The Blow-Up and Global Existence of Solution to Caputo-Hadamard Fractional Partial Differential Equation with Fractional Laplacian
The Blow-Up and Global Existence of Solution to Caputo-Hadamard Fractional Partial Differential Equation with Fractional Laplacian
复制标题
分数阶拉普拉斯CaputoâHadamard分数阶偏微分方程解的放大与整体存在性
DOI：
10.1007/s00332-021-09736-y
复制
发表时间：
2021-10-01
影响因子：
3
通讯作者：
Li, Zhiqiang
中科院分区：
文献类型：
--
作者：
Li, Changpin;Li, Zhiqiang
本文研究了一类时间导数为Caputo-Hadamard意义，空间导数为分数阶拉普拉斯式的半线性时空分数阶扩散方程解的爆破性和整体存在性。得到了所考虑的半线性方程的温和解的卷积形式，其中基本解用Fox h函数表示。然后，应用收缩映射原理，证明了温和解的局部存在唯一性，并证明了温和解是一个弱解。用不动点参数证明了该半线性方程在有限时间内解的爆破性和解的整体存在性。最后，通过数值模拟验证了该解在有限时间内的爆破性。
This paper is devoted to studying the blow-up and global existence of the solution to a semilinear time-space fractional diffusion equation, where the time derivative is in the Caputo-Hadamard sense and the spatial derivative is the fractional Laplacian. The mild solution of the considered semilinear equation by a convolution form is obtained, where the fundamental solutions are denoted by Fox H-functions. Then, applying contraction mapping principle, the local existence and uniqueness of the mild solution are shown, and the mild solution is proved to be a weak solution. The blow-up in a finite time and global existence of the solution to this semilinear equation are displayed by a fixed point argument. Finally, the blow-up of solution in a finite time is verified by numerical simulations.