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In this talk, I will introduce the functional analytic approach which was proposed by Tosio Kato to obtain well-posedness results for various quasi-linear partial differential equations of evolution that appear mostly in mathematical physics. The main idea of the approach which is based on semigroup theory is to choose an appropriate infinite dimensional Banach space and define a corresponding operator acting on it so that the evolution equation is converted into an ordinary differential equation depending on time variable in the following form: \[u_t + A(u)u = f(u).\] Here, the space derivatives and other restrictions of the evolution equation are captured either by the Banach space or the operator $A(u)$. Throughout the talk, conditions on the linear operator $A(u)$ and the function $f(u)$ will be discussed and the results attained for the solutions of the corresponding initial value problems will be explained on examples.