scholarly journals Long-time behavior for the porous medium equation with small initial energy

2021 ◽  
pp. 108029
Author(s):  
Lorenzo Brasco ◽  
Bruno Volzone
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Initial Energy ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Medium Equation ◽  
Long Time

POROUS MEDIUM FLOW IN A TUBE: TRAVELING WAVES AND KPP BEHAVIOR

2007 ◽  
Vol 09 (05) ◽  
pp. 731-751 ◽  
Author(s):  
JUAN LUIS VÁZQUEZ
Keyword(s):  
Porous Medium ◽  
Traveling Waves ◽  
Porous Medium Equation ◽  
Time Behavior ◽  
Middle Region ◽  
Medium Equation ◽  
Nonnegative Solutions ◽  
Medium Flow ◽  
Long Time ◽  
Porous Medium Flow

We study the long-time behavior of the solutions of the Porous Medium Equation ut = Δ um, m > 1, posed in a tube Ω = ℝ × D, where D is a bounded domain in ℝn, and with homogeneous Dirichlet conditions on the lateral boundary. We show that the asymptotic behavior of general nonnegative solutions follows the KPP pattern in suitable rescaled variables. We proceed as follows: we pass to the renormalized problem and show that this problem admits a wave solution that travels along the tube with constant speed with respect to the new time (which is logarithmic in the old time scale). This solution has a bounded free boundary as a forward front. We also show that the universal asymptotic pattern for solutions with compactly supported data is described in the renormalized form by two of these traveling waves going out in different directions, joined by a stationary profile in the middle region. We compare this situation with the heat equation case ut = Δu that behaves quite differently.

scholarly journals Large time behavior of a two phase extension of the porous medium equation

2019 ◽  
Vol 21 (2) ◽  
pp. 199-229 ◽  
Author(s):  
Ahmed Ait Hammou Oulhaj ◽  
Clément Cancès ◽  
Claire Chainais-Hillairet ◽  
Philippe Laurençot
Keyword(s):  
Porous Medium ◽  
Large Time ◽  
Porous Medium Equation ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Two Phase ◽  
Medium Equation ◽  
Phase Extension

scholarly journals Large time behavior for the porous medium equation with convection

Meccanica ◽  
2017 ◽  
Vol 52 (13) ◽  
pp. 3255-3260 ◽  
Author(s):  
Daniele Andreucci ◽  
Anatoli F. Tedeev
Keyword(s):  
Porous Medium ◽  
Large Time ◽  
Porous Medium Equation ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Medium Equation

scholarly journals Long-time asymptotics for a 1D nonlocal porous medium equation with absorption or convection

2019 ◽  
Vol 22 (03) ◽  
pp. 1950015
Author(s):  
Filomena Feo ◽  
Yanghong Huang ◽  
Bruno Volzone
Keyword(s):  
Porous Medium ◽  
Exponential Convergence ◽  
Porous Medium Equation ◽  
Entropy Method ◽  
Nonlocal Diffusion ◽  
One Dimensional ◽  
Medium Equation ◽  
Long Time ◽  
Porous Medium Equations ◽  
Long Time Asymptotics

In this paper, the long-time asymptotic behaviors of one-dimensional porous medium equations with a fractional pressure and absorption or convection are studied. In the parameter regimes when the nonlocal diffusion is dominant, the entropy method is adapted to derive the exponential convergence of relative entropy of solutions in similarity variables.

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