Gradient estimates for Yamabe type equations under different curvature conditions and applications (Q6616027)

The authors provide several (Hamilton) gradient estimates for positive solutions of the following time-dependent equation: \N\[\Nu_t=\Delta u -\langle \Delta f,\nabla u\rangle+a(x,t)u+b(x,t)u^\alpha \tag{1}\N\]\Non \((M^n,g,\exp(-f)dv)\), an \(n\)-dimensional complete non-compact Riemannian manifold endowed with a weighted measure \(\exp(-f)dv\) for \(f\) a smooth function on \(M^n\), \(a(\cdot,t)\) is a \(C^2\) function on \(M^n\) for \(t>0\) and \(b(x,\cdot)\) is a \(C^1\) function on \((0,\infty)\) for \(x\in M^n\), and \(\alpha\) is a real constant. Let \(m\ge 0\), the \(m\)-dimensional Bakry-Émery Ricci tensor is defined as \(\text{Ric}_f^m=\text{Ric}+\text{Hess}f+\frac{1}{m}\nabla f\otimes\nabla f\). \N\NBy assuming that \(\text{Ric}_f^m\ge -(m+n-1)K\) for some positive constant \(K\) in \(B(x_0,2R)\), the ball of center \(x_0\in M^n\) and of radius \(2R\), and if a solution \(u(x,t)\) of \((1)\) belongs to \([\delta,B]\) such that \(\delta\) and \(B\) are constants and \((x,t)\in Q(2R,T)=B(x_0,2R)\times(0,T]\) for \(T>0\), then the gradient estimate of \(u(x,t)\) is upper bounded by a function given in terms of \(m\), \(n\), \(R\), \(\delta\), \(B\), \(\sup_{Q(2R,T)}|a(x,t)|\), \(\sup_{Q(2R,T)}|\nabla a(x,t)|\), \(\sup_{Q(2R,T)}|b(x,t)|\), \(\sup_{Q(2R,T)}|\nabla b(x,t)|\), and a universal positive constant (Theorem 1.1). \N\NThe second result provides the Hamilton type gradient estimate of a strictly positive bounded solution of \((1)\) and the case when the manifold is evolved under the \((k,\infty)\)-super Perelman-Ricci flow is also treated (Theorems 1.2 and 1.3). Also, the authors obtain Hamilton gradient estimate under the integral Ricci curvature conditions (Theorem 1.4).