For Riemannian manifolds with a measure,we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below,generalizing the classical ones of Yau(i.e.,when f is constant).
In this paper,we study a simplified Othmer-Stevens model with reproduction term. By making use of a smart function transformation,the comparative method and some special mathe-matical analysis,we prove the existence of global,blow-up or quenching solutions of the problem on different conditions. More interesting results are reached. The result of the paper not only verifies a real biological phenomenon,but also provides a theo-retical groundwork for numerical problems of the chemotaxis model.
CHEN Xueyong, LIU Weian School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
In this paper, we consider a nonlinear size-structured population model with functional response, which describes the dynamics of a predator-prey system living in a common habitat. We present a kind of functional response for the prey being a plant or algae, and explain its biological meanings. When the vital rates depend both on the individual’s size and on the total population or only depend on the former, we obtain the existence of equilibrium solutions.
LIU Keying,LIU Weian School of Mathematics and Statistics,Wuhan University,Wuhan 430072,Hubei,China
Let Ω be a connected bounded domain in Rn. Denote by λi the i-th eigenvalue of the Laplacian operator with any order p:{0(-△)pu =λu u =■u/→■n =···=■→■np-1 p-u1=0 in Ω,on ■Ω.In this article, we give some expressions for upper bound of the (k + 1)-th eigenvalue λk+1 in terms of the first k eigenvalues.
