scholarly journals Study of a family of higher order nonlocal degenerate parabolic equations: From the porous medium equation to the thin film equation

2015 ◽  
Vol 259 (11) ◽  
pp. 5782-5812 ◽  
Author(s):  
Rana Tarhini
Keyword(s):  
Thin Film ◽  
Porous Medium ◽  
Parabolic Equations ◽  
Porous Medium Equation ◽  
Higher Order ◽  
Degenerate Parabolic Equations ◽  
Medium Equation ◽  
Thin Film Equation ◽  
Degenerate Parabolic

LYAPUNOV FUNCTIONALS IN SINGULAR LIMITS FOR PERTURBED QUASILINEAR DEGENERATE PARABOLIC EQUATIONS

2003 ◽  
Vol 01 (04) ◽  
pp. 351-385 ◽  
Author(s):  
MANUELA CHAVES ◽  
VICTOR A. GALAKTIONOV
Keyword(s):  
Porous Medium ◽  
Parabolic Equations ◽  
Quasilinear Parabolic Equation ◽  
Porous Medium Equation ◽  
Degenerate Parabolic Equations ◽  
Radial Solutions ◽  
Medium Equation ◽  
Singular Limits ◽  
Self Similar ◽  
Degenerate Parabolic

As a key example, we study the asymptotic behaviour near finite focusing time t=T of radial solutions of the porous medium equation with absorption [Formula: see text] with bounded compactly supported initial data u(x,0)=u0(|x|), and exponents m>1 and p>pc, where pc=pc(m,N)∈(-m,0) is a critical exponent. We show that under certain assumptions, the behaviour of the solution as t→T- near the origin is described by self-similar Graveleau solutions of the porous medium equation ut=Δum. In the rescaled variables, we deal with an exponential non-autonomous perturbation of a quasilinear parabolic equation, which is shown to admit an approximate Lyapunov functional. The result is optimal, and in the critical case p=pc an extra ln (T-t) scaling of the Graveleau asymptotics is shown to occur. Other types of self-similar and non self-similar focusing patterns are discussed.

scholarly journals On a fractional thin film equation

2020 ◽  
Vol 9 (1) ◽  
pp. 1516-1558
Author(s):  
Antonio Segatti ◽  
Juan Luis Vázquez
Keyword(s):  
Thin Film ◽  
Porous Medium ◽  
Porous Medium Equation ◽  
Order Theory ◽  
Compactly Supported ◽  
Limit Value ◽  
Medium Equation ◽  
Thin Film Equation ◽  
Self Similar ◽  
Similar Solutions

Abstract This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value α = 4 while the Porous Medium Equation is the limit α = 2. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.

scholarly journals Null-controllability of perturbed porous medium gas flow

2020 ◽  
Vol 26 ◽  
pp. 85
Author(s):  
Borjan Geshkovski
Keyword(s):  
Porous Medium ◽  
Gas Flow ◽  
Porous Medium Equation ◽  
Null Controllability ◽  
Spectral Techniques ◽  
Medium Equation ◽  
Self Similar Solution ◽  
Linearized Problem ◽  
Thin Film Equation ◽  
Controllability Result

In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation, which is the equation satisfied by a perturbation around the self-similar solution of the porous medium equation in Lagrangian-like coordinates. We prove a local null-controllability result for a regularized version of the nonlinear problem, in which singular terms have been removed from the nonlinearity. We use spectral techniques and the source-term method to deal with the linearized problem and the conclusion follows by virtue of a Banach fixed-point argument. The spectral techniques are also used to prove a null-controllability result for the linearized thin-film equation, a degenerate fourth order analog of the problem under consideration.

scholarly journals Decreasing of the L^1 norm and mass conservation for Porous Medium Equations with advection

2018 ◽  
Vol 4 (2) ◽  
pp. 67-77
Author(s):  
Nicolau Matiel Lunardi Diehl ◽  
Lucinéia Fabris
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Mass Conservation ◽  
Degenerate Parabolic Equations ◽  
Conservation Property ◽  
The Cauchy Problem ◽  
Porous Medium Equations ◽  
Degenerate Parabolic

In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the formu_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u),   x \in R^n , t > 0,where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.

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