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Volume 17, Issue 1
The Nonexistence of Global Solutions for a Damped Time Fractional Diffusion Equation with Nonlinear Memory

Qian Qian & Yaning Li

J. Info. Comput. Sci. , 17 (2022), pp. 003-018.

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In this paper, we study the non-global existence of solutions to the following time fractional nonlinear diffusion equations $$\begin{cases} ^cD^{\alpha}_{0|t}u-\Delta u+(1+t)^ru_t=I^{\beta}_{0|t}(|u|^{p-1}u), \ x\in \mathbb{R}^n, \ t>0 \\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x), \ x\in\mathbb{R}^n, \end{cases}$$ where $1<\alpha<2$, $\beta\in(0,1)$, $1<\alpha+\beta<2$, $r\in (-1,1)$, $p>1$, $u_0, u_1\in L^q(\mathbb{R}^n)(q>1)$ and $^cD^{\alpha}_{0|t}u$ denotes left Caputo fractional derivative of order $\alpha$. By using the test function method, we prove that the problem admits no global weak solution with suitable initial data when $p$ falls in different intervals. Our results generalize that in [4].

  • Keywords

fractional derivative, blow-up, test function, nonlinear memory.

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COPYRIGHT: © Global Science Press

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@Article{JICS-17-003, author = {Qian , Qian and Li , Yaning}, title = {The Nonexistence of Global Solutions for a Damped Time Fractional Diffusion Equation with Nonlinear Memory}, journal = {Journal of Information and Computing Science}, year = {2022}, volume = {17}, number = {1}, pages = {003--018}, abstract = {

In this paper, we study the non-global existence of solutions to the following time fractional nonlinear diffusion equations $$\begin{cases} ^cD^{\alpha}_{0|t}u-\Delta u+(1+t)^ru_t=I^{\beta}_{0|t}(|u|^{p-1}u), \ x\in \mathbb{R}^n, \ t>0 \\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x), \ x\in\mathbb{R}^n, \end{cases}$$ where $1<\alpha<2$, $\beta\in(0,1)$, $1<\alpha+\beta<2$, $r\in (-1,1)$, $p>1$, $u_0, u_1\in L^q(\mathbb{R}^n)(q>1)$ and $^cD^{\alpha}_{0|t}u$ denotes left Caputo fractional derivative of order $\alpha$. By using the test function method, we prove that the problem admits no global weak solution with suitable initial data when $p$ falls in different intervals. Our results generalize that in [4].

}, issn = {3080-180X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jics/22357.html} }
TY - JOUR T1 - The Nonexistence of Global Solutions for a Damped Time Fractional Diffusion Equation with Nonlinear Memory AU - Qian , Qian AU - Li , Yaning JO - Journal of Information and Computing Science VL - 1 SP - 003 EP - 018 PY - 2022 DA - 2022/01 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jics/22357.html KW - fractional derivative, blow-up, test function, nonlinear memory. AB -

In this paper, we study the non-global existence of solutions to the following time fractional nonlinear diffusion equations $$\begin{cases} ^cD^{\alpha}_{0|t}u-\Delta u+(1+t)^ru_t=I^{\beta}_{0|t}(|u|^{p-1}u), \ x\in \mathbb{R}^n, \ t>0 \\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x), \ x\in\mathbb{R}^n, \end{cases}$$ where $1<\alpha<2$, $\beta\in(0,1)$, $1<\alpha+\beta<2$, $r\in (-1,1)$, $p>1$, $u_0, u_1\in L^q(\mathbb{R}^n)(q>1)$ and $^cD^{\alpha}_{0|t}u$ denotes left Caputo fractional derivative of order $\alpha$. By using the test function method, we prove that the problem admits no global weak solution with suitable initial data when $p$ falls in different intervals. Our results generalize that in [4].

Qian , Qian and Li , Yaning. (2022). The Nonexistence of Global Solutions for a Damped Time Fractional Diffusion Equation with Nonlinear Memory. Journal of Information and Computing Science. 17 (1). 003-018. doi:
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