scholarly journals Finite Speed of Propagation and Waiting Time Phenomena for Degenerate Parabolic Equations with Linear Growth Lagrangian

2015 ◽  
Vol 47 (3) ◽  
pp. 2426-2441 ◽  
Author(s):  
Lorenzo Giacomelli
Keyword(s):  
Waiting Time ◽  
Parabolic Equations ◽  
Linear Growth ◽  
Degenerate Parabolic Equations ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Degenerate Parabolic ◽  
Speed Of Propagation

scholarly journals A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

1996 ◽  
Vol 126 (4) ◽  
pp. 739-767 ◽  
Author(s):  
B. H. Gilding ◽  
R. Kersner
Keyword(s):  
Travelling Wave ◽  
Spatial Variable ◽  
Degenerate Parabolic Equations ◽  
Initial Time ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Special Cases ◽  
Degenerate Parabolic ◽  
Speed Of Propagation ◽  
Necessary And Sufficient

A degenerate parabolic partial differential equation with a time derivative and first- and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

Complete quenching phenomenon and instantaneous shrinking of support of solutions of degenerate parabolic equations with nonlinear singular absorption

2019 ◽  
Vol 149 (5) ◽  
pp. 1323-1346 ◽  
Author(s):  
Nguyen Anh Dao ◽  
Jesus Ildefonso Díaz ◽  
Huynh Van Kha
Keyword(s):  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Degenerate Parabolic Equations ◽  
One Dimensional ◽  
Finite Speed Of Propagation ◽  
Nonnegative Solutions ◽  
Degenerate Parabolic ◽  
Speed Of Propagation ◽  
The One ◽  
Quenching Phenomenon

AbstractThis paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.

scholarly journals Finite speed of propagation and waiting time for a thin-film Muskat problem

2017 ◽  
Vol 147 (4) ◽  
pp. 813-830 ◽  
Author(s):  
Philippe Laurençot ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Thin Film ◽  
Initial Data ◽  
Waiting Time ◽  
Scale Invariance ◽  
Sufficient Conditions ◽  
Two Phase ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Muskat Problem ◽  
Speed Of Propagation

Propagation at a finite speed is established for non-negative weak solutions to a thin-film approximation of the two-phase Muskat problem. The expansion rate of the support matches the scale invariance of the system. Moreover, we determine sufficient conditions on the initial data for the occurrence of waiting time phenomena.

scholarly journals Asymptotic Properties of Solutions to the Cauchy Problem for Degenerate Parabolic Equations with Inhomogeneous Density on Manifolds

2021 ◽  
Author(s):  
Daniele Andreucci ◽  
Anatoli F. Tedeev
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Degenerate Parabolic Equations ◽  
Isoperimetric Function ◽  
Finite Speed Of Propagation ◽  
The Cauchy Problem ◽  
Inhomogeneous Density ◽  
Degenerate Parabolic

AbstractWe consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

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