scholarly journals Well-posedness results for a sixth-order logarithmic Boussinesq equation

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 3985-4000
Author(s):  
Erhan Pişkin ◽  
Nazlı Irkıl
Keyword(s):  
Global Existence ◽  
Galerkin Method ◽  
Sobolev Inequality ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Existence Of Solutions ◽  
Sixth Order ◽  
Well Posedness ◽  
Decay Estimates ◽  
Global Existence Of Solutions

The main goal of this paper is to study for a sixth-order logarithmic Boussinesq equation. We obtain several results: Firstly, by using Feado-Galerkin method and a logaritmic Sobolev inequality, we proved global existence of solutions. Later, we proved blow up property in infinity time of solutions. Finally, we showed the decay estimates result of the solutions.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

scholarly journals On the System of Coupled Nondegenerate Kirchhoff Equations with Distributed Delay: Global Existence and Exponential Decay

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Abdelbaki Choucha ◽  
Salah Mahmoud Boulaaras ◽  
Djamel Ouchenane ◽  
Salem Alkhalaf ◽  
Ibrahim Mekawy ◽  
...  
Keyword(s):  
Global Existence ◽  
Exponential Stability ◽  
Galerkin Method ◽  
Exponential Decay ◽  
Energy Method ◽  
Existence Of Solutions ◽  
Distributed Delay ◽  
Stability Result ◽  
Kirchhoff Equations ◽  
Global Existence Of Solutions

This paper studies the system of coupled nondegenerate viscoelastic Kirchhoff equations with a distributed delay. By using the energy method and Faedo-Galerkin method, we prove the global existence of solutions. Furthermore, we prove the exponential stability result.

Global existence and blow-up for the generalized sixth-order Boussinesq equation

2012 ◽  
Vol 75 (11) ◽  
pp. 4325-4338 ◽  
Author(s):  
Amin Esfahani ◽  
Luiz Gustavo Farah ◽  
Hongwei Wang
Keyword(s):  
Global Existence ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Sixth Order

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.

scholarly journals Remarks on the critical nonlinear high-order heat equation

2019 ◽  
Vol 26 (1/2) ◽  
pp. 127-152
Author(s):  
Tarek Saanouni
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
High Order ◽  
Well Posedness ◽  
Potential Well ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
Nonlinear High Order

The initial value problem for a semi-linear high-order heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

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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