A Study On Several Types Of Heat Equations In Different Spaces

This paper is concerned with parabolic equations in diferent spaces. Here wemainly consider three kinds of spaces. The first one is a general metric space, and frac-tal space is one of the typical representive types of metric spaces, here we investigatethe inhomogeneous parabolic equation, which is considered in Chapter2; the secondone is the Euclidean space, we consider the more generalized parabolic system in theEuclidean space, which is considered in Chapter3; the third one is the Riemannianmanifold, we study the Lichnerowicz type equation, which is considered in Chapter4.In the first part, firstly we give some motivation for studying the parabolic equationin general metric space, then we establish the non-existence result by using the heatkernel method; By using the upper-regular measure in[31], we establish Propositions2.2and2.3, then combining with the contraction mapping theorem, we have provedthe global existence under some appropriate conditions. Finally, we obtain the Ho¨lderregularity results.In the second part, we study the parabolic system (3-1) in the Euclidean space,and have obtained the critical exponent for the system. At the end, we leave some openproblems.In the third part, we study the Lichnerowicz type equation on compact Riemmani-an manifolds. We have established the existence result for the Lichnerowicz equation,and obtained the solution under diferent sense depending on whether b(x) has zeropoints. At last, several special cases are investigated there.In this thesis, we mainly use the following five methods:The first one is the heat kernel method. We use a similar technique as inWeissler[70,71], by expressing the solution using semigroup and heat kernel estimate,we could obtain the existence and nonexistence results, which are stated in Chpater2,3, and also the regularity in Chpater2.The second method is the contraction mapping theorem, which is used in Chapter 2,3. This method is used to obtain the existence of weak solutions to the partial dif-ferential equation. Before using the contraction mapping theorem, we need to choosesome special Banach spaces. To do this, we need to establish some apriori estimates,which are more technical, see in the proof of Theorem3.1in Chapter3, where wedistinguish four diferent cases.The third method is the test function method. We introduce some test functions in(3-15), Chapther2. These test functions will help us to deal with the case successfullywhen a(x), b(x) have zero points. However, there is no test function in general metricspace, for example, in many classical fractal spaces. It seems to work well only togeneral manifolds.The forth method is the monotone convergence theorem, which we have appliedto obtain the local existence in Chapter2, and also the global existence in Chapter4.The fifth method is the sub-sup solutions method. Motivated by Sattinger[61], weestablish the sub-sup solutions method on compact manifolds. This method is used inChapter4to obtain the existence of the solution.