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**The occurrence of interfaces in nonlinear diffusion-advection processes.**
*(English)*
Zbl 0672.76094

The nonlinear diffusion-advection processes referred to in the title of this paper are those described by the equation (*) \(u_ t=(a(u))_{xx}+(b(u))_ x\), in which subscripts denote partial differentiation. The functions a and b belong to \(C([0,\infty))\cap C^ 2(0,\infty)\), and are such that \(a''\) and \(b''\) are locally Hölder continuous on (0,\(\infty)\), and \(a'(s)>0\) for \(s>0\). Furthermore, without any loss of generality, it is supposed that \(a(0)=0\) and \(b(0)=0\). Because of its resemblance to the celebrated equation arising in statistical mechanics, (*) is often referred to as the nonlinear Fokker- Planck equation. Equation (*) models a number of different physical phenomena. For instance, when u denotes unsaturated soil-moisture content, the equation describes the infiltration of water in a homogeneous porous medium. It also appears with \(a(s)=s^ 4\) and \(b(s)=- s^ 3\) in the theory of the flow of a thin viscous film over an inclined bed.

Equation (*) is parabolic when \(u>0\), but may degenerate for \(u=0\). Hence the equation need not admit classical solutions. Under appropriate conditions though, the equation is known to possess a unique generalized solution which is nonnegative and continuous. This solution is a classical solution of (*) in a neighbourhood of any point where it is positive. It is trivially also a classical solution in the interior of the set of points where it is zero. However, the derivatives of the solution may be undefined or discontinuous at points separating a region where the solution is positive from one where it is zero. The characteristic that equation (*) admit solutions possessing interfaces separating a region where the solution is positive from one where it is zero, is, itself, a peculiarity associated with the degeneracy of the equation. For instance, solutions of the linear heat equation do not display such behaviour. Given an initial-boundary value problem for the linear heat equation with nontrivial nonnegative initial data, the solution is positive everywhere in the problem domain. In this context, the linear heat equation is often said to propagate perturbations with infinite speed. An equation which does admit solutions possessing interfaces of the type described is said to have finite speed of propagation of perturbations.

In this paper we shall establish necessary and sufficient conditions for equation (*) to admit solutions possessing interfaces separating a region where the solution is positive from one where it is zero. We also study some properties characterizing such an interface. For convenience, we restrict the discussion to the Cauchy problem for equation (*) and to interfaces which provide an upper bound for the support of a solution. Nonetheless, our results may be extended to the Cauchy-Dirichlet problem and the first boundary-value problem for equation (*), and to general interfaces.

Equation (*) is parabolic when \(u>0\), but may degenerate for \(u=0\). Hence the equation need not admit classical solutions. Under appropriate conditions though, the equation is known to possess a unique generalized solution which is nonnegative and continuous. This solution is a classical solution of (*) in a neighbourhood of any point where it is positive. It is trivially also a classical solution in the interior of the set of points where it is zero. However, the derivatives of the solution may be undefined or discontinuous at points separating a region where the solution is positive from one where it is zero. The characteristic that equation (*) admit solutions possessing interfaces separating a region where the solution is positive from one where it is zero, is, itself, a peculiarity associated with the degeneracy of the equation. For instance, solutions of the linear heat equation do not display such behaviour. Given an initial-boundary value problem for the linear heat equation with nontrivial nonnegative initial data, the solution is positive everywhere in the problem domain. In this context, the linear heat equation is often said to propagate perturbations with infinite speed. An equation which does admit solutions possessing interfaces of the type described is said to have finite speed of propagation of perturbations.

In this paper we shall establish necessary and sufficient conditions for equation (*) to admit solutions possessing interfaces separating a region where the solution is positive from one where it is zero. We also study some properties characterizing such an interface. For convenience, we restrict the discussion to the Cauchy problem for equation (*) and to interfaces which provide an upper bound for the support of a solution. Nonetheless, our results may be extended to the Cauchy-Dirichlet problem and the first boundary-value problem for equation (*), and to general interfaces.

### MSC:

76R50 | Diffusion |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

### Keywords:

nonlinear diffusion-advection processes; locally Hölder continuous; nonlinear Fokker-Planck equation; perturbations; Cauchy problem; Cauchy- Dirichlet problem; first boundary-value problem
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\textit{B. H. Gilding}, Arch. Ration. Mech. Anal. 100, No. 3, 243--263 (1988; Zbl 0672.76094)

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