Academic Report
Title: On existence of positive global solutions to a semilinear parabolic equation with a potential term on Riemannian manifolds
Reporter: A.Prof. SUN Yuhua
Time: Oct 28, 2019 (Monday) PM 14:30-15:30
Location: A1101# room, Innovation Park Building
Contact: WANG Wendong (tel:84708351-8139)
Abstract: We study the following Cauchy problem \partial_t u = \Delta u -V(x)u+ W(x)u^p}, {u(x,0)= {u_0(x)}} in M,（*）where $M$ is a connected non-compact geodesically complete Riemannian manifold.We prove that then (*) admits no global positive solution under some conditions. This result covers in a unified way most of the previous results in this subject (See \cite{Ishi08, Pinsky09, Zhang-duk99,Zhang01}).We also prove that if $M$ has nonnegative Ricci curvature, $V$ is a Green bounded nonnegative function, $W=1$, and for some $\epsilon>0$\mu (B(x_0, r)) \geq Cr^{\frac{2}{p-1}}(\ln r)^{\frac{1}{p-1}+\epsilon},\quad\mbox{for all large enough $r$,}then (*) has a global positive solution for some $u_0$.
The brief introduction to the reporter: Sun Yuhua, associate professor of Nankai University, main research direction: Analysis on manifold and partial differential equation. Published 14 papers in CPAM, cvpde, PAMS, jmaa, PJM and other journals. We have completed one NSFC project, and now we have participated in one NSFC International (regional) cooperation and exchange project and one general project.