Interfaces in diffusion-absorption processes in nonhomogeneous media

We study the Cauchy problem for the nonlinear parabolic equation ? ( x ) u t = ( a ( x ) ? x ( u ) ) x - b ( x ) h ( u ) in? R × ( 0 , T with nonnegative coefficients ? ( x ) , a ( x ) and b ( x ) . It is assumed that ? ( 0 ) = 0 , ? ' ( s ) 0 , ? ' ( s ) / s ? L 1 ( 0 , ? ) for some ? 0 , h ( s ) ? 0 and h ( s ) / s is nondecreasing for s ? 0 . The solution of this problem may possess the property of finite speed of propagation of disturbances from the data, which leads to formation of interfaces that bound the support of the solution. It is proved that the behavior of interfaces can be characterized in terms of convergence or divergence of the integrals ? x 0 x ? ( s ) ( ? x 0 s d z a ( z ) ) d s , J x 0 ( x ) = b ( x ) ? ( x ) ? x 0 x ( ? 0 s ? ( z ) a ( z ) d z ) d s , b ( x ) ? ( x ) J x 0 ( x ) , ? x 0 x ? ( s ) d s as x ? ∞ and ? ? d s h ( s ) , ? ? ? ( s ) h ( s ) d s as? ? ? 0 + . We derive two-sided a priori bounds for the interface location, establish sufficient and necessary conditions for disappearance of interfaces in a finite time (the interface blow-up), and derive the integral equation for the interface.

[1]  J. Vázquez,et al.  The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates , 1996 .

[2]  P. Rosenau,et al.  Nonlinear thermal evolution in an inhomogeneous medium , 1982 .

[3]  L. Peletier,et al.  Diffusion in inhomogeneous media: Localization and positivity , 1985 .

[4]  Mark A. Peletier,et al.  Disappearing interfaces in nonlinear diffusion , 1997 .

[5]  Juan Luis Vázquez,et al.  The Cauchy problem for the inhomogeneous porous medium equation , 2006, Networks Heterog. Media.

[6]  A. Tedeev.  The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations , 2007 .

[7]  G. Grillo,et al.  Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincar\'e inequalities , 2012, 1204.6159.

[8]  Ariel G. S'anchez,et al.  Asymptotic behavior for the heat equation in nonhomogeneous media with critical density , 2012, 1206.1167.

[9]  R. Iagar,et al.  Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations , 2013 .

[10]  Sergey Shmarev,et al.  Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity , 2008 .

[11]  B. Gilding,et al.  The characterization of reaction-convection-diffusion processes by travelling waves , 1993 .

[12]  Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density , 2012 .

[13]  Robert Kersner,et al.  Disappearance of interfaces in finite time , 1993 .

[14]  Sergey Shmarev,et al.  A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions , 2005 .

[15]  Philip Rosenau,et al.  Non‐linear diffusion in a finite mass medium , 1982 .

[16]  Guillermo Reyes,et al.  Disappearance of interfaces for the porous medium equation with variable density and absorption , 2003 .

[17]  Philip Rosenau,et al.  Propagation of thermal waves in an inhomogeneous medium , 1981 .

[18]  S. Shmarev,et al.  Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology , 2009 .

[19]  D. Hilhorst,et al.  Regularity of Interfaces for an Inhomogeneous Filtration Equation , 2001 .

[20]  The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium , 1990 .

[21]  Sergei I. Shmarev,et al.  Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension , 2005 .

[22]  V. V. Pukhnachov,et al.  Evolution Equations and Lagrangian Coordinates , 1997 .

[23]  Ugur G. Abdulla.  Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption , 2002 .

[24]  J. Vázquez,et al.  Long time behavior for the inhomogeneous PME in a medium with slowly decaying density , 2008 .

[25]  J. L. Vásquez,et al.  Lagrangian coordinates and regularity of interfaces in reaction-diffusion equations , 1995 .

[26]  R. Kersner,et al.  On the Cauchy problem for a class of parabolic equations with variable density , 2013 .

[27]  John R. King,et al.  Interface Development and Local Solutions to Reaction-Diffusion Equations , 2000, SIAM J. Math. Anal..

[28]  Robert Kersner,et al.  Travelling Waves in Nonlinear Diffusion-Convection Reaction , 2004 .

[29]  A. S. Kalashnikov.  Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations , 1987 .

[30]  S. Kamin,et al.  The filtration equation in a class of functions decreasing at infinity , 1994 .

[31]  J. Vázquez.  The Porous Medium Equation , 2006 .