Let $$(M^{n},g,e^{-f}dv)$$ be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation $$\begin{aligned} \left( \varDelta _{f}-\frac{\partial }{\partial t}\right) u(x,t)+q(x,t)u(x,t)+au(x,t)(\ln u(x,t))^{\alpha }=0, \end{aligned}$$where $$(x,t)\in M\times (-\infty ,\infty )$$ and a, $$\alpha $$ are arbitrary constants. Under the assumption that the $$\infty $$-Bakry-Emery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton’s type and Souplet–Zhang’s type) gradient estimates to positive smooth solutions of this equation.

中文翻译：

---

加权非线性抛物线方程的梯度估计

令 $$(M^{n},g,e^{-f}d​​v)$$ 是一个完整的平滑度量空间。我们证明了加权非线性抛物线方程正解的椭圆梯度估计 $$\begin{aligned} \left( \varDelta _{f}-\frac{\partial }{\partial t}\right) u(x,t )+q(x,t)u(x,t)+au(x,t)(\ln u(x,t))^{\alpha }=0, \end{aligned}$$where $$( x,t)\in M\times (-\infty ,\infty )$$ 和 a, $$\alpha $$ 是任意常数。在 $$\infty $$-Bakry-Emery Ricci 曲率从下方有界的假设下，我们获得了该方程的正平滑解的局部椭圆（Hamilton 型和 Souplet-Zhang 型）梯度估计。