Blow-up and global existence of solutions for a time fractional diffusion equation

Yaning Li; Quanguo Zhang

doi:10.1515/fca-2018-0085

Article

Blow-up and global existence of solutions for a time fractional diffusion equation

Yaning Li and Quanguo Zhang

Published/Copyright: February 9, 2019

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From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 6

Abstract

In this paper, we study the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations

0CDtαu−△u=0It1−γ(|u|p−1u),x∈RN,t>0,u(0,x)=u0(x),x∈RN,

where 0 < α < γ < 1, p > 1, u₀ ∈ C₀(ℝ^N), 0Itθ denotes left Riemann-Liouville fractional integrals of order θ. 0CDtαu=∂∂t0It1−α (u(t, x) − u(0, x))}. Let β = 1 − γ. We prove that if 1 < p < p^∗ = max{1+βα,1+2(α+β)αN} , the solutions of (1.1) blows up in a finite time. If N < 2(α+β)β , p ≥ p^∗ or N ≥ 2(α+β)β , p > p^∗, and ∥u₀∥_{L^q_c(ℝ^N)} is sufficiently small, where qc=Nα(p−1)2(α+β) , the solutions of (1.1) exists globally.

MSC 2010: Primary 26A33; Secondary 34A08; 33K15

Key Words and Phrases: fractional calculus; blow-up; global existence; nonlinear memory

Acknowledgements

This paper has been partially supported by NSF of China (11801276, 11626132, 11601216, 71501101), NSF of JiangSu Province(BK20150928) and the Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (2015SJB063).

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Received: 2018-01-02

Published Online: 2019-02-09

Published in Print: 2018-12-19

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Articles in the same Issue

https://doi.org/10.1515/fca-2018-0085

Keywords for this article

fractional calculus; blow-up; global existence; nonlinear memory