scholarly journals Finite speed of propagation for the thin-film equation and other higher-order parabolic equations with general nonlinearity

2001 ◽  
pp. 233-264 ◽  
Author(s):  
Daniele Andreucci ◽  
Anatoli Tedeev
Keyword(s):  
Thin Film ◽  
Parabolic Equations ◽  
Higher Order ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Thin Film Equation ◽  
Speed Of Propagation ◽  
Higher Order Parabolic Equations ◽  
General Nonlinearity

Source-type solutions to thin-film equations in higher dimensions

1997 ◽  
Vol 8 (5) ◽  
pp. 507-524 ◽  
Author(s):  
RAUL FERREIRA ◽  
FRANCISCO BERNIS
Keyword(s):  
Thin Film ◽  
Negative Solution ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
First Order ◽  
Source Type ◽  
Thin Film Equations ◽  
Thin Film Equation ◽  
Self Similar ◽  
Speed Of Propagation

We prove that the thin film equation ht+div (hn grad (Δh))=0 in dimension d[ges ]2 has a unique C1 source-type radial self-similar non-negative solution if 0<n<3 and has no solution of this type if n[ges ]3. When 0<n3 the solution h has finite speed of propagation and we obtain the first order asymptotic behaviour of h at the interface or free boundary separating the regions where h>0 and h=0. (The case d=1 was already known [1]).

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 Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption

1986 ◽  
Vol 104 (1-2) ◽  
pp. 1-19 ◽  
Author(s):  
F. Bernis
Keyword(s):  
Decay Rates ◽  
Dirichlet Boundary Conditions ◽  
Weighted Energy Method ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Speed Of Propagation ◽  
Image Position ◽  
Higher Order Parabolic Equations ◽  
Quasilinear Elliptic ◽  
Uniformly Bounded

SynopsisThe “energy solutions” to the equationhave finite speed of propagation if l < q < 2 or l < r < 2. If 1 <r <2 (Vq <1) support u(· t) is uniformly bounded for t >0 (localisation property) and if q<2 ≦ r, sharp upper bounds of the interface (or free boundary) are obtained. We use a weighted energy method, the weights being powers of the distance to a variable half-space. We also study decay rates as t→∞ and extinction in finite time for bounded and unbounded domains (with null Dirichlet boundary conditions). Our equation includes the porous media equation with absorption. Analogous results hold if (−Δ)m is replaced by an appropriate quasilinear elliptic operator.

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