scholarly journals Global Existence of Solutions to the Fowler Equation in a Neighbourhood of Travelling-Waves

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Afaf Bouharguane
Keyword(s):  
Fluid Flow ◽  
Global Existence ◽  
Diffusion Equation ◽  
Existence Of Solutions ◽  
Travelling Waves ◽  
Sand Dunes ◽  
Fractional Diffusion ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Fowler Equation

We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation.

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 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.

scholarly journals Global and non global solutions for a class of coupled parabolic systems

2020 ◽  
Vol 9 (1) ◽  
pp. 1383-1401 ◽  
Author(s):  
T. Saanouni
Keyword(s):  
Global Existence ◽  
Exponential Decay ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Parabolic Systems ◽  
Heat Equations ◽  
Well Posedness ◽  
Potential Well ◽  
Global Existence Of Solutions ◽  
Coupled Parabolic Systems

Abstract In the present paper, we investigate the global well-posedness and exponential decay for some coupled non-linear heat equations. Moreover, we discuss the global and non global existence of solutions using the potential well method.

Global and non Global Solutions for Some Fractional Heat Equations With Pure Power Nonlinearity

2017 ◽  
Vol 69 (4) ◽  
pp. 854-872
Author(s):  
Tarek Saanouni
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Heat Equations ◽  
Well Posedness ◽  
Potential Well ◽  
Initial Value ◽  
Fractional Heat Equation ◽  
Global Existence Of Solutions

AbstractThe initial value problem for a semi-linear fractional 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.

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.

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