分数阶超扩散方程的Fujita临界指标

Fractional diffusion equation is a research field which has developed rapidly in recent years. It is meaningful to study the dynamic behavior for the fractional diffusion equation. This project will study the Fujita exponent of fractional superdiffusion equation. First, by using the test function method, we study the global existence and blow up of solutions for time fractional superdiffusion equation with different class of nonlinear term, and then obtain their Fujita exponents. Based on this, by using the properties of solution operators, test function method and the energy method, we further consider global existence and blow up of solutions for time-space fractional superdiffusion equation. These researches will help us understand the influence of the nonlocal and singularity of the fractional differential operator on the properties of solutions, and will further reveal the difference and relationship between the fractional superdiffusion equation and the classical heat equation and wave equation.

分数阶扩散方程是近年来发展非常迅速的一个研究领域，深入研究分数阶扩散方程的动力学行为具有重要意义。本项目将研究一类时间分数阶超扩散方程的Fujita临界指标。首先，通过检验函数法讨论初值在不同空间时，非线性项具有指数函数形式、幂函数形式以及非局部性质的时间分数阶超扩散方程解的全局存在与爆破，进而得到其对应的Fujita临界指标。在此基础上，利用解算子的性质、检验函数法以及能量法，进一步考虑时间空间分数阶超扩散方程解的全局存在与爆破。这些问题的研究，将帮助我们更深刻的理解分数阶算子的非局部性和奇异性对解的性质的影响，揭示分数阶超扩散方程与热传导方程和波动方程之间的区别与联系。

分数阶扩散方程的动力学行为无论是在纯粹数学还是在应用学科中，都得到了非常广泛的重视。本项目首先运用解算子方法并通过构造合适的检验函数，研究了一类非线性项带有非局部项的时间分数阶扩散方程适度解的局部存在性、弱解的有限时间爆破以及全局存在性，从而得到了其对应的Fujita临界指标。其次讨论了一类半线性时间空间分数阶扩散方程解的有限时间爆破以及全局存在性。最后还运用临界点理论和变分法得到一类分数阶微分方程边值问题解的存在性。