Parabolic equations with concave non-linearity

TL;DR

This paper establishes the existence and uniqueness of positive solutions for semilinear parabolic equations with concave nonlinearities, extending results to general measure spaces and various geometric settings.

Contribution

It introduces a novel approach for proving solutions to parabolic equations with concave nonlinearities, broadening applicability beyond traditional convex cases.

Findings

Existence and uniqueness of solutions for concave G(u)

Applicable to diverse geometric and measure space settings

Results hold with damping terms under relaxed assumptions

Abstract

In this paper we prove the existence and uniqueness of positive mild solutions for the semilinear parabolic equations of the form $u_t+\mathcal{L}u=f+h\cdot G(u)$, where $h$ is a positive function and $G$ a positive concave function (for example, $G(u)=u^\alpha$ for $0<\alpha<1$). In contrast with the case of convex $G$, where the Fujita exponent appears, and only existence of a positive solution for special data is achieved, in this concave case we obtain the existence and uniqueness for any positive data. The method relies on proving the existence and uniqueness of solutions for a certain class of Hammerstein-Volterra-type integral equations with a concave non-linear term, in the very general setting of arbitrary measure spaces. As a consequence, we obtain results for the existence and uniqueness of mild solutions to semilinear parabolic equations with concave nonlinearities under rather general assumptions on $f, h$ and $G$, in a variety of settings: e.g. for $\mathcal{L}$ being the Laplacian on complete Riemannian manifolds (e.g., symmetric spaces of any rank) or H\"ormander sum of squares on general Lie groups. We also study the corresponding equations in the presence of a damping term, in which case assumptions can be relaxed even further.