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 - 2024
DA - 2024/01
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 - <p style="text-align: justify;">In this paper, we study the non-global existence of solutions to the following time fractional
nonlinear diffusion equations
$$\begin{cases} ^cD^{\alpha}_{0|t}u-\Delta u+(1+t)^ru_t=I^{\beta}_{0|t}(|u|^{p-1}u), \ x\in \mathbb{R}^n, \ t&gt;0 \\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x), \ x\in\mathbb{R}^n, \end{cases}$$ where $1&lt;\alpha&lt;2$, $\beta\in(0,1)$, $1&lt;\alpha+\beta&lt;2$, $r\in (-1,1)$, $p&gt;1$, $u_0, u_1\in L^q(\mathbb{R}^n)(q&gt;1)$ 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].</p>