In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (△-e/et)u(x,t)+q(x,t)u^p(x,t)=0 and nonlinear parabolic equations (△-e/et)u(x,...In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (△-e/et)u(x,t)+q(x,t)u^p(x,t)=0 and nonlinear parabolic equations (△-e/et)u(x,t)+h(x,t)u^p(x,t)=0(p 〉 1) on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045-1053) and the author ([2], Nonlinear Anal. 74 (2011), 5141-5146).展开更多
基金supported by the National Science Foundation of China(41275063 and 11401575)
文摘In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (△-e/et)u(x,t)+q(x,t)u^p(x,t)=0 and nonlinear parabolic equations (△-e/et)u(x,t)+h(x,t)u^p(x,t)=0(p 〉 1) on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045-1053) and the author ([2], Nonlinear Anal. 74 (2011), 5141-5146).