Triviality of bounded solutions and gradient estimates for nonlinear \(f\)-heat equations on complete smooth metric measure spaces
The authors analyze qualitative properties of positive smooth solutions to a generalized nonlinear parabolic equation equation involving the \(f\)-Laplacian \(\mathcal{L}_f\), given by \[ \left(\mathcal{L}_f-q(x,t)-\frac{\partial}{\partial t}\right)w(x,t) = \mathcal{G}(w(x,t)) , \] on complete smooth metric measure spaces \(M_f\times (-\infty,\infty)\), with or without boundary. They assume that the potential function \(q(x,t)\) is at least \(C^1\) in \(x\) and at least \(C^0\) in \(t\), and that the nonlinear sourcing term \(\mathcal{G}(w(x,t))\) is at least \(C^1\) in \(w>0\). The article is divided into two parts. In the first part, the authors consider spaces without boundary and establish local and global type space only (elliptic type) gradient estimates for the equation under the condition that the Bakry-Émery Ricci curvature tensor is bounded from below. They then obtain a parabolic Harnack inequality and some Liouville-type theorems for bounded ancient and eternal solutions using these gradient estimates. In the second part, the authors obtain similar results for weighted manifolds with compact boundary by imposing a Dirichlet boundary condition and assuming lower bounds on the weighted mean curvature of the boundary and the Bakry-Émery Ricci curvature tensor.
scientific article; zbMATH DE number 7603381
scientific article