By combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results for semilinear differential equations involving a dissipative linear operator, generating an extendable compact $C_0$-semigroup of contractions, and a Carathéodory nonlinearity $f\colon [0,T] \times E \to F$, with $E$ and $F$ two real Banach spaces such that $E \subseteq F$, besides imposing other conditions. The case $E\neq F$ allows to treat, as an application, parabolic equations with continuous superlinear nonlinearities which satisfy a sign condition.
"Existence results for evolution equations with superlinear growth." Topol. Methods Nonlinear Anal. 54 (2B) 917 - 936, 2019. https://doi.org/10.12775/TMNA.2019.101