Viscosity solutions for quasilinear degenerate parabolic equations of porous medium type

2005 ◽  
Vol 54 (3) ◽  
pp. 817-860 ◽  
Author(s):  
Juan Luis Vazquez ◽  
Cristina Brandle
Keyword(s):  
Porous Medium ◽  
Viscosity Solutions ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Medium Type

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.

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.

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.

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