| Summary: | The existence (and uniqueness) results on mild solutions of the abstract
semilinear Cauchy problems in Banach spaces are well known.
Following the results of Tartar (2008) and Burazin (2008) in the case of
decoupled hyperbolic systems, we give an alternative proof, which enables
us to derive an estimate on the mild solution and its time of existence.
The nonlinear term in the equation is allowed to be time-dependent.
We discuss the optimality of the derived estimate by testing it on three
examples: the linear heat equation, the semilinear heat equation that
models dynamic deflection of an elastic membrane, and the semilinear
Schrodinger equation with time-dependent nonlinearity, that appear
in the modelling of numerous physical phenomena.
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