes to the mind is the semigroups theory, this theory deals with the Cuachy problems of the form $$\frac{{\partial u}}{{\partial t}} = Au$$ where $A$ is an operator acts ob Banach spaces to another. My question is : is there any theory which deals with the evolution problems of the form $$\frac{{\partial u}}{{\partial t}} = A(t)u$$ this time $A$ depends on $t$ ? thank you.

Answer 1

Yes, these are known as nonautonomous problems, and have been studied extensively, including through the lens of semigroup theory. See for example Lunardi's book Analytic Semigroups and Optimal Regularity in Parabolic Problems, in particular chapter 6.

Answer 2

The appendix in the book Evolution Equations in Thermoelasticity is another source. This book presents a theory for nonhomogeneous linear equations having the form $$U_t+A(t)U=F(t)$$ which enable us to treat nonlinear evolution equations via fixed-point method.

For example, the nonlinear equation $$u_{tt}-[a(u_{x})]_x=0$$ can be written as $$u_{tt}-a'(u_{x})u_{xx}=0.\tag{$1$}$$

Setting a "nice" function $\tilde{u}$, we obtain (from the theory in the said book) existence of solution for the linear equation $$u_{tt}-a'(\tilde{u}_{x})u_{xx}=0\tag{2}.$$

In this case, the nonautonomous operator is $$A(t)=\left(\begin{matrix} 0 &-I\\ -a'(\tilde{u}_x(t))\partial_{xx}&0\end{matrix}\right).$$

From the Banach fixed-point theorem applied to the map $$\tilde{u}\mapsto\text{solution of }(2)$$ we obtain existence of local solution for $(1)$.

(Of course, to apply the method we need some regularity conditions on the nonlinearity $a$ as well as on the initial data.)