| Abstract: |
| This talk is devoted to the study of existence results for semilinear parabolic partial differential equations with nonlocal initial conditions and superlinear growth. The class of nonlocal conditions considered includes, as special cases, multipoint Cauchy problems, weighted mean value conditions, and periodic problems.
The analysis is carried out by applying a Leray-Schauder continuation principle, reformulating the problem as an ordinary differential equation in an abstract Banach space setting. In this framework, Lebesgue spaces provide the natural setting in which parabolic equations can be rewritten as ordinary differential equations.
A major difficulty arises from the presence of superlinear nonlinearities, since the associated Nemytskii operator is not, in general, continuous on Lebesgue spaces unless sublinear growth conditions are satisfied, as highlighted by Vainberg`s theorem.
This issue is addressed by exploiting the compactness and regularizing properties of the semigroup generated by the linear part of the equation, together with the construction of a suitable approximation scheme. These techniques allow us to establish existence results beyond the classical sublinear framework. |
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