Parabolic differential inclusions with strongly elliptic differential operators and superlinear growths
Irene Benedetti, Fatima Fennour, Eleonora Palazzoni · Discrete and Continuous Dynamical Systems - S · 2025
We investigate the existence of mild solutions for a class of semilinear parabolic partial differential equations on bounded domains $ \Omega \subset \mathbb{R}^n $. The linear component, represented by a diffusion term in divergence form, generates a strongly elliptic differential operator. These equations are of interest as models for reaction-diffusion processes in various contexts. The nonlinearity is described by a multivalued map, allowing us to account for real phenomena with a certain degree of uncertainty in the collected data. By transforming the dynamics into an abstract setting and combining an approximation technique with the Leray-Schauder continuation principle, we establish global existence results. Specifically, through a suitable approximation argument, we handle nonlinearities with superlinear growth. Additionally, we prove the existence of at least one mild solution on the half-line and examine the stability of the zero solution.