# Blowing-up solutions of a time-space fractional semi-linear equation with a structural damping and a nonlocal in time nonlinearity

Source: [Blowing-up solutions of a time-space fractional semi-linear equation with a structural damping and a nonlocal in time nonlinearity](<https://www.sota2.com/research/paper/68fa56b21e209131ce99efbd>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T14:18:43.217Z

Authors: K. Bouguetof

Published: 2020-02-22T14:16:57.000Z

## Abstract

In this paper, we investigate the semilinear equation with a time-space fractional structural damping and a nonlocal in time nonlinearity \\begin{equation\*} {\\mathbf{D}}\_{0\|t}^{1+\\alpha\_1}u + (-\\Delta)^\\sigma u+(-\\Delta )^\\delta\\mathbf{D}\_{0\|t}^{\\alpha \_2} u = I\_{0\|t}^{1-\\gamma }\|u\|^{p}, \\qquad (t,x)\\in (0,\\infty) \\times \\mathbb{R}^N, \\end{equation\*} where $p\>1$, $\\alpha\_i, \\gamma\\ in (0,1)$, $\\delta, \\sigma \\in (0,1)$, ${\\mathbf{D}}\_{0\|t}^{\\alpha\_i}$ is the Caputo fractional derivative and $I\_{0\|t}^{1-\\gamma }$ is the Riemann-Liouville fractional integral of order $1-\\gamma$. We prove the non-existence of global solutions if \\begin{equation\*} 1\<p\\leqslant \\frac{2(2+\\alpha\_1-\\gamma)}{(\\frac{\\alpha\_1+1}{\\sigma} N+2\\gamma-2\\alpha\_1-2)\_+ }+1, \\end{equation\*} for any space dimension $N\\geqslant 1.$ Then, we extend the result to the system \\begin{align\*} &{\\mathbf{D}}\_{0\|t}^{1+\\alpha\_1}u + (-\\Delta)^{\\sigma\_1} u + (-\\Delta )^{\\delta\_1}{\\mathbf{D}}\_{0\|t}^{\\alpha\_2} u = I\_{0\|t}^{1-\\gamma \_{1}}\|v\|^{p},\\qquad (t,x)\\in (0,\\infty) \\times \\mathbb{R}^N, \\\\ &{\\mathbf{D}}\_{0\|t}^{1+\\beta \_{1}}v+(-\\Delta)^{\\sigma\_2} v + (-\\Delta )^{\\delta\_2}{\\mathbf{D}}\_{0\|t}^{\\beta\_2}v = I\_{0\|t}^{1-\\gamma\_2}\|u\|^{q},\\qquad (t,x)\\in (0,\\infty )\\times \\mathbb{R}^{N}, \\end{align\*} where $p,q\>1$, $0\<\\delta\_i$, $\\sigma\_i\<1$ and $\\gamma\_2\\in (0,1)$. Also, we present the necessary conditions for the existence of local or global solutions.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2002.09704>)

[PDF](<https://arxiv.org/pdf/2002.09704.pdf>)