Some Dynamic Properties For Semilinear Heat Equations On Hyperbolic Space

It is well known that every simply connected, complete, N-dimensional Riemannian manifold with constant sectional curva-ture is similar to either the N-dimensional hyperbolic space HN, the sphere SN or Euclidean space RN [46, Theorem8.6.2]. Peo-ple pay many attentions to the works of PDEs on the hyperbol-ic space, such as the wave equations [47], Schrodinger equation-s [7,8,2], elliptic equations [38.11,5] and heat equations on the whole hyperbolic space or some domain of HN [6,45]. These works have been or will become to the central issues of the recent studies. As some important properties of the PDEs on RN are ex-tended to similar PDEs on the hyperbolic space, people find some special phenomena only for the PDEs on the hyperbolic space.In this thesis, we study the dynamical properties for semilin-ear heat equations on hyperbolic space where p>1, α>0, u∈C(HN)∩L∞(HN) and u0(x)≥0.In the first part of the thesis, we consider the relation of the globally geometric properties of the space and the long time be-havior of the solutions, especially for solutions with small initial data, of the problem (1). Notice that in the case of α=0there exist Fujita exponents for the problems posed on RN or complete manifolds with non-negative Ricci curvature [20,53]. However, there does not exist Fujita exponent in this case for the prob-lem (1). In the case of α>0, Bandle et al.[6] considered the non-negative solutions of the problem (1), and studied the Fujita exponent. They proved that there exists an exponent p*H=1+α/λ0with λ0=(N-1/2)2such that (ⅰ) if1<p<p*H then all the non-trivial solutions blow up in finite time.(ⅱ) if p>p*H,or assume that p=p*H and α≥2/3λ0, then the problem (1) possesses both global solutions for small initial data and blowing up solutions for large initial data.(ⅲ) if p=p*H and α≥2/3λ0, then there are blowing up solutions for large initial data. We prove that there are global positive solutions in case p=p*H and α≥2/3λ0.This theorem and the results in [6] give a complete characterization of Fujita exponent for the problem (1). The main difficulty is to find appropriate global sup-solution of problem (1). We transform the equation, and obtain an equivalent equation without explicit variable t on the right hand side of the equivalent equation. Thus we can use the properties of elliptic equations on the hyperbolic space to construct proper sup-solutions.In the second part, we study the relations of the locally geo-metric properties of HN and the properties of the solutions of the problem (1). We show that some of the results of the semilinear heat equation on RN, which were determined by locally properties of the space, can be extended to the hyperbolic space, e.g., the life span problem of blowing up solutions, the blow-up set and the continuation of the blow up solutions. As far as we know, this is the first time to consider such kinds of problems on the hyper-bolic space. For more details, we study the life span problem in Chapter3. In Chapter4we consider the blow-up set, and prove that there is only a single point in the blow-up set if the initial datum is a decreasing radial function. In Chapter5we discuss the continuation of blow-up solutions after the blow-up time. We give a sufficient condition for the complete blow-up, which means that there are no bonded continuation for the blow-up solution. These results show that a phenomenon of hyperbolic space and RN is similar, if the phenomenon is determined by localty and short time blowing up.There are two differences between the problems on M.N and the problems on H. First, the recursion expression of the heat kernel on H^is not convenient for our study, and the useful ap-proximation expression of the heat kernel is complicated with ad-ditional nonlinear terms for the heat kernel on RN. Second, there is no proper scaling property of the equation on HN, and we can not use the approaches depending on the relation between the scaling of equations and the scaling of the space. We use more elaborate estimates, the semi-group property of the heat kernel and the local upper solution to prove our results.