图上带吸收项的热方程整体解的存在性研究

The analysis on graphs is an emerging research area, and especially the theory of differential equation on graphs has extensive applications in the fields of complex networks, computer graphics and data mining, etc. In this project, we study the existence of global solutions of the initial value problem for the Fujita-type equation with critical exponent on graphs, and analyze the effect on the existence of global solutions of the heat equations on graphs by changes of absorption terms, we also discuss the existence and nonexistence of global solutions for heat equations with absorption on graphs via unbounded Laplacian and fractional Laplacian. .We devote to find some new methods to deal with an unsolved problem which was put forward in the initial period of our research project. Also, we will change the traditional ideas to new ideas, and apply the results of the heat kernel estimate on graphs to investigate the existence of global solutions for heat equations with absorption. Moreover, we will explore some new types of the Fujita-type equation by extending the absorption terms and the Laplacian operators in different form. .The expected results of this project are as follows: Some new versions and some improved versions of the estimates of heat kernels on graphs are established, some results on the existence and nonexistence of global solutions for heat equations with absorption on graphs are obtained. By combining the partial differential equation theory and the analysis on graphs, we provide an approach to reveal the similarities and differences of behavior of solutions between the partial differential equations on graphs and the partial differential equations in Riemannian manifolds and Euclidean space. The present investigation would help to promote the theoretical and applied research on the continuous case of partial differential equations.

图上的分析是一个新兴的研究领域，其中图上的偏微分方程理论在复杂网络、计算机图形学和数据挖掘等学科中有广泛的应用。本项目研究图上临界指数的Fujita型方程的初值问题的整体解的存在情况；分析吸收项的变化对图上热方程整体解的存在性的影响；讨论图上带无界拉普拉斯算子和分数阶拉普拉斯算子的Fujita型方程的整体解的存在性问题。本项目旨在探索新方法，解决前期研究中未解决的难题；突破传统方法，通过图上热核估计来研究图上带吸收项的热方程整体解的存在情况；开拓新内容，将Fujita型方程的吸收项和拉普拉斯算子进行不同形式的推广。预期成果包括，建立和改进图上热核估计的表达式；给出图上带吸收项的热方程整体解的存在性与非存在性的若干结果。本项目将图上的分析与偏微分方程相结合，揭示图上偏微分方程解的性质与黎曼流形和欧氏空间中偏微分方程解的性质的某些异同，研究成果对连续情形的偏微分方程理论及应用研究具有促进作用。

图上的分析是近些年兴起的研究，它是微分几何、调和分析、概率论、图论和偏微分方程的交叉，其中图上的偏微分方程理论在复杂网络、计算机图形学和数据挖掘等学科中具有广泛的应用。本项目研究图上的半线性抛物方程（系统）初值问题解的爆破性与整体存在性，具体包括研究图上临界指数的Fujita型方程的初值问题的整体解的存在情况；研究吸收项的变化对图上半线性抛物方程初值问题的解的整体存在性的影响；研究分数阶算子，并将图上Fujita型方程的结论推广至分数阶算子及方程组的情形。项目负责人在前期的研究基础上，得到Fujita型方程的推广形式在有限时刻爆破的条件，以及在指数临界时Fujita型方程的初值问题的解爆破等重要结果，目前已发表与该项目相关的论文6篇，其中SCI收录5篇，在项目周期内培养硕士研究生2名，参加国内外学术交流10余人次。本项目通过将图上的分析与偏微分方程理论相结合，揭示图上偏微分方程解的性质与黎曼流形和欧氏空间中偏微分方程解的性质的异同点。研究成果不仅丰富了图上分析的研究，而且对连续情形的偏微分方程理论及应用研究具有促进作用。项目负责人已按照原计划完成了全部研究任务，同时也为下一阶段研究做了相关铺垫与准备。