scholarly journals Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian

2019 ◽  
Vol 18 (3) ◽  
pp. 1205-1226 ◽  
Author(s):  
Ronghua Jiang ◽  
◽  
Jun Zhou ◽  
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Blow Up ◽  
Existence Of Solutions ◽  
Global Existence Of Solutions ◽  
Fraction I

scholarly journals Blow-up and global existence for a kinetic equation of swarm formation

2017 ◽  
Vol 27 (06) ◽  
pp. 1153-1175 ◽  
Author(s):  
Mirosław Lachowicz ◽  
Henryk Leszczyński ◽  
Martin Parisot
Keyword(s):  
Asymptotic Behavior ◽  
Kinetic Equation ◽  
Global Existence ◽  
Integral Operator ◽  
Blow Up ◽  
Existence Of Solutions ◽  
Interaction Rate ◽  
Global Existence Of Solutions ◽  
Nonlinear Integral Operator ◽  
The Right

In this paper we study a kinetic equation that describes swarm formations. The right-hand side of this equation contains nonlinear integro-differential terms responsible for two opposite tendencies: dissipation and swarming. The nonlinear integral operator describes the changes of velocities (orientations) of interacting individuals. The interaction rate is assumed to be dependent of velocities of interacting individuals. Although the equation seems to be rather simple it leads to very complicated dynamics. In this paper, we study possible blow-ups versus global existence of solutions and provide results on the asymptotic behavior. The complicated dynamics and possibility of blow-ups can be directly related to creation of swarms.

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

2018 ◽  
Vol 21 (6) ◽  
pp. 1619-1640 ◽  
Author(s):  
Yaning Li ◽  
Quanguo Zhang
Keyword(s):  
Global Existence ◽  
Nonlinear Diffusion ◽  
Blow Up ◽  
Existence Of Solutions ◽  
Fractional Diffusion ◽  
Fractional Integrals ◽  
Nonlinear Diffusion Equations ◽  
Global Existence Of Solutions ◽  
Time Fractional Diffusion Equation ◽  
Alpha Beta

AbstractIn this paper, we study the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations$$\begin{array}{} \displaystyle \left\{\begin{array}{l}{_0^CD_t^\alpha u}-\triangle u={_0I_t^{1-\gamma}}(|u|^{p-1}u), \, \, x\in \mathbb{R}^N,\, \, t\gt 0,\\ u(0,x)=u_0(x),\, \, x\in \mathbb{R}^N, \end{array}\right. \end{array}$$where 0 <α<γ< 1,p> 1,u0∈C0(ℝN),$\begin{array}{} {_0I_t^{\theta}} \end{array}$denotes left Riemann-Liouville fractional integrals of orderθ.$\begin{array}{} {_0^CD_t^\alpha u}=\frac{\partial}{\partial t}{_0I_t^{1-\alpha}} \end{array}$(u(t,x) −u(0,x))}. Letβ= 1 −γ. We prove that if 1 <p<p∗=$\begin{array}{} \max\big\{1+\frac{\beta}{\alpha},1+\frac{2(\alpha+\beta)}{\alpha N}\big\} \end{array}$, the solutions of (1.1) blows up in a finite time. IfN<$\begin{array}{} \frac{2(\alpha+\beta)}{\beta} \end{array}$,p≥p∗orN≥$\begin{array}{} \frac{2(\alpha+\beta)}{\beta} \end{array}$,p>p∗, and ∥u0∥Lqc(ℝN)is sufficiently small, where$\begin{array}{} q_c=\frac{N\alpha(p-1)}{2(\alpha+\beta)} \end{array}$, the solutions of (1.1) exists globally.

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