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Bounded solutions for impulsive semilinear evolution equations with non-local conditions

Abstract:

For semilinear evolution equations in function spaces, difficulties arise when the nonlinear term consists of a composition operator, commonly called Nemytskiis operator, which almost never maps a function space into itself unless the generator function is affine. In this chapter, the authors generalize the results to the case of semilinear impulsive evolution equations with non-local conditions and nonlinear term involving spatial derivative such as the well-known Burges equation. First, they provide the definition of a sectorial operator and some preliminary results on compact analytic semi-groups. The authors then prove the main result, i.e. the existence of bounded mild solutions to semilinear evolution equation. Among other things, the results obtained can be applied to the following semilinear Burgers equation with impulses and non-local condition.