非局部扩散方程（组）解的渐近行为

In recent years，nonlinear nonlocal diffusion equation mainly deriving from biology, physics, chemistry, etc, becomes a hot topic in the field of partial differential equations， and has an important practical value and theoretical significance. In this project，we plan to study the asymptotic behavior of solutions to nonlinear nonlocal diffusion, mainly concern with critical Fujita and second critical exponent, asymptotic profiles for global solutions , and life span for nonglobal solutions, including (1) studing a coupled nonlinear nonlocal diffusion equations , planning to generalize the results of single equation, and trying to delete the monotonicity conditions of kernel function , which is essential in the proof of critical Fujita exponent in the single equation; (2)studing a nonlocal diffusion with weighted source, concerning the influence of weighted function on asymptotic behavior of solutions. Via rescaled test function method, improving Kaplan’s method, interpolation inequality, contractive mapping and etc, we aim to overcome the essential difficulties coming from nonlocal diffusion, nonlinear coupled terms, and nonlinear weighted source.

非线性非局部扩散方程源于生物、物理、化学等诸多领域，是近年来偏微分方程研究领域中颇受关注的研究课题，具有重要的实际意义和理论价值。 本项目拟研究此类方程解的渐近行为，涉及Fujita临界指标与第二临界指标、整体解的渐近性态与非整体解的生命跨度等问题。内容包括：（1）研究一类非局部扩散非线性耦合组，推广单个方程的结果，并去除前人在处理单个方程时所必需用到的核函数的单调性条件。（2）研究一类具加权非线性源的非局部扩散方程，考虑权函数对解的渐近行为的影响。拟采用rescaled 检验函数方法、Kaplan方法、插值不等式、不动点定理等手段来克服扩散项的非局部性、非线性耦合项、加权非线性源带来的本质困难。

本项目研究了一类非局部扩散非线性耦合方程组解的渐近行为，以及完全渐近严格拟-phi-伪压缩算子不动点的混杂投影迭代算法。具体内容如下：对于非线性非局部扩散方程组，我们不仅研究Fujita临界曲线，推广了前人单个方程的结果，并研究了其未涉及的第二临界指标问题。发现虽然扩散是非局部的，但结果仍与经典的非线性热方程组保持一致。 特别地，我们采用新方法去掉前人对于核函数的单调性的假设条件。对于Banach空间中完全渐近严格拟-phi-伪压缩算子，我们考虑该算子不动点的混杂投影迭代算法，并由此讨论了均衡问题中出现的一些半连续凸泛函的极小化子。