scholarly journals On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

2015 ◽  
Vol 35 (12) ◽  
pp. 5927-5962 ◽  
Author(s):  
Gabriele Grillo ◽  
Matteo Muratori ◽  
Fabio Punzo
Keyword(s):  
Porous Medium ◽  
Asymptotic Behaviour ◽  
Porous Medium Equation ◽  
Variable Density ◽  
Medium Equation ◽  
Asymptotic Behaviour Of Solutions

scholarly journals Asymptotic behaviour of solutions for porous medium equation with periodic absorption

2001 ◽  
Vol 26 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Yin Jingxue ◽  
Wang Yifu
Keyword(s):  
Porous Medium ◽  
Periodic Solution ◽  
Boundary Value Problem ◽  
Asymptotic Behaviour ◽  
Small Neighborhood ◽  
Porous Medium Equation ◽  
Direct Consequence ◽  
Medium Equation ◽  
Asymptotic Behaviour Of Solutions ◽  
Nontrivial Periodic Solutions

This paper is concerned with porous medium equation with periodic absorption. We are interested in the discussion of asymptotic behaviour of solutions of the first boundary value problem for the equation. In contrast to the equation without sources, we show that the solutions may not decay but may be “attracted” into any small neighborhood of the set of all nontrivial periodic solutions, as time tends to infinity. As a direct consequence, the null periodic solution is “unstable.” We have presented an accurate condition on the sources for solutions to have such a property. Whereas in other cases of the sources, the solutions might decay with power speed, which implies that the null periodic solution is “stable.”

scholarly journals Global existence of solutions and smoothing effects for classes of reaction–diffusion equations on manifolds

2021 ◽  
Author(s):  
Gabriele Grillo ◽  
Giulia Meglioli ◽  
Fabio Punzo
Keyword(s):  
Porous Medium ◽  
Global Existence ◽  
Source Term ◽  
Reaction Diffusion ◽  
Porous Medium Equation ◽  
Variable Density ◽  
Reaction Diffusion Equations ◽  
Medium Equation ◽  
Global Existence Of Solutions ◽  
Smoothing Effects

AbstractWe consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in $${{\mathbb {R}}}^n$$ R n .

Maximal viscosity solutions of the modified porous medium equation and their asymptotic behaviour

1996 ◽  
Vol 7 (5) ◽  
pp. 453-471 ◽  
Author(s):  
Josephus Hulshof ◽  
Juan Luis Vazquez
Keyword(s):  
Porous Medium ◽  
Asymptotic Behaviour ◽  
Viscosity Solutions ◽  
Propagation Speed ◽  
Porous Medium Equation ◽  
Main Theme ◽  
Optimal Regularity ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Similar Solutions

We construct a theory for maximal viscosity solutions of the Cauchy problem for the modified porous medium equation ut + γ|ut| = Δ(um) with γ∈(−1, 1) and m > 1. We investigate the existence, uniqueness, finite propagation speed and optimal regularity of these solutions. As a second main theme, we prove that the asymptotic behaviour is given by a certain family of self-similar solutions of the so-called second kind with anomalous similarity exponents.

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