arXiv subjects

Liujie Guo

Publications and source records attributed to Liujie Guo.

4 recordsLinked to original sources

arXiv2022

Global boundedness and asymptotic behavior of time-space fractional nonlocal reaction-diffusion equation

The global boundedness and asymptotic behavior are investigate for the solution of time-space fractional non-local reaction-diffusion equation (TSFNRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=-(-Δ)^{s} u+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty),$$ where $s\in(0,1),α\in(0,1), N \leq 2$. The operator $\partial_{t}^{α}$ is the Caputo fractional derivative, which $-(-Δ)^{s}$ is the fractional Laplacian operator. For appropriate assumptions on $J$, it is proved that for homogeneous Dirichlet boundary condition, this problem admits a global bounded weak solution for $N=1$, while for $N=2$, global bounded weak solution exists for large $k$ values by Gagliardo-Nirenberg inequality and fractional differential inequality. With further assumptions on the initial datum, for small $μ$ values, the solution is shown to converge to $0$ exponentially or locally uniformly as $t \rightarrow \infty$. Furthermore, under the condition of $J \equiv 1$, it is proved that the nonlinear TSFNRDE has a unique weak solution which is global bounded in fractional Sobolev space with the nonlinear fractional diffusion terms $-(-Δ)^{s} u^{m}\, (2-\frac{2}{N}<m<1)$.

math.AP↗

arXiv2022

Global existence, uniqueness and $L^{\infty}$-bound of weak solutions of fractional time-space Keller-Segel system

This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in $\mathbb{R}^{n}$, $n\geq 2$. The global existence and $L^{\infty}$-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) $b>1-\fracα{n}$, for any initial value and birth rate; (ii) $0 q$, the $L^{h}$-norms of the weak solution blow up at the same time.

math.AP↗

arXiv2022

Global boundedness and Allee effect for a nonlocal time fractional p-Laplacian reaction-diffusion equation

The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional p-Laplacian reaction-diffusion equation (NTFPLRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=Δ_{p} u+μu^{2}(1-kJ*u) -γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty)$$ with $0<α<1,β, μ,k>0,N\leq 2$ and $Δ_{p}u =div(\left| \bigtriangledown u \right|^{p-2}\bigtriangledown u)$. Under appropriate assumptions on $J$ and the conditions of $1<p<2$, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if $k^{*}=0$ for $N=1$ or $k^{*}=(μC^{2}_{GN}+1)η^{-1}$ for $N=2$, where $C_{GN}$ is the constant in Gagliardo-Nirenberg inequality. With further assumptions on the initial datum, for small $μ$ values, the solution is shown to converge to $0$ exponentially or locally uniformly as $t \rightarrow \infty$, which is referred as the Allee effect in sense of Caputo derivative. Moreover, under the condition of $J \equiv 1$, it is proved that the nonlinear NTFPLRDE has a global bounded solution in any dimensional space with the nonlinear p-Laplacian diffusion terms $Δ_{p} u^{m}\, (2-\frac{2}{N}< m\leq 3)$.

math.AP↗

arXiv2021

Global boundedness and Allee effect for a nonlocal time fractional reaction-diffusion equation

The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional reaction-diffusion equation (NTFRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=Δu+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty)$$ with $0<α<1,β, μ,k>0,N\leq 2$ and $u(x,0)=u_{0}(x)$. Under appropriate assumptions on $J$ and the property of time fractional derivative, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if $k^{*}=0$ for $N=1$ or $k^{*}=(μC^{2}_{GN}+1)η^{-1}$ for $N=2$, where $C_{GN}$ is the constant in Gagliardo-Nirenberg inequality. With further assumptions on the initial datum, for small $μ$ values, the solution is shown to converge to $0$ exponentially or locally uniformly as $t \rightarrow \infty$, which is referred as the Allee effect in sense of Caputo derivative. Moreover, under the condition of $J \equiv 1$, it is proved that the nonlinear NTFRDE has a global bounded solution in any dimensional space with the nonlinear diffusion terms $Δu^{m}\, (2-\frac{2}{N}< m\leq 3)$.

math.AP↗