Journal of Nonlinear Science ( IF 3 ) Pub Date : 2021-07-30 , DOI: 10.1007/s00332-021-09736-y Changpin Li 1 , Zhiqiang Li 1
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.
中文翻译:
带分数拉普拉斯算子的 Caputo-Hadamard 分数偏微分方程解的爆炸和全局存在性
本文致力于研究半线性时空分数扩散方程解的膨胀和全局存在性,其中时间导数为 Caputo-Hadamard 意义,空间导数为分数拉普拉斯算子。得到了所考虑的半线性方程的卷积形式的温和解,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,证明了温和解是弱解。这个半线性方程的解在有限时间内的爆发和全局存在由不动点参数显示。最后,通过数值模拟验证了有限时间内解的膨胀。