Advances in Mathematical Physics

Advances in Mathematical Physics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 606757 | 26 pages | https://doi.org/10.1155/2011/606757

On the Solution of a Hyperbolic One-Dimensional Free Boundary Problem for a Maxwell Fluid

Academic Editor: Luigi Berselli
Received11 Mar 2011
Revised12 May 2011
Accepted14 Jun 2011
Published31 Jul 2011

Abstract

We study a hyperbolic (telegrapher's equation) free boundary problem describing the pressure-driven channel flow of a Bingham-type fluid whose constitutive model was derived in the work of Fusi and Farina (2011). The free boundary is the surface that separates the inner core (where the velocity is uniform) from the external layer where the fluid behaves as an upper convected Maxwell fluid. We present a procedure to obtain an explicit representation formula for the solution. We then exploit such a representation to write the free boundary equation in terms of the initial and boundary data only. We also perform an asymptotic expansion in terms of a parameter tied to the rheological properties of the Maxwell fluid. Explicit formulas of the solutions for the various order of approximation are provided.

1. Introduction

In this paper we study the well posedness of a hyperbolic free boundary problem arisen from a one-dimensional model for the channel flow of a rate-type fluid with stress threshold presented in [1]. The model describes the one-dimensional flow of a fluid which behaves as a nonlinear viscoelastic fluid if the stress is above a certain threshold ๐œ๐‘œ and like a rate type fluid if the stress is below that threshold. The problem investigated here belongs to a series of extensions of the classical Bingham model we have proposed in recent years (see [2โ€“5]).

In particular, in [1] we describe the one-dimensional flow of such a fluid in an infinite channel, assuming that in the outer part of the channel the material behaves as a viscoelastic upper convected Maxwell fluid, while in the inner core as a rate-type Oldroyd-B fluid. The general mathematical model is derived within the framework of the theory of natural configurations developed by Rajagopal and Srinivasa (see [6]). The constitutive equations are obtained imposing how the system stores and dissipates energy and exploiting the criterion of the maximization of the dissipation rate.

The main practical motivation behind this study comes from the analysis of materials like asphalt or bitumen which exhibit a stress threshold beyond which they change its rheological properties. Indeed from the papers [7โ€“9], it is clear that such materials have a viscoelastic behaviour (for instance, upper convected Maxwell fluid) which is observed if the applied stress is greater than a certain threshold (see, in particular, [7]).

The mathematical formulation for the channel flow driven by a constant pressure gradient consists in a free boundary problem involving a hyperbolic telegrapher's equation (Maxwell fluid) and a third-order equation (Oldroyd-B fluid). The free boundary is the surface dividing the two domains: the inner channel core and the external layer. Due to the high complexity of the general problem, here we have considered a simplified version which arises when the order of magnitude of some physical parameters involved in the general model ranges around particular values. In such a case we have that the velocity of the inner core is constant in space and time, while the outer part behaves as a viscoelastic upper convected Maxwell fluid (see [1] for more details). The mathematical formulation turns out to be a hyperbolic free boundary problem which, in the authors knowledge, is new since it involves a telegrapher's equation coupled with an ODE describing the evolution of the interface.

The paper is structured as follows. In Section 2 we formulate the problem, namely, problem (2.1), and specify the basic assumptions. In Section 3 we give an equivalent formulation of the problem which leads to a nonlinear integrodifferential equation for the free boundary. We prove local existence and uniqueness for such an equation (see Theorem 3.3), under specific assumptions on the data.

The interesting aspect of the mathematical analysis lies on the technique we employ to reduce the complete problem to a single integrodifferential equation from which some mathematical properties can be derived (the free boundary equation can be solved autonomously from the governing equation of the velocity field). Such a methodology is a generalization of a technique already introduced in [2].

In Section 4 we perform an asymptotic expansion in terms of a coefficient ๐œ” (representing the ration between the elastic characteristic time and the relaxation time of the viscoelastic material), which typically is of the order ๐‘‚(10โˆ’1). This procedure allows to obtain approximations of the actual solution up to any order through an iterative procedure. We do not prove the convergence of the asymptotic approximations to the actual solution (whose existence is proved in Theorem 3.3), limiting ourselves to develop only the formal procedure. Indeed, the main advantage of this procedure is that, for each order of approximation, the governing equation for the velocity field is the โ€œstandardโ€ wave equation, which is by far easier to handle than the telegrapher's equation. We end the paper with few conclusive remarks.

2. Mathematical Formulation

In this section we state the mathematical problem. We refer the reader to [1] for all the details describing how this simplified model was derived from the general one. In the general case, in the region [0,๐‘ ], the fluid behaves as an Oldroyd-B type fluid. The problem we are studying here is a particular case of such a model, which stems from some specific assumptions on the physical parameters (fulfilled by materials like asphalt and bitumen). Under such assumptions, the inner core [0,๐‘ ] moves with uniform constant velocity ๐‘‰๐‘œ.

We consider an orthogonal coordinate system ๐‘ฅ๐‘œ๐‘ฆ and assume that the fluid is confined between two parallel plates placed at distance 2๐ฟ. We assume that the motion takes place along the ๐‘ฅ-direction and that the velocity field has the form โƒ—โƒ—๐‘–๐‘ฃ(๐‘ฆ,t)=๐‘ฃ(๐‘ฆ,๐‘ก). We rescale the problem (in a nondimensional form) and, because of symmetry, we study the upper part of the layer ๐‘ฆโˆˆ[0,1] (the space variable is rescaled by ๐ฟ). The geometry of the system we investigate is depicted in Figure 1.

The mathematical model is written for the velocity field ๐‘ฃ(๐‘ฆ,๐‘ก) in the viscoelastic region which is separated from the region with zero strain rate (uniform velocity) by the moving interface ๐‘ฆ=๐‘ (๐‘ก).

The nondimensional formulation is the following:๐‘ฃ๐‘ก๐‘ก+2๐œ”๐‘ฃ๐‘ก=๐‘ฃ๐‘ฆ๐‘ฆ+๐›ฝ2๐‘ฃ๐‘ฆโˆˆ(๐‘ ,1),๐‘ก>0,(๐‘ฆ,0)=๐‘ฃ๐‘œ๎€ท๐‘ (๐‘ฆ)๐‘ฆโˆˆ๐‘œ๎€ธ,๐‘ฃ,1๐‘ก(๎€ท๐‘ ๐‘ฆ,0)=0๐‘ฆโˆˆ๐‘œ๎€ธ,๐‘ฃ,1๐‘ฃ(1,๐‘ก)=0๐‘ก>0,(๐‘ ,๐‘ก)=๐‘‰๐‘œ๐‘ฃ๐‘ก>0,๐‘ฆ(๐‘ ,๐‘ก)+ฬ‡๐‘ ๐‘ฃ๐‘ก(๐‘ ,๐‘ก)=โˆ’๐›ฝ2๐‘ ๐™ฑ๐š—๐‘ก>0,(0)=๐‘ ๐‘œ,๐‘ ๐‘œโˆˆ(0,1).(2.1) where (i)๐œŒ is the material density, (ii)๐œ‚ is the viscosity of the fluid, (iii)๐œ‡ is the elastic modulus, (iv)๐›ฝ2 is a positive parameter depending on the viscosity ๐œ‚ (see [1]), (v)๐™ฑ๐š— is the Bingham number, (vi)๐‘‰๐‘œ is the velocity of the inner core, (vii)2๐œ”=๐‘ก๐‘’/๐‘ก๐‘Ÿ, (viii)๐‘ก๐‘’โˆš=๐ฟ๐œŒ/๐œ‡ is the characteristic elastic time, (ix)๐‘ก๐‘Ÿ=๐œ‚/2๐œ‡ is the relaxation time.

In the case of asphalt typical values are (see [8, 9])๐œ‡=1MPa,๐œŒ=1.5ร—103K๐‘”/m3,๐œ‚=102MPaโ‹…s.(2.2)

Taking ๐ฟ=500m we get๐‘ก๐‘’=15s,๐‘ก๐‘Ÿ=50s,โŸน๐œ”=0.15.(2.3)

Remark 2.1. In [1] we have proved that problem (2.1) admits a stationary solution provided ๐‘‰๐‘œโฉฝ๐›ฝ2๎‚€12๎‚+๐™ฑ๐š—(2.4) and that the stationary solution is given by ๐‘ฃโˆž๐›ฝ(๐‘ฆ)=โˆ’22(๐‘ โˆ’๐™ฑ๐š—โˆ’๐‘ฆ)2+๐›ฝ2๐™ฑ๐š—2+๐‘‰๐‘œ,๐‘ โˆž๎ƒŽ=1+๐™ฑ๐š—โˆ’๐™ฑ๐š—2+2๐‘‰๐‘œ๐›ฝ2.(2.5)

3. An Equivalent Formulation

Before proceeding in proving analytical results of problem (2.1) we introduce the new coordinate system๐‘ฅ=1โˆ’๐‘ฆ,โŸบ๐‘ฆ=1โˆ’๐‘ฅ,(3.1) and the new variable๐‘ˆ(๐‘ฅ,๐‘ก)=exp(๐œ”๐‘ก)๐‘ฃ(1โˆ’๐‘ฅ,๐‘ก),โŸบ๐‘ฃ(๐‘ฆ,๐‘ก)=๐‘ˆ(1โˆ’๐‘ฆ)exp(โˆ’๐œ”๐‘ก).(3.2) With transformations (3.1)-(3.2), problem (2.1) becomes๐‘ˆ๐‘ฅ๐‘ฅโˆ’๐‘ˆ๐‘ก๐‘ก+๐œ”2๐‘ˆ=โˆ’๐›ฝ2exp(๐œ”๐‘ก),๐‘ฅโˆˆ(0,๐œ‰),๐‘ก>0,๐‘ˆ(๐‘ฅ,0)=๐‘ˆ๐‘œ๎€ท(๐‘ฅ),๐‘ฅโˆˆ0,๐œ‰๐‘œ๎€ธ,๐‘ˆ๐‘ก(๐‘ฅ,0)=๐‘ˆ1๎€ท(๐‘ฅ),๐‘ฅโˆˆ0,๐œ‰๐‘œ๎€ธ,๐‘ˆ๐‘ˆ(0,๐‘ก)=0,๐‘ก>0,(๐œ‰,๐‘ก)=exp(๐œ”๐‘ก)๐‘‰๐‘œ๐‘ˆ,๐‘ก>0,๐‘ฅฬ‡(๐œ‰,๐‘ก)+๐œ‰๐‘ˆ๐‘กฬ‡(๐œ‰,๐‘ก)โˆ’๐œ‰๐œ”๐‘ˆ(๐œ‰,๐‘ก)=exp(๐œ”๐‘ก)๐›ฝ2๐™ฑ๐š—,๐‘ก>0,๐œ‰(0)=๐œ‰๐‘œ,๐œ‰๐‘œโˆˆ(0,1),(3.3)

where๐œ‰(๐‘ก)=1โˆ’๐‘ (๐‘ก),๐œ‰๐‘œ=1โˆ’๐‘ ๐‘œ,๐‘ˆ๐‘œ(๐‘ฅ)=๐‘ฃ๐‘œ(1โˆ’๐‘ฅ),๐‘ˆ1(๐‘ฅ)=๐œ”๐‘ˆ๐‘œ(๐‘ฅ).(3.4) Notice that, by means of (3.2), the evolution equation for the new variable ๐‘ˆ(๐‘ฅ,๐‘ก) has become a nonhomogeneous Klein-Gordon equation [10].

The domain of problem (3.3) is depicted in Figure 2. We begin by considering the domain ๐ท๐ผ (see Figure 3). Here the solution has the representation formula (see [11])1๐‘ˆ(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๐‘œ(๐‘ฅโˆ’๐‘ก)+๐‘ˆ๐‘œ(๎€ป+1๐‘ฅ+๐‘ก)2๎€œ๐‘ฅ+๐‘ก๐‘ฅโˆ’๐‘ก๎€บ๐‘…(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ1(๐œ)โˆ’๐‘…๐œƒ(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ(๎€ป+๐›ฝ๐œ)๐‘‘๐œ22๎€œ๐‘ก0๎€œexp(๐œ”๐œƒ)๐‘‘๐œƒ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ,(3.5) where ๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ) is the Riemann's function that solves the problem (see again Figure 3)๐‘…๐œ๐œโˆ’๐‘…๐œƒ๐œƒ+๐œ”2๐‘…[],[],๐‘…=0(๐œ,๐œƒ)โˆˆฮฉ(๐‘ฅ,๐‘ก),(๐‘ฅ,๐‘ก;๐‘ฅ+๐‘กโˆ’๐œƒ,๐œƒ)=1๐œƒโˆˆ0,๐‘ก๐‘…(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก+๐œƒ,๐œƒ)=1๐œƒโˆˆ0,๐‘กฮฉ(๐‘ฅ,๐‘ก)={(๐‘ฅ,๐‘ก)โˆถ๐‘ฅโˆ’๐‘ก+๐œƒโฉฝ๐œโฉฝ๐‘ฅ+๐‘กโˆ’๐œƒ,0โฉฝ๐œƒโฉฝ๐‘ก}.(3.6)

To determine the solution of problem (3.6) we set๎”๐‘ง=(๐‘กโˆ’๐œƒ)2โˆ’(๐œโˆ’๐‘ฅ)2,(3.7) where๐‘ง2๐œƒโˆ’๐‘ง2๐œ๐‘ง=1,๐œƒ๐œƒโˆ’๐‘ง๐œ๐œ=1๐‘ง.(3.8)

By means of (3.7) problem (3.6) becomes๐‘…๎…ž๎…ž๐‘…(๐‘ง)+๎…ž(๐‘ง)๐‘งโˆ’๐œ”2๐‘…๐‘…(๐‘ง)=0(0)=1,(3.9) where (3.9)(1) is the modified Bessel equation of zero order. The solution of (3.9) is given by๐‘…(๐‘ง)=๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ)=๐ผ๐‘œ๎‚ต๐œ”๎”(๐‘กโˆ’๐œƒ)2โˆ’(๐œโˆ’๐‘ฅ)2๎‚ถ,(3.10) where ๐ผ๐‘œ is the modified Bessel function of zero order. It is easy to prove that the function defined by (3.10) satisfies problem (3.6). Moreover, since [12]๐ผ๎…ž๐‘œ(๐‘ฅ)๐‘ฅ=12๎€บ๐ผ๐‘œ(๐‘ฅ)โˆ’๐ผ2๎€ป(๐‘ฅ),(3.11)

one can prove that๐‘…๐œƒ๐œ”(๐‘ฅ,๐‘ก;๐œ,๐œƒ)=2(๐œƒโˆ’๐‘ก)2๎‚ธ๐ผ๐‘œ๎‚ต๐œ”๎”(๐‘กโˆ’๐œƒ)2โˆ’(๐œโˆ’๐‘ฅ)2๎‚ถโˆ’๐ผ2๎‚ต๐œ”๎”(๐‘กโˆ’๐œƒ)2โˆ’(๐œโˆ’๐‘ฅ)2,๎‚ถ๎‚น(3.12) where ๐ผ2 is the modified Bessel function of second order. Recalling that ๐‘ˆ(๐‘ฅ,0)=๐‘ˆ๐‘œ(๐‘ฅ) and, by (3.3)(3), (3.4), that๐‘ˆ๐‘ก(๐‘ฅ,0)=๐œ”๐‘ˆ๐‘œ(๐‘ฅ),(3.13)we see that๎€บ๐‘…(๐‘ฅ,๐‘ก;๐œ,0)๐œ”โˆ’๐‘…๐œƒ๎€ป๐‘ˆ(๐‘ฅ,๐‘ก;๐œ,0)๐‘œ=๎‚ธ๐ผ(๐œ)๐‘œ๎‚ต๐œ”๎”๐‘ก2โˆ’(๐œโˆ’๐‘ฅ)2๐œ”๎‚ถ๎‚ต๐œ”+2๐‘ก2๎‚ถโˆ’๐œ”2๐‘ก2๐ผ2๎‚ต๐œ”๎”๐‘ก2โˆ’(๐œโˆ’๐‘ฅ)2๐‘ˆ๎‚ถ๎‚น๐‘œ(๐œ),(3.14)and representation formula (3.5) can be rewritten as 1๐‘ˆ(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๐‘œ(๐‘ฅโˆ’๐‘ก)+๐‘ˆ๐‘œ๎€ป+1(๐‘ฅ+๐‘ก)2๎€œ๐‘ฅ+๐‘ก๐‘ฅโˆ’๐‘ก๎‚ธ๐ผ๐‘œ๎‚ต๐œ”๎”๐‘ก2โˆ’(๐œโˆ’๐‘ฅ)2๐œ”๎‚ถ๎‚ต๐œ”+2๐‘ก2๎‚ถโˆ’๐œ”2๐‘ก2๐ผ2๎‚ต๐œ”๎”๐‘ก2โˆ’(๐œโˆ’๐‘ฅ)2๐‘ˆ๎‚ถ๎‚น๐‘œ+๐›ฝ(๐œ)๐‘‘๐œ22๎€œ๐‘ก0๎‚€exp๐‘˜๐œƒ2๎‚๎€œ๐‘‘๐œƒโ‹…๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐ผ๐‘œ๎‚ต๐œ”๎”(๐‘กโˆ’๐œƒ)2โˆ’(๐œโˆ’๐‘ฅ)2๎‚ถ๐‘‘๐œ.(3.15)

Let us now write a representation formula for ๐‘ˆ(๐‘ฅ,๐‘ก) in the domain ๐ท๐ผ๐ผ (see Figure 4). We once again make use of (3.6), where now ๐‘ˆ๐‘œ has to be extended to the domain [โˆ’๐œ‰๐‘œ,0]. Following [2], we extend ๐‘ˆ๐‘œ imposing condition (3.3)(4), that is, ๐‘ˆ(0,๐œƒ)=0. From the representation formula we get10=2๎€บ๐‘ˆ๐‘œ๎€ทโˆ’๐‘กโˆ—๎€ธ+๐‘ˆ๐‘œ๎€ท๐‘กโˆ—+1๎€ธ๎€ป2๎€œ๐‘กโˆ—โˆ’๐‘กโˆ—๎€บ๐‘…๎€ท0,๐‘กโˆ—๎€ธ;๐œ,0๐œ”โˆ’๐‘…๐œƒ๎€ท0,๐‘กโˆ—๐‘ˆ;๐œ,0๎€ธ๎€ป๐‘œ+๐›ฝ(๐œ)๐‘‘๐œ22๎€œ๐‘กโˆ—0๎€œexp(๐œ”๐œƒ)๐‘‘๐œƒ๐‘กโˆ—โˆ’๐œƒโˆ’๐‘กโˆ—+๐œƒ๐‘…๎€ท0,๐‘กโˆ—๎€ธ;๐œ,๐œƒ๐‘‘๐œ,(3.16) where ๐‘กโˆ—=๐‘กโˆ’๐‘ฅ(3.17)is the coordinate of the intersection of the characteristic ๐œ=๐‘ฅโˆ’๐‘ก+๐œƒ with ๐œ=0. Relation (3.16) can be rewritten as10=2๎€บ๐‘ˆ๐‘œ(๐‘ฅโˆ’๐‘ก)+๐‘ˆ๐‘œ(๎€ป+1๐‘กโˆ’๐‘ฅ)2๎€œ๐‘กโˆ’๐‘ฅ๐‘ฅโˆ’๐‘ก๎€บ๐‘…(0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐œ”โˆ’๐‘…๐œƒ(๎€ป๐‘ˆ0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐‘œ(+๐›ฝ๐œ)๐‘‘๐œ22๎€œ0๐‘กโˆ’๐‘ฅ๎€œexp(๐œ”๐œƒ)๐‘‘๐œƒ๐‘กโˆ’๐‘ฅโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…(0,๐‘กโˆ’๐‘ฅ;๐œ,๐œƒ)๐‘‘๐œ.(3.18) From (3.18), the extended function ๐‘ˆ๐‘ x๐‘œ(๐‘ฅ), defined in [โˆ’๐œ‰๐‘œ,0], fulfills the following Volterra integral equation of second type: ๐‘ˆ๐‘œ๐‘ ๐‘ฅ(๎€œ๐‘ฅโˆ’๐‘ก)โˆ’0๐‘ฅโˆ’๐‘ก๎€บ๐‘…(0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐œ”โˆ’๐‘…๐œƒ(๎€ป๐‘ˆ0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐‘œ๐‘ ๐‘ฅ(๐œ)๐‘‘๐œ=โˆ’๐‘ˆ๐‘œ(๎€œ๐‘กโˆ’๐‘ฅ)+0๐‘กโˆ’๐‘ฅ๎€บ๐‘…(0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐œ”โˆ’๐‘…๐œƒ(๎€ป๐‘ˆ0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐‘œ(โˆ’๐›ฝ๐œ)๐‘‘๐œ22๎€œ0๐‘กโˆ’๐‘ฅ๎€œexp(๐œ”๐œƒ)๐‘‘๐œƒ๐‘กโˆ’๐‘ฅโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…(0,๐‘กโˆ’๐‘ฅ;๐œ,๐œƒ)๐‘‘๐œ.(3.19) Equation (3.19) can be put in the more compact form๐‘ˆ๐‘œ๐‘ ๐‘ฅ๎€œ(๐œ’)โˆ’๐œ’0๐พ๐‘ ๐‘ฅ(๐œ’,๐œ)๐‘ˆ๐‘œ๐‘ ๐‘ฅ(๐œ)๐‘‘๐œ=๐น๐‘ ๐‘ฅ(๐œ’),(3.20)where ๐œ’=๐‘ฅโˆ’๐‘กโˆˆ[โˆ’๐œ‰๐‘œ,0] and ๐พ๐‘ ๐‘ฅ(=๎‚ธ๐ผ๐œ’,๐œ)๐‘œ๎‚€๐œ”โˆš๐œ’2โˆ’๐œ2๎‚๎‚ต๐œ”๐œ”โˆ’2๐œ’2๎‚ถ+๐œ”2๐œ’2๐ผ2๎‚€๐œ”โˆš๐œ’2โˆ’๐œ2๎‚๎‚น,๐น๐‘ ๐‘ฅ(๐œ’)=โˆ’๐‘ˆ๐‘œ๎€œ(โˆ’๐œ’)+0โˆ’๐œ’๎€บ๐‘…(0,โˆ’๐œ’;๐œ,0)๐œ”โˆ’๐‘…๐œƒ๎€ป๐‘ˆ(0,โˆ’๐œ’;๐œ,0)๐‘œโˆ’๐›ฝ(๐œ)๐‘‘๐œ22๎€œ0โˆ’๐‘ฅ๎€œexp(๐œ”๐œƒ)๐‘‘๐œƒโˆ’๐‘ฅโˆ’๐œƒ๐‘ฅ+๐œƒ๐‘…(0,โˆ’๐‘ฅ;๐œ,๐œƒ)๐‘‘๐œ.(3.21)

Due to the regularity of the kernel ๐พ๐‘ ๐‘ฅ(๐œ’,๐œ) the function ๐‘ˆ๐‘œ๐‘ ๐‘ฅ(๐œ’) (which can be determined using the iterated kernels method, [13]) is a smooth function. Thus we extend ๐‘ˆ๐‘œ(๐‘ฅ) as ๐‘ˆ๐‘œ(๎ƒฏ๐‘ˆ๐‘ฅ)=๐‘œ๐‘ ๐‘ฅ๎€บ(๐‘ฅ),๐‘ฅโˆˆโˆ’๐œ‰๐‘œ๎€ป,๐‘ˆ,0๐‘œ๎€บ(๐‘ฅ),๐‘ฅโˆˆ0,๐œ‰๐‘œ๎€ป,(3.22)and the solution ๐‘ˆ(๐‘ฅ,๐‘ก) in the domain ๐ท๐ผ๐ผ is given by 1๐‘ˆ(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๐‘œ(๐‘ฅ+๐‘ก)โˆ’๐‘ˆ๐‘œ(๎€ป+1๐‘กโˆ’๐‘ฅ)2๎€œ๐‘ฅโˆ’๐‘ก๐‘กโˆ’๐‘ฅ๎€บ๐‘…(0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐œ”โˆ’๐‘…๐œƒ(๎€ป๐‘ˆ0,๐‘กโˆ’๐‘ฅ;๐œ,0)๐‘œ(+1๐œ)๐‘‘๐œ2๎€œ๐‘ฅ+๐‘ก๐‘ฅโˆ’๐‘ก๎€บ๐‘…(๐‘ฅ,๐‘ก;๐œ,0)๐œ”โˆ’๐‘…๐œƒ๎€ป๐‘ˆ(๐‘ก,๐‘ฅ;๐œ,0)๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+22๎€œ0๐‘กโˆ’๐‘ฅ๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘กโˆ’๐‘ฅโˆ’๐œƒ+๐›ฝ๐‘…(0,๐‘กโˆ’๐‘ฅ;๐œ,๐œƒ)๐‘‘๐œ22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ.(3.23)

Remark 3.1. We notice that, considering the representation formulae (3.5) and (3.23), lim๐‘ฅโ†’๐‘ก+๐‘ˆ(๐‘ฅ,๐‘ก)=lim๐‘ฅโ†’๐‘กโˆ’๐‘ˆ(๐‘ฅ,๐‘ก).(3.24) Moreover, taking the first derivatives (with respect to time ๐‘ก and space ๐‘ฅ) of ๐‘ˆ(๐‘ฅ,๐‘ก) for the domains ๐ท๐ผ and ๐ท๐ผ๐ผ it is easy to prove that, assuming the compatibility condition ๐‘ˆ๐‘œ(0)=0, lim๐‘ฅโ†’๐‘ก+๐‘ˆ๐‘ฅ(๐‘ฅ,๐‘ก)=lim๐‘ฅโ†’๐‘กโˆ’๐‘ˆ๐‘ฅ๎‚ธ๐œ‰(๐‘ฅ,๐‘ก),๐‘กโˆˆ0,๐‘œ2๎‚น,lim๐‘ฅโ†’๐‘ก+๐‘ˆ๐‘ก(๐‘ฅ,๐‘ก)=lim๐‘ฅโ†’๐‘กโˆ’๐‘ˆ๐‘ก๎‚ธ๐œ‰(๐‘ฅ,๐‘ก),๐‘กโˆˆ0,๐‘œ2๎‚น,(3.25) where the derivatives in limits on the l.h.s. of (3.25) are evaluated using (3.5), while the ones on the r.h.s. using (3.23). This implies that the solution is ๐ถ1 across the characteristic ๐‘ฅ=๐‘ก, that is, the line that separates the domains ๐ท๐ผ and ๐ท๐ผ๐ผ.

We now write the representation formula for ๐‘ˆ(๐‘ฅ,๐‘ก) in the domain ๐ท๐ผ๐ผ๐ผ. We proceed as in [2] assuming that the velocity of the free boundary ๐‘ฅ=๐œ‰(๐‘ก) is less than the velocity of the characteristics (i.e., |ฬ‡๐œ‰|<1) and extending ๐‘ˆ๐‘œ to the domain [๐œ‰๐‘œ,๐œ‰(๐œ‰๐‘œ)+๐œ‰๐‘œ] (see Figure 2) in a way such that ๐‘ˆ(๐œ‰,๐‘ก)=exp(๐œ”๐‘ก)๐‘‰๐‘œ (i.e., imposing the free boundary condition (3.3)(5)).

Given a point (๐‘ฅ,๐‘ก) in the domain ๐ท๐ผ๐ผ๐ผ we define the point (๐œ‰โˆ—,๐‘กโˆ—) as the intersection of the characteristic (with negative slope) passing from (๐‘ฅ,๐‘ก) and the free boundary ๐‘ฅ=๐œ‰(๐‘ก) (see Figure 5). It is easy to check that ๐œ‰โˆ—+๐‘กโˆ—=๐‘ฅ+๐‘ก,โŸน๐‘กโˆ—=๐‘กโˆ—(๐‘ฅ,๐‘ก),(3.26)๐œ•๐‘กโˆ—=1๐œ•๐‘กฬ‡๐œ‰(๐‘กโˆ—,)+1๐œ•๐‘กโˆ—=1๐œ•๐‘ฅฬ‡๐œ‰(๐‘กโˆ—.)+1(3.27)We consider once again the representation formula (3.5) and impose condition (3.3)(5), getting2๐‘’๐œ”๐‘กโˆ—๐‘‰๐‘œ=๐‘ˆ๐‘œ๎€ท๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธ+๐‘ˆ๐‘œ๎€ท๐œ‰โˆ—+๐‘กโˆ—๎€ธ+๎€œ๐œ‰โˆ—+๐‘กโˆ—๐œ‰โˆ—โˆ’๐‘กโˆ—๎€บ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ;๐œ,0๐œ”โˆ’๐‘…๐œƒ๎€ท๐œ‰โˆ—,๐‘กโˆ—๐‘ˆ;๐œ,0๎€ธ๎€ป๐‘œ(๐œ)๐‘‘๐œ+๐›ฝ2๎€œ๐‘กโˆ—0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰โˆ—+๐‘กโˆ—๐œ‰โˆ’๐œƒโˆ—โˆ’๐‘กโˆ—+๐œƒ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ;๐œ,๐œƒ๐‘‘๐œ.(3.28)From (3.28) we see that the extension ๐‘ˆ๐‘œ๐‘‘๐‘ฅ to the domain [๐œ‰๐‘œ,๐œ‰(๐œ‰๐‘œ)+๐œ‰๐‘œ] is the solution of the following Volterra integral equation of second kind: ๐‘ˆ๐‘œ๐‘‘๐‘ฅ๎€ท๐œ‰โˆ—+๐‘กโˆ—๎€ธ+๎€œ๐œ‰โˆ—+๐‘กโˆ—๐œ‰๐‘œ๎€บ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ;๐œ,0๐œ”โˆ’๐‘…๐œƒ๎€ท๐œ‰โˆ—,๐‘กโˆ—๐‘ˆ;๐œ,0๎€ธ๎€ป๐‘œ๐‘‘๐‘ฅ(๐œ)๐‘‘๐œ=2๐‘’๐œ”๐‘กโˆ—๐‘‰๐‘œโˆ’๐‘ˆ๐‘œ๎€ท๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธโˆ’๎€œ๐œ‰๐‘œ๐œ‰โˆ—โˆ’๐‘กโˆ—๎€บ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ;๐œ,0๐œ”โˆ’๐‘…๐œƒ๎€ท๐œ‰โˆ—,๐‘กโˆ—๐‘ˆ;๐œ,0๎€ธ๎€ป๐‘œ(๐œ)๐‘‘๐œโˆ’๐›ฝ2๎€œ๐‘กโˆ—0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰โˆ—+๐‘กโˆ—๐œ‰โˆ’๐œƒโˆ—โˆ’๐‘กโˆ—+๐œƒ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ;๐œ,๐œƒ๐‘‘๐œ.(3.29)Recalling (3.26) and proceeding as for the domain ๐ท๐ผ๐ผ, the above can be rewritten as๐‘ˆ๐‘œ๐‘‘๐‘ฅ๎€œ(๐œ’)+๐œ’๐œ‰๐‘œ๐พ๐‘‘๐‘ฅ(๐œ’,๐œ)๐‘ˆ๐‘œ๐‘‘๐‘ฅ(๐œ)๐‘‘๐œ=๐น๐‘‘๐‘ฅ(๐œ’),(3.30)where ๐œ’=๐‘ฅ+๐‘ก and๐พ๐‘‘๐‘ฅ๎‚ธ๐ผ(๐œ’,๐œ)=๐‘œ๎‚ต๐œ”๎”(๐‘กโˆ—(๐œ’))2โˆ’๎€ท๐œโˆ’๐œ‰โˆ—๎€ธ(๐œ’)2๐œ”๎‚ถ๎‚ต๐œ”+2๐‘กโˆ—(๐œ’)2๎‚ถโˆ’๐œ”2๐‘กโˆ—(๐œ’)2๐ผ2๎‚ต๐œ”๎”(๐‘กโˆ—(๐œ’))2โˆ’๎€ท๐œ‰โˆ—๎€ธ(๐œ’)โˆ’๐œ2,๐น๎‚ถ๎‚น๐‘‘๐‘ฅ๎‚ธ(๐œ’)=2๐‘’๐œ”๐‘ก๐‘‰๐‘œโˆ’๐‘ˆ๐‘œ๎€œ(๐‘ฅโˆ’๐‘ก)โˆ’๐œ‰๐‘œ๐‘ฅโˆ’๐‘ก๎€บ๐‘…(๐‘ฅ,๐‘ก;๐œ,0)๐œ”โˆ’๐‘…๐œƒ๎€ป๐‘ˆ(๐‘ฅ,๐‘ก;๐œ,0)๐‘œ(๐œ)๐‘‘๐œโˆ’๐›ฝ2๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๎‚น||||๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ(๐‘ฅ=๐œ‰โˆ—(๐œ’),๐‘ก=๐‘กโˆ—(๐œ’)).(3.31) Once again the regularity of the kernel ๐พ๐‘‘๐‘ฅ(๐œ’,๐œ) ensures the regularity of the solution ๐‘ˆ๐‘œ๐‘‘๐‘ฅ(๐œ’). The function ๐‘ˆ๐‘œ(๐‘ฅ) can thus be defined in the interval [โˆ’๐œ‰๐‘œ,๐œ‰(๐œ‰๐‘œ)+๐œ‰๐‘œ] as๐‘ˆ๐‘œโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ˆ(๐‘ฅ)=๐‘œ๐‘ ๐‘ฅ๎€บ(๐‘ฅ),๐‘ฅโˆˆโˆ’๐œ‰๐‘œ๎€ป,๐‘ˆ,0๐‘œ๎€บ(๐‘ฅ),๐‘ฅโˆˆ0,๐œ‰๐‘œ๎€ป,๐‘ˆ๐‘œ๐‘‘๐‘ฅ(๎€บ๐œ‰๐‘ฅ),๐‘ฅโˆˆ๐‘œ๎€ท๐œ‰,๐œ‰๐‘œ๎€ธ+๐œ‰๐‘œ๎€ป.(3.32) The solution in the domain ๐ท๐ผ๐ผ๐ผ is thus given by๐‘ˆ(๐‘ฅ,๐‘ก)=๐‘’๐œ”๐‘กโˆ—๐‘‰๐‘œ+12๎€บ๐‘ˆ๐‘œ(๐‘ฅโˆ’๐‘ก)โˆ’๐‘ˆ๐‘œ๎€ท๐œ‰โˆ—โˆ’๐‘กโˆ—+1๎€ธ๎€ป2๎€œ๐‘ฅ+๐‘ก๐‘ฅโˆ’๐‘ก๐‘ƒ(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ(โˆ’1๐œ)๐‘‘๐œ2๎€œ๐œ‰๐‘ฅ+๐‘กโˆ—โˆ’๐‘กโˆ—๐‘ƒ๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ๐‘ˆ;๐œ,0๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒโˆ’๐›ฝ๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ22๎€œ๐‘กโˆ—0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰๐‘ฅ+๐‘กโˆ’๐œƒโˆ—โˆ’๐‘กโˆ—+๐œƒ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ;๐œ,๐œƒ๐‘‘๐œ,(3.33) where for simplicity of notation we have introduced๐‘ƒ(๐‘ฅ,๐‘ก;๐œ,๐œƒ)=๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐œ”โˆ’๐‘…๐œƒ(๐‘ฅ,๐‘ก;๐œ,๐œƒ),(3.34)and where ๐‘ˆ๐‘œ(๐‘ฅ) is given by (3.32). Therefore for any fixed ๐ถ1 function ๐œ‰(๐‘ก) with |ฬ‡๐œ‰|<1 we have that the solution to problem (3.3)(1โ€“5) is given by (3.5), (3.23), (3.33) with ๐‘ˆ๐‘œ defined by (3.32). At this point we make use of (3.3)(6) to determine the evolution equation of the free boundary ๐‘ฅ=๐œ‰(๐‘ก). We begin writing the derivatives ๐‘ˆ๐‘ก(๐‘ฅ,๐‘ก) and ๐‘ˆ๐‘ฅ(๐‘ฅ,๐‘ก). To this aim we exploit formula (3.33) since ๐‘ˆ๐‘ก and ๐‘ˆ๐‘ฅ have to be evaluated on ๐‘ฅ=๐œ‰(๐‘ก), which belongs to domain ๐ท๐ผ๐ผ๐ผ. Differentiating (3.33) with respect to ๐‘ฅ we get๐‘ˆ๐‘ฅ1(๐‘ฅ,๐‘ก)=2๎ƒฌ๐‘ˆ๎…ž๐‘œ(๐‘ฅโˆ’๐‘ก)โˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘กโˆ—๎€ธโˆ’1ฬ‡๐œ‰(๐‘กโˆ—๎ƒญ+๎ƒฉ)+1๐œ”๐‘‰๐‘œ๐‘’๐œ”๐‘กโˆ—ฬ‡๐œ‰โˆ—๎ƒช+1+12๎€บ๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅ+๐‘ก,0)๐‘ˆ๐‘œ(๐‘ฅ+๐‘ก)โˆ’๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก,0)๐‘ˆ๐‘œ๎€ปโˆ’1(๐‘ฅโˆ’๐‘ก)2๎ƒฌ๐‘ƒ๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ๐‘ˆ;๐‘ฅ+๐‘ก,0๐‘œ๎€ท๐œ‰(๐‘ฅ+๐‘ก)โˆ’๐‘ƒโˆ—,๐‘กโˆ—;๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธ๐‘ˆ,0๐‘œ๎€ท๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘กโˆ—๎€ธโˆ’1ฬ‡๐œ‰(๐‘กโˆ—)๎ƒญโˆ’1+12๎€œ๐œ‰๐‘ฅ+๐‘กโˆ—โˆ’๐‘กโˆ—๎€บ๐‘ƒ๐‘ฅ๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘ก;๐œ,0โˆ—๎€ธ+๐‘ƒ๐‘ก๎€ท๐œ‰โˆ—,๐‘กโˆ—๐‘ˆ;๐œ,0๎€ธ๎€ป๐‘œ(๐œ)ฬ‡๐œ‰(๐‘กโˆ—+1)+1๐‘‘๐œ2๎€œ๐‘ฅ+๐‘ก๐‘ฅโˆ’๐‘ก๐‘ƒ๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…๐‘ฅโˆ’๐›ฝ(๐‘ฅ,๐‘ก;๐œ๐œƒ)๐‘‘๐œ22๎€œ๐‘กโˆ—0๐‘’๐œ”๐œƒ๎ƒฌ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ๎€ท๐œ‰;๐‘ฅ+๐‘กโˆ’๐œƒ,๐œƒโˆ’๐‘…โˆ—,๐‘กโˆ—;๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘ก+๐œƒ,๐œƒโˆ—๎€ธโˆ’1ฬ‡๐œ‰(๐‘กโˆ—๎ƒญโˆ’๐›ฝ)+1๐‘‘๐œƒ22๎€œ๐‘กโˆ—0๐‘’๐œ”๐œƒ๎€œ๐œ‰๐‘ฅ+๐‘กโˆ’๐œƒโˆ—โˆ’๐‘กโˆ—+๐œƒ๎€บ๐‘…๐‘ฅ๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘ก;๐œ,๐œƒโˆ—๎€ธโˆ’๐‘…๐‘ก๎€ท๐œ‰โˆ—,๐‘กโˆ—;๐œ,๐œƒ๎€ธ๎€ป๐‘‘๐œฬ‡๐œ‰(๐‘กโˆ—,)+1(3.35)while, differentiating (3.33) with respect to ๐‘ก, we obtain ๐‘ˆ๐‘ก1(๐‘ฅ,๐‘ก)=2๎ƒฌโˆ’๐‘ˆ๎…ž๐‘œ(๐‘ฅโˆ’๐‘ก)โˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘กโˆ—๎€ธโˆ’1ฬ‡๐œ‰(๐‘กโˆ—๎ƒญ+๎ƒฉ)+1๐œ”๐‘‰๐‘œ๐‘’๐œ”๐‘กโˆ—ฬ‡๐œ‰โˆ—๎ƒช+1+12๎€บ๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅ+๐‘ก,0)๐‘ˆ๐‘œ(๐‘ฅ+๐‘ก)+๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก,0)๐‘ˆ๐‘œ๎€ปโˆ’1(๐‘ฅโˆ’๐‘ก)2๎ƒฌ๐‘ƒ๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ๐‘ˆ;๐‘ฅ+๐‘ก,0๐‘œ๎€ท๐œ‰(๐‘ฅ+๐‘ก)โˆ’๐‘ƒโˆ—,๐‘กโˆ—;๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธ๐‘ˆ,0๐‘œ๎€ท๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘กโˆ—๎€ธโˆ’1ฬ‡๐œ‰(๐‘กโˆ—)๎ƒญโˆ’1+12๎€œ๐œ‰๐‘ฅ+๐‘กโˆ—โˆ’๐‘กโˆ—๎€บ๐‘ƒ๐‘ฅ๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘ก;๐œ,0โˆ—๎€ธ+๐‘ƒ๐‘ก๎€ท๐œ‰โˆ—,๐‘กโˆ—๐‘ˆ;๐œ,0๎€ธ๎€ป๐‘œ(๐œ)ฬ‡๐œ‰(๐‘กโˆ—)+1๐‘‘๐œ+๐›ฝ2๎€œ๐‘ก0๐‘’๐œ”๐œƒ+1๐‘‘๐œƒ2๎€œ๐‘ฅ+๐‘ก๐‘ฅโˆ’๐‘ก๐‘ƒ๐‘ก(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…๐‘กโˆ’๐›ฝ(๐‘ฅ,๐‘ก;๐œ๐œƒ)๐‘‘๐œ22๎€œ๐‘กโˆ—0๐‘’๐œ”๐œƒ๎ƒฌ๐‘…๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธ๎€ท๐œ‰;๐‘ฅ+๐‘กโˆ’๐œƒ,๐œƒโˆ’๐‘…โˆ—,๐‘กโˆ—;๐œ‰โˆ—โˆ’๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘ก+๐œƒ,๐œƒโˆ—๎€ธโˆ’1ฬ‡๐œ‰(๐‘กโˆ—๎ƒญโˆ’๐›ฝ)+1๐‘‘๐œƒ22๎€œ๐‘กโˆ—0๐‘’๐œ”๐œƒ๎€œ๐œ‰๐‘ฅ+๐‘กโˆ’๐œƒโˆ—โˆ’๐‘กโˆ—+๐œƒ๎€บ๐‘…๐‘ฅ๎€ท๐œ‰โˆ—,๐‘กโˆ—๎€ธฬ‡๐œ‰๎€ท๐‘ก;๐œ,๐œƒโˆ—๎€ธโˆ’๐‘…๐‘ก๎€ท๐œ‰โˆ—,๐‘กโˆ—;๐œ,๐œƒ๎€ธ๎€ป๐‘‘๐œฬ‡๐œ‰(๐‘กโˆ—.)+1(3.36) Notice that ๐œ”๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅ+๐‘ก,0)=๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก,0)=๐œ”+2๐‘ก2,๐‘…(๐‘ฅ,๐‘ก;๐‘ฅ+๐‘กโˆ’๐œƒ,๐œƒ)โˆ’๐‘…(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก+๐œƒ,๐œƒ)=0.(3.37) Now we evaluate (3.35) and (3.36) on the free boundary ๐‘ฅ=๐œ‰(๐‘ก), that is,๐‘ˆ๐‘ฅ๐‘ˆ(๐œ‰,๐‘ก)=๎…ž๐‘œ(๐œ‰โˆ’๐‘ก)ฬ‡โˆ’๎‚ต๐œ”๐œ‰+1๐œ”+2๐‘ก2๎‚ถ๐‘ˆ๐‘œ1(๐œ‰โˆ’๐‘ก)ฬ‡+๎‚ต๐œ‰+1๐œ”๐‘‰๐‘œ๐‘’๐œ”๐‘กฬ‡๎‚ถ+1๐œ‰+12๎€œ๐œ‰+๐‘ก๐œ‰โˆ’๐‘ก๎€บ๐‘ƒ๐‘ฅ(๐œ‰,๐‘ก;๐œ,0)โˆ’๐‘ƒ๐‘ก๎€ป๐‘ˆ(๐œ‰,๐‘ก;๐œ,0)๐‘œ(๐œ)๐‘‘๐œฬ‡๐œ‰+1โˆ’๐›ฝ2๎€œ๐‘ก0๐‘’๐œ”๐œƒฬ‡+๐›ฝ๐œ‰+1๐‘‘๐œƒ22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐œ‰+๐‘กโˆ’๐œƒ๐œ‰โˆ’๐‘ก+๐œƒ๎€บ๐‘…๐‘ฅ(๐œ‰,๐‘ก;๐œ,๐œƒ)โˆ’๐‘…๐‘ก๎€ป(๐œ‰,๐‘ก;๐œ,๐œƒ)๐‘‘๐œฬ‡,๐‘ˆ๐œ‰+1๐‘ก๐‘ˆ(๐œ‰,๐‘ก)=โˆ’๎…ž๐‘œ(ฬ‡๐œ‰๐œ‰โˆ’๐‘ก)ฬ‡+๎‚ต๐œ”๐œ‰+1๐œ”+2๐‘ก2๎‚ถ๐‘ˆ๐‘œฬ‡๐œ‰(๐œ‰โˆ’๐‘ก)ฬ‡+๎‚ต๐œ‰+1๐œ”๐‘‰๐‘œ๐‘’๐œ”๐‘กฬ‡๎‚ถ+1๐œ‰+12๎€œ๐œ‰+๐‘ก๐œ‰โˆ’๐‘ก๎€บ๐‘ƒ๐‘ก(๐œ‰,๐‘ก;๐œ,0)โˆ’๐‘ƒ๐‘ฅ(๎€ปฬ‡๐œ‰,๐‘ก;๐œ,0)๐œ‰๐‘ˆ๐‘œ(๐œ)๐‘‘๐œฬ‡๐œ‰+1+๐›ฝ2๎€œ๐‘ก0๐‘’๐œ”๐œƒฬ‡๐œ‰ฬ‡+๐›ฝ๐œ‰+1๐‘‘๐œƒ22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐œ‰+๐‘กโˆ’๐œƒ๐œ‰โˆ’๐‘ก+๐œƒ๎€บ๐‘…๐‘ก(๐œ‰,๐‘ก;๐œ,๐œƒ)โˆ’๐‘…๐‘ฅ(๎€ปฬ‡๐œ‰,๐‘ก;๐œ,๐œƒ)๐œ‰๐‘‘๐œฬ‡.๐œ‰+1(3.38) At this point we insert (3.38), (3.3)(5) in (3.3)(6), obtaining ๎€ทฬ‡๎€ธ๐œ”๐œ‰โˆ’1๎‚ธ๎‚ต๐œ”+2๐‘ก2๎‚ถ๐‘ˆ๐‘œ(๐œ‰โˆ’๐‘ก)โˆ’๐‘ˆ๎…ž๐‘œ๐›ฝ(๐œ‰โˆ’๐‘ก)+2๐œ”๎€ท๐‘’๐œ”๐‘ก๎€ธ+1โˆ’12๎€œ๐œ‰+๐‘ก๐œ‰โˆ’๐‘ก๎€บ๐‘ƒ๐‘ก(๐œ‰,๐‘ก;๐œ,0)โˆ’๐‘ƒ๐‘ฅ๎€ป๐‘ˆ(๐œ‰,๐‘ก;๐œ,0)๐‘œ+๐›ฝ(๐œ)๐‘‘๐œ22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰+๐‘กโˆ’๐œƒ๐œ‰โˆ’๐‘ก+๐œƒ๎€บ๐‘…๐‘ก(๐œ‰,๐‘ก;๐œ,๐œƒ)โˆ’๐‘…๐‘ฅ๎€ป๎‚น(๐œ‰,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ=๐‘’๐œ”๐‘ก๐›ฝ2๐™ฑ๐š—,(3.39) which is a nonlinear integrodifferential equation of the first order and where ๐‘ˆ๐‘œ is defined by (3.32). Equation (3.39) is the free boundary equation which, as we mentioned in the introduction, does no longer depend on the velocity field ๐‘ˆ(๐‘ฅ,๐‘ก).

Next we remark that (3.39) can be further simplified. Indeed, recalling (3.10) and (3.12), ๐‘…๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,๐œƒ)=โˆ’๐‘…๐œ(๐‘ฅ,๐‘ก;๐œ,๐œƒ),๐‘…๐œƒ๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,๐œƒ)=โˆ’๐‘…๐œƒ๐œ(๐‘ฅ,๐‘ก;๐œ,๐œƒ),(3.40)so that, on (๐œ‰(๐‘ก),๐‘ก;๐œ,๐œƒ), we have ๎€œ๐œ‰+๐‘กโˆ’๐œƒ๐œ‰โˆ’๐‘ก+๐œƒ๐‘…๐‘ฅ๎€œ๐‘‘๐œ=โˆ’๐œ‰+๐‘กโˆ’๐œƒ๐œ‰โˆ’๐‘ก+๐œƒ๐‘…๐œ๐‘‘๐œ=๐‘…(๐œ‰,๐‘ก;๐œ‰+๐‘กโˆ’๐œƒ,๐œƒ)โˆ’๐‘…(๐œ‰,๐‘ก;๐œ‰โˆ’๐‘ก+๐œƒ,๐œƒ)=0,(3.41)while, on (๐œ‰(๐‘ก),๐‘ก;๐œ,๐œƒ),๎€œ๐œ‰+๐‘ก๐œ‰โˆ’๐‘ก๐‘ƒ๐‘ฅ๎€œ๐‘‘๐œ=โˆ’๐œ‰+๐‘ก๐œ‰โˆ’๐‘ก๐‘…๐œ๎€œ๐œ”๐‘‘๐œ+๐œ‰+๐‘ก๐œ‰โˆ’๐‘ก๐‘…๐œƒ๐œ[]+๎€บ๐‘…๐‘‘๐œ=๐œ”๐‘…(๐œ‰,๐‘ก;๐œ‰โˆ’๐‘ก,0)โˆ’๐‘…(๐œ‰,๐‘ก;๐œ‰+๐‘ก,0)๐œƒ(๐œ‰,๐‘ก;๐œ‰+๐‘ก,0)โˆ’๐‘…๐œƒ๎€ป(๐œ‰,๐‘ก;๐œ‰โˆ’๐‘ก,0)=0.(3.42)Hence (3.39) reduces to ๎€ทฬ‡๎€ธ๐œ”๐œ‰โˆ’1๎‚ธ๎‚ต๐œ”+2๐‘ก2๎‚ถ๐‘ˆ๐‘œ(๐œ‰โˆ’๐‘ก)โˆ’๐‘ˆ๎…ž๐‘œ๐›ฝ(๐œ‰โˆ’๐‘ก)+2๐œ”๎€ท๐‘’๐œ”๐‘ก๎€ธ+1โˆ’12๎€œ๐œ‰+๐‘ก๐œ‰โˆ’๐‘ก๐‘ƒ๐‘ก(๐œ‰,๐‘ก;๐œ,0)๐‘ˆ๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰+๐‘กโˆ’๐œƒ๐œ‰โˆ’๐‘ก+๐œƒ๐‘…๐‘ก๎‚น(๐œ‰,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ=๐‘’๐œ”๐‘ก๐›ฝ2๐™ฑ๐š—.(3.43)

Remark 3.2. The function ๐‘ˆ(๐‘ฅ,๐‘ก) is continuous across the characteristic ๐‘ฅ+๐‘ก=๐œ‰๐‘œ. Indeed lim๐‘ฅ+๐‘กโ†’๐œ‰+๐‘œ๐‘ˆ(๐‘ฅ,๐‘ก)=lim๐‘ฅ+๐‘กโ†’๐œ‰โˆ’๐‘œ๐‘ˆ(๐‘ฅ,๐‘ก),(3.44)where the limit lim๐‘ฅ+๐‘กโ†’๐œ‰+๐‘œ is evaluated using (3.33) and the limit lim๐‘ฅ+๐‘กโ†’๐œ‰โˆ’๐‘œ using (3.5) or (3.23). If we evaluate the derivatives ๐‘ˆ๐‘ฅ and ๐‘ˆ๐‘ก on the characteristic ๐‘ฅ+๐‘ก=๐œ‰๐‘œ we get two different results depending on whether we are evaluating such derivatives in ๐ท๐ผ or ๐ท๐ผ๐ผ๐ผ. We can prove that lim๐‘ฅ+๐‘กโ†’๐œ‰โˆ’๐‘œ๐‘ˆ๐‘ฅ1(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๎…ž๐‘œ(๐‘ฅโˆ’๐‘ก)+๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธ๎€ท+๐‘ƒ๐‘ฅ,๐‘ก;๐œ‰๐‘œ๎€ธ๐‘ˆ,0๐‘œ๎€ท๐œ‰๐‘œ๎€ธโˆ’๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก,0)๐‘ˆ๐‘œ๎€ป+1(๐‘ฅโˆ’๐‘ก)2๎€œ๐œ‰๐‘œ๐‘ฅโˆ’๐‘ก๐‘ƒ๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰๐‘œโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ,(3.45)lim๐‘ฅ+๐‘กโ†’๐œ‰+๐‘œ๐‘ˆ๐‘ฅ1(๐‘ฅ,๐‘ก)=2๎‚ธ๐‘ˆ๎…ž๐‘œ(๐‘ฅโˆ’๐‘ก)โˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธฬ‡๐œ‰๐‘œโˆ’1ฬ‡๐œ‰๐‘œ๎€ท+1+๐‘ƒ๐‘ฅ,๐‘ก;๐œ‰๐‘œ๎€ธ๐‘ˆ,0๐‘œ๎€ท๐œ‰๐‘œ๎€ธโˆ’๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก,0)๐‘ˆ๐‘œ๎‚น+(๐‘ฅโˆ’๐‘ก)๐œ”๐‘‰๐‘œฬ‡๐œ‰๐‘œ+1+12๎€œ๐œ‰๐‘œ๐‘ฅโˆ’๐‘ก๐‘ƒ๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ(๐›ฝ๐œ)๐‘‘๐œ+22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰๐‘œโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ,(3.46)lim๐‘ฅ+๐‘กโ†’๐œ‰โˆ’๐‘œ๐‘ˆ๐‘ก1(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธโˆ’๐‘ˆ๎…ž๐‘œ๎€ท(๐‘ฅโˆ’๐‘ก)+๐‘ƒ๐‘ฅ,๐‘ก;๐œ‰๐‘œ๎€ธ๐‘ˆ,0๐‘œ๎€ท๐œ‰๐‘œ๎€ธ+๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก,0)๐‘ˆ๐‘œ๎€ป(๐‘ฅโˆ’๐‘ก)+๐›ฝ2๎€œ๐‘ก0๐‘’๐œ”๐œƒ1๐‘‘๐œƒ+2๎€œ๐œ‰๐‘œ๐‘ฅโˆ’๐‘ก๐‘ƒ๐‘ก(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐œ‰๐‘œโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…๐‘ก(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ,(3.47)lim๐‘ฅ+๐‘กโ†’๐œ‰+๐‘œ๐‘ˆ๐‘ก1(๐‘ฅ,๐‘ก)=2๎‚ธโˆ’๐‘ˆ๎…ž๐‘œ(๐‘ฅโˆ’๐‘ก)โˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธฬ‡๐œ‰๐‘œโˆ’1ฬ‡๐œ‰๐‘œ๎€ท+1+๐‘ƒ๐‘ฅ,๐‘ก;๐œ‰๐‘œ๎€ธ๐‘ˆ,0๐‘œ๎€ท๐œ‰๐‘œ๎€ธโˆ’๐‘ƒ(๐‘ฅ,๐‘ก;๐‘ฅโˆ’๐‘ก,0)๐‘ˆ๐‘œ๎‚น+(๐‘ฅโˆ’๐‘ก)๐œ”๐‘‰๐‘œฬ‡๐œ‰๐‘œ+1+12๎€œ๐œ‰๐‘œ๐‘ฅโˆ’๐‘ก๐‘ƒ๐‘ก(๐‘ฅ,๐‘ก;๐œ,0)๐‘ˆ๐‘œ(๐œ)๐‘‘๐œ+๐›ฝ2๎€œ๐‘ก0๐‘’๐œ”๐œƒ+๐›ฝ๐‘‘๐œƒ22๎€œ๐‘ก0๐‘’๐œ”๐œƒ๎€œ๐‘‘๐œƒ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘…๐‘ฅ(๐‘ฅ,๐‘ก;๐œ,๐œƒ)๐‘‘๐œ,(3.48) where ฬ‡๐œ‰๐‘œ=ฬ‡๐œ‰(0). It is easy to check that, imposing that (3.45) equals (3.46) and that (3.47) equals (3.48), we get the following condition: ฬ‡๐œ‰๐‘œ๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธ=๐œ”๐‘ˆ๐‘œ๎€ท๐œ‰๐‘œ๎€ธ=๐œ”๐‘‰๐‘œ,(3.49) which is the condition that must be fulfilled if we want the first derivatives of ๐‘ˆ(๐‘ฅ,๐‘ก) to be continuous across the characteristic ๐‘ฅ+๐‘ก=๐œ‰๐‘œ.

If we assume that the free boundary equation (3.3)(6) holds up to ๐‘ก=0 we get๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธ=๐›ฝ2๐™ฑ๐š—.(3.50) Moreover, from (3.43) we have that, when ๐‘ก=0,๎€ทฬ‡๐œ‰๐‘œโˆ’1๎€ธ๎€บ๐œ”๐‘ˆ๐‘œ๎€ท๐œ‰๐‘œ๎€ธโˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธ๎€ป=๐›ฝ2๐™ฑ๐š—.(3.51) We can therefore prove the following.

Theorem 3.3. If one assumes that compatibility condition ๐‘ˆ๐‘œ(๐œ‰๐‘œ)=๐‘‰๐‘œ and hypotheses (3.49), (3.50), (3.51) hold, then necessarily either ๐‘‰๐‘œ=0 or ๐œ”=0 and problem (3.3) admits a unique local ๐ถ1 solution (๐‘ˆ,๐œ‰), such that ฬ‡๐œ‰(0)=0. If one does not assume hypothesis (3.49) (meaning that the first derivatives of ๐‘ˆ are not continuous along the characteristic ๐‘ฅ+๐‘ก=๐œ‰๐‘œ), then problem (3.3) admits a unique local solution (๐‘ˆ,๐œ‰), such that ฬ‡โˆ’1<๐œ‰(๐‘ก)<0 if and only if ๐‘‰๐‘œ<๐›ฝ2๐™ฑ๐š—2๐œ”.(3.52)

Proof. If we suppose that (3.49), (3.50), (3.51) hold then we have ๎€ทฬ‡๐œ‰๐‘œ๐‘ˆโˆ’1๎€ธ๎€บ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธฬ‡๐œ‰๐‘œโˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธ๎€ป=๐›ฝ2ฬ‡๐œ‰๐™ฑ๐š—,โŸน๐‘œฬ‡๐œ‰=0or๐‘œ=2.(3.53) The initial velocity ฬ‡๐œ‰๐‘œ=2 is not physically acceptable since existence of a solution requires that |ฬ‡๐œ‰|<1. Therefore ฬ‡๐œ‰๐‘œ=0 and, recalling (3.49), we have either ๐‘‰๐‘œ=0 or ๐œ”=0, since ๐‘ˆ๎…ž๐‘œ(๐œ‰๐‘œ)=๐›ฝ2๐™ฑ๐š—โ‰ ยฑโˆž. If, on the other hand, we suppose that condition (3.49) does not hold, but we assume (3.52), then it is easy to show that โˆ’1<๐œ”๐‘‰๐‘œ๐œ”๐‘‰๐‘œโˆ’๐›ฝ2๐›ฝ๐™ฑ๐š—=1+2๐™ฑ๐š—๐œ”๐‘‰๐‘œโˆ’๐›ฝ2=ฬ‡๐œ‰๐™ฑ๐š—๐‘œ<0,(3.54) so that for a sufficiently small time ๐‘ก>0 there exists a unique solution with ฬ‡โˆ’1<๐œ‰<0. The existence of such a solution can be proved using classical tools like iterated kernels method (see [13]).

Remark 3.4. Let us consider the limit case in which ๐œ”=0 and ๐›ฝ2=0. In this particular situation the Riemann's function ๐‘…(๐‘ฅ,๐‘ก;๐œ,๐œƒ)โ‰ก1 and the solution ๐‘ˆ(๐‘ฅ,๐‘ก) is given by โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ1๐‘ˆ(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๐‘œ(๐‘ฅ+๐‘ก)+๐‘ˆ๐‘œ๎€ป(๐‘ฅโˆ’๐‘ก),in๐ท๐ผ,12๎€บ๐‘ˆ๐‘œ(๐‘ฅ+๐‘ก)โˆ’๐‘ˆ๐‘œ๎€ป(๐‘กโˆ’๐‘ฅ),in๐ท๐ผ๐ผ,12๎€บ๐‘ˆ๐‘œ(๐‘ฅโˆ’๐‘ก)โˆ’๐‘ˆ๐‘œ๎€ป(๐œ‰โˆ’๐‘ก)+๐‘‰๐‘œ,in๐ท๐ผ๐ผ๐ผ,(3.55) and the free boundary equation is the characteristic with positive slope passing through (๐œ‰๐‘œ,0), that is, ๎€ทฬ‡๎€ธ๐‘ˆ๐œ‰โˆ’1๎…ž๐‘œ(๐œ‰โˆ’๐‘ก)=0โŸน๐‘ˆ๐‘œ(๐œ‰โˆ’๐‘ก)=๐‘ˆ๐‘œ๎€ท๐œ‰๐‘œ๎€ธ,(3.56) namely, ๐œ‰(๐‘ก)=๐œ‰๐‘œฬ‡+๐‘กโŸน๐œ‰(๐‘ก)=1.(3.57) So, setting ๐‘ก๐‘œ=1โˆ’๐œ‰๐‘œ, for ๐‘กโ‰ฅ๐‘ก๐‘œ, the region with uniform velocity (the inner core) has disappeared. For ๐‘กโ‰ฅ๐‘ก๐‘œ, the solution ๐‘ˆ(๐‘ฅ,๐‘ก) is thus found solving ๐‘ˆ๐‘ฅ๐‘ฅ=๐‘ˆ๐‘ก๐‘ก,0โฉฝ๐‘ฅโฉฝ1,๐‘กโฉพ1โˆ’๐œ‰๐‘œ๐‘ˆ๎€ท๐‘ฅ,1โˆ’๐œ‰๐‘œ๎€ธ=๐‘ˆโˆ—๐‘œ๐‘ˆ(๐‘ฅ),0โฉฝ๐‘ฅโฉฝ1,๐‘ก๎€ท๐‘ฅ,1โˆ’๐œ‰๐‘œ๎€ธ=๐‘ˆโˆ—1(๐‘ฅ),0โฉฝ๐‘ฅโฉฝ1,๐‘ˆ(0,๐‘ก)=0๐‘กโฉพ1โˆ’๐œ‰๐‘œ,๐‘ˆ(1,๐‘ก)=๐‘‰๐‘œ๐‘กโฉพ1โˆ’๐œ‰๐‘œ,(3.58) where ๐‘ˆโˆ—๐‘œ(๐‘ฅ) and ๐‘ˆโˆ—1(๐‘ฅ) are determined evaluating (3.55) at time ๐‘ก=๐‘ก๐‘œ. To solve problem (3.58) we introduce the new variable ๐‘Š(๐‘ฅ,๐‘ก)=๐‘ˆ(๐‘ฅ,๐‘ก)โˆ’๐‘ฅ๐‘‰๐‘œ(3.59) and rescale time with ๐œƒ=๐‘กโˆ’๐‘ก๐‘œ.(3.60) Problem (3.58) becomes ๐‘Š๐‘ฅ๐‘ฅ=๐‘Š๐œƒ๐œƒ,0โฉฝ๐‘ฅโฉฝ1,๐œƒโฉพ0,๐‘Š(๐‘ฅ,0)=๐‘ˆโˆ—๐‘œ(๐‘ฅ)โˆ’๐‘‰๐‘œ๐‘Š(๐‘ฅ),0โฉฝ๐‘ฅโฉฝ1,๐œƒ(๐‘ฅ,0)=๐‘Š1(๐‘ฅ)=๐‘ˆโˆ—1(๐‘ฅ),0โฉฝ๐‘ฅโฉฝ1,๐‘Š(0,๐œƒ)=0,๐œƒโฉพ0,๐‘Š(1,๐œƒ)=0,๐œƒโฉพ0,(3.61) whose solution is [11] ๐‘Š(๐‘ฅ,๐œƒ)=โˆž๎“๐‘–=1๎€บ๐ด๐‘›cos(๐œ‹๐‘›๐œƒ)+๐ต๐‘›๎€ปsin(๐œ‹๐‘›๐œƒ)sin(๐œ‹๐‘›๐‘ฅ),(3.62) where ๐ด๐‘›๎€œ=2๐œƒ0๐‘Š๐‘œ(๐‘ง)sin(๐œ‹๐‘›๐‘ง)๐‘‘๐‘ง,๐ต๐‘›=2๎€œ๐œ‹๐‘›๐œƒ0๐‘Š1๐‘ˆ๎€ท(๐‘ง)sin(๐œ‹๐‘›๐‘ง)๐‘‘๐‘ง,(๐‘ฅ,๐‘ก)=๐‘Š๐‘ฅ,๐‘กโˆ’1+๐œ‰๐‘œ๎€ธ+๐‘ฅ๐‘‰๐‘œ.(3.63)

4. Asymptotic Expansion

In this section we look for a solution to problem (3.3)(1) in the following form: ๐‘ˆ(๐‘ฅ,๐‘ก)=โˆž๎“๐‘–=0๐œ”๐‘–๐‘ˆ(๐‘–)(๐‘ฅ,๐‘ก).(4.1) This allows to obtain a sequence of problems for each ๐‘–=0,1,2โ€ฆ with the free boundary being given by ๐œ‰(๐‘–)(๐‘ก). (We remark that the sequence {๐œ‰(๐‘–)(๐‘ก)} is not, in general, an asymptotic sequence.) Such an analysis is motivated by the fact that, in practical cases (asphalt and bitumen), ๐œ”=๐‘‚(10โˆ’1) (see (2.3)). Hence, it makes sense to look for a โ€œperturbativeโ€ approach for the system (3.3).

We do not discuss the issue of the convergence of series (4.1) and of the sequence {๐œ‰(๐‘–)(๐‘ก)}, which is beyond the scope of the present paper. We limit ourselves to a formal derivation of the free boundary problems that can be obtained plugging (4.1)(1) into (3.3): โˆž๎“๐‘–=0๎‚ƒ๐œ”๐‘–๐‘ˆ(๐‘–)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐œ”๐‘–๐‘ˆ(๐‘–)๐‘ก๐‘ก(๐‘ฅ,๐‘ก)+๐œ”๐‘–+2๐‘ˆ(๐‘–)(๎‚„๐‘ฅ,๐‘ก)=โˆ’๐›ฝ2โˆž๎“๐‘–=0(๐œ”๐‘ก)๐‘–.๐‘–!(4.2) Hence, for each ๐‘–=0,1,2,โ€ฆ, we have ๐‘–=0,๐‘ˆ(๐‘œ)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐‘ˆ(๐‘œ)๐‘ก๐‘ก(๐‘ฅ,๐‘ก)=โˆ’๐›ฝ2,๐‘–=1,๐‘ˆ(1)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐‘ˆ(1)๐‘ก๐‘ก(๐‘ฅ,๐‘ก)=โˆ’๐›ฝ2๐‘ก,๐‘–=2,๐‘ˆ(2)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐‘ˆ(2)๐‘ก๐‘ก(๐‘ฅ,๐‘ก)=โˆ’๐›ฝ2๐‘ก22!โˆ’๐‘ˆ(๐‘œ)โ‹ฎ(๐‘ฅ,๐‘ก),๐‘–>2,๐‘ˆ(๐‘–)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐‘ˆ(๐‘–)๐‘ก๐‘ก(๐‘ฅ,๐‘ก)=โˆ’๐›ฝ2๐‘ก๐‘–๐‘–!โˆ’๐‘ˆ(๐‘–โˆ’2)(๐‘ฅ,๐‘ก)(4.3)

and the following free boundary problems โŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽฉ๐‘ˆ๐‘–=0,(๐‘œ)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐‘ˆ(๐‘œ)๐‘ก๐‘ก(๐‘ฅ,๐‘ก)=โˆ’๐›ฝ2๐‘ˆ(๐‘œ)(๐‘ฅ,0)=๐‘ˆ๐‘œ(๐‘ˆ๐‘ฅ),๐‘ก(๐‘œ)๐‘ˆ(๐‘ฅ,0)=0,(๐‘œ)๐‘ˆ(0,๐‘ก)=0,(๐‘œ)๎€ท๐œ‰(๐‘œ)๎€ธ,๐‘ก=๐‘‰๐‘œ๐‘ˆ๐‘ฅ(๐‘œ)๎€ท๐œ‰(๐‘œ)๎€ธ+ฬ‡๐œ‰,๐‘ก(๐‘œ)๐‘ˆ๐‘ก(๐‘œ)๎€ท๐œ‰(๐‘œ)๎€ธ,๐‘ก=๐›ฝ2๐œ‰๐™ฑ๐š—,(๐‘œ)(0)=๐œ‰๐‘œ,โŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽฉ๐‘ˆ(4.4)๐‘–=1,(1)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐‘ˆ(1)๐‘ก๐‘ก(๐‘ฅ,๐‘ก)=โˆ’๐‘ก๐›ฝ2๐‘ˆ(1)๐‘ˆ(๐‘ฅ,0)=0,๐‘ก(1)(๐‘ฅ,0)=๐‘ˆ๐‘œ๐‘ˆ(๐‘ฅ),(1)๐‘ˆ(0,๐‘ก)=0,(1)๎€ท๐œ‰(1)๎€ธ,๐‘ก=๐‘‰๐‘œ๐‘ก๐‘ˆ๐‘ฅ(1)๎€ท๐œ‰(1)๎€ธ+ฬ‡๐œ‰,๐‘ก(1)๐‘ˆ๐‘ก(1)๎€ท๐œ‰(1)๎€ธโˆ’ฬ‡๐œ‰,๐‘ก(1)๐‘‰๐‘œ=๐‘ก๐›ฝ2๐œ‰๐™ฑ๐š—,(1)(0)=๐œ‰๐‘œ,โŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽชโŽฉ๐‘ˆ(4.5)๐‘–โ‰ฅ2,(๐‘–)๐‘ฅ๐‘ฅ(๐‘ฅ,๐‘ก)โˆ’๐‘ˆ(๐‘–)๐‘ก๐‘ก๐‘ก(๐‘ฅ,๐‘ก)=โˆ’๐‘–๐›ฝ๐‘–!2โˆ’๐‘ˆ(๐‘–โˆ’2)๐‘ˆ(๐‘ฅ,๐‘ก)(๐‘–)๐‘ˆ(๐‘ฅ,0)=0,๐‘ก(๐‘–)(๐‘ˆ๐‘ฅ,0)=0,(๐‘–)๐‘ˆ(0,๐‘ก)=0,(๐‘–)๎€ท๐œ‰(๐‘–)๎€ธ=๐‘ก,๐‘ก๐‘–๐‘‰๐‘–!๐‘œ๐‘ˆ๐‘ฅ(๐‘–)๎€ท๐œ‰(๐‘–)๎€ธ+ฬ‡๐œ‰,๐‘ก(๐‘–)๐‘ˆ๐‘ก(๐‘–)๎€ท๐œ‰(๐‘–)๎€ธโˆ’ฬ‡๐œ‰,๐‘ก(๐‘–)๐‘‰๐‘œ๐‘ก๐‘–โˆ’1=๐‘ก(๐‘–โˆ’1)!๐‘–๐›ฝ๐‘–!2๐œ‰๐™ฑ๐š—,(๐‘–)(0)=๐œ‰๐‘œ.(4.6) We immediately remark that, in each problem, the governing equation is no longer a telegrapher's equation, but a nonhomogeneous wave equation. Hence, using classical d'Alembert formula, we can write the representation formula for each domain ๐ท๐ผ, ๐ท๐ผ๐ผ, ๐ท๐ผ๐ผ๐ผ and for each order of approximation ๐‘–=0,1,.... In particular, in ๐ท๐ผ we have ๐ท๐ผโŽงโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽฉ๐‘ˆ(๐‘œ)1(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๐‘œ(๐‘ฅ+๐‘ก)+๐‘ˆ๐‘œ๎€ป+๐›ฝ(๐‘ฅโˆ’๐‘ก)2๐‘ก2,๐‘ˆ2!(1)1(๐‘ฅ,๐‘ก)=2๎€œ๐‘ฅ+๐‘ก๐‘ฅโˆ’๐‘ก๐‘ˆ๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+2๐‘ก3,โ‹ฎ๐‘ˆ3!(๐‘–)๐›ฝ(๐‘ฅ,๐‘ก)=22๎€œ๐‘ก0๎€œ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘ˆ(๐‘–โˆ’2)๐›ฝ(๐œ,๐œƒ)๐‘‘๐œ๐‘‘๐œƒ+2๐‘ก๐‘–+2(๐‘–+2)!,๐‘–โฉพ2,(4.7) while in ๐ท๐ผ๐ผ๐ท๐ผ๐ผโŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽฉ๐‘ˆ(๐‘œ)1(๐‘ฅ,๐‘ก)=2๎€บ๐‘ˆ๐‘œ(๐‘ฅ+๐‘ก)โˆ’๐‘ˆ๐‘œ๎€ป+๐›ฝ(๐‘กโˆ’๐‘ฅ)2๐‘ก2โˆ’๐›ฝ2!2(๐‘ฅโˆ’๐‘ก)2,๐‘ˆ2!(1)1(๐‘ฅ,๐‘ก)=2๎€œ๐‘ฅ+๐‘ก๐‘กโˆ’๐‘ฅ๐‘ˆ๐‘œ๐›ฝ(๐œ)๐‘‘๐œ+2๐‘ก3โˆ’๐›ฝ3!2(๐‘กโˆ’๐‘ฅ)3,โ‹ฎ๐‘ˆ3!(๐‘–)๐›ฝ(๐‘ฅ,๐‘ก)=22๎€œ๐‘ก0๎€œ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘กโˆ’๐‘ฅ+๐œƒ๐‘ˆ(๐‘–โˆ’2)๐›ฝ(๐œ,๐œƒ)๐‘‘๐œ๐‘‘๐œƒ+22๎€œ๐‘ก0๎€œ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘ˆ(๐‘–โˆ’2)+๐›ฝ(๐œ,๐œƒ)๐‘‘๐œ๐‘‘๐œƒ2๐‘ก๐‘–+2โˆ’๐›ฝ(๐‘–+2)!2(๐‘กโˆ’๐‘ฅ)๐‘–+2(๐‘–+2)!,๐‘–โฉพ2,(4.8) and in ๐ท๐ผ๐ผ๐ผ๐ท๐ผ๐ผ๐ผโŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽฉ๐‘ˆ(๐‘œ)(๐‘ฅ,๐‘ก)=๐‘‰๐‘œ+12๎€บ๐‘ˆ๐‘œ(๐‘ฅโˆ’๐‘ก)โˆ’๐‘ˆ๐‘œ๎€ท๐œ‰(๐‘œ)โˆ—โˆ’๐‘กโˆ—+๐›ฝ๎€ธ๎€ป2๐‘ก2โˆ’๐›ฝ2!2๐‘กโˆ—2,๐‘ˆ2!(1)(๐‘ฅ,๐‘ก)=๐‘‰๐‘œ๐‘กโˆ—+12๎€œ๐œ‰(1)โˆ—โˆ’๐‘กโˆ—๐‘ฅโˆ’๐‘ก๐‘ˆ๐‘œ(๐›ฝ๐œ)๐‘‘๐œ+2๐‘ก3โˆ’๐›ฝ3!2๐‘กโˆ—3,โ‹ฎ๐‘ˆ3!(๐‘–)(๐›ฝ๐‘ฅ,๐‘ก)=22๎€œ๐‘ก0๎€œ๐‘ฅ+๐‘กโˆ’๐œƒ๐‘ฅโˆ’๐‘ก+๐œƒ๐‘ˆ(๐‘—โˆ’2)(๐›ฝ๐œ,๐œƒ)๐‘‘๐œ๐‘‘๐œƒโˆ’22๎€œ๐‘กโˆ—0๎€œ๐œ‰๐‘ฅ+๐‘กโˆ’๐œƒ(๐‘–)โˆ—โˆ’๐‘กโˆ—+๐œƒ๐‘ˆ(๐‘–โˆ’2)(๐œ,๐œƒ)๐‘‘๐œ๐‘‘๐œƒ+๐‘‰๐‘œ๎€ท๐‘กโˆ—๎€ธ๐‘–+๐›ฝ๐‘–!2๐‘ก๐‘–+2โˆ’๐›ฝ(๐‘–+2)!2๎€ท๐‘กโˆ—๎€ธ๐‘–+2(๐‘–+2)!.๐‘–โ‰ฅ2.(4.9) Proceeding as in Section 3 we can show that the evolution equations of the free boundary ๐‘ฅ=๐œ‰(๐‘–)(๐‘ก) at each step are given by๎‚€ฬ‡๐œ‰๐‘–=0,(0)๎‚๎€บ๐›ฝโˆ’12๐‘กโˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰(๐‘œ)โˆ’๐‘ก๎€ธ๎€ป=๐›ฝ2๎‚€ฬ‡๐œ‰๐™ฑ๐š—,(4.10)๐‘–=1,(1)๎‚๎‚ธ๐‘ˆโˆ’1๐‘œ๎€ท๐œ‰(1)๎€ธโˆ’๐‘กโˆ’๐‘‰๐‘œ+๐›ฝ2๐‘ก2๎‚น2!=๐‘ก๐›ฝ2๎‚€ฬ‡๐œ‰๐™ฑ๐š—,(4.11)๐‘–โฉพ2,(๐‘–)๎‚๎‚ธ๐›ฝโˆ’12(๐‘–+2)๐‘ก๐‘–+1โˆ’๐‘‰(๐‘–+1)!๐‘œ๐‘ก๐‘–โˆ’1(๐‘–โˆ’1)!+๐›ฝ2๎€œ๐‘ก0๐‘ˆ(๐‘–โˆ’2)๎€ท๐œ‰(๐‘–)๎€ธ๎‚น=๐‘กโˆ’๐‘ก+๐œƒ,๐œƒ๐‘‘๐œƒ๐‘–๐›ฝ๐‘–!2๐™ฑ๐š—.(4.12) At the zero order, assuming the compatibility condition ๐‘ˆ๎…ž(๐œ‰๐‘œ)=๐›ฝ2Bn (see problem (4.4)), we have ๎‚€ฬ‡๐œ‰1โˆ’๐‘œ(๐‘œ)๎‚๐‘ˆ๎…ž๐‘œ๎€ท๐œ‰๐‘œ๎€ธ=๐›ฝ2ฬ‡๐œ‰๐™ฑ๐š—,โŸน๐‘œ(๐‘œ)=0.(4.13)

At the first order (see problem (4.5)), we assume that the compatibility condition of second order holds in the corner (๐œ‰๐‘œ,0). This means that we can differentiate the free boundary equation (4.5)(6) and take the limit for ๐‘กโ†’0. We have ๐‘ˆ(1)๐‘ฅ๐‘ฅ๎€ท๐œ‰(1)๎€ธฬ‡๐œ‰,๐‘ก(1)+๐‘ˆ(1)๐‘ฅ๐‘ก๎€ท๐œ‰(1)๎€ธ+ฬˆ๐œ‰,๐‘ก(1)๐‘ˆ๐‘ก(1)๎€ท๐œ‰(1)๎€ธ+ฬ‡๐œ‰,๐‘ก(1)2๐‘ˆ(1)๐‘ฅ๐‘ก๎€ท๐œ‰(1)๎€ธโˆ’ฬˆ๐œ‰,๐‘ก(1)๐‘‰๐‘œ=๐›ฝ2๐™ฑ๐š—,(4.14)

which, when ๐‘กโ†’0, reduces to๐‘ˆโ€ฒ๐‘œ๎€ท๐œ‰๐‘œ๎€ธ๎‚€ฬ‡๐œ‰1+(1)2๐‘œ๎‚=๐›ฝ2ฬ‡๐œ‰๐™ฑ๐š—,โŸน๐‘œ(1)=0.(4.15)

For the generic ๐‘–th order (see problem (4.6)), we assume that the compatibility conditions in the corner (๐œ‰๐‘œ,0) hold up to order (๐‘–โˆ’1). Therefore we can take the (๐‘–โˆ’1)th derivative of (4.6)(6), obtaining ๐‘‘๐‘–โˆ’1๐‘‘๐‘ก๐‘–โˆ’1๎‚ƒ๐‘ˆ๐‘ฅ(๐‘–)๎€ท๐œ‰(๐‘–)๎€ธ๎‚„+๐‘‘,๐‘ก๐‘–โˆ’1๐‘‘๐‘ก๐‘–โˆ’1๎‚ƒฬ‡๐œ‰(๐‘–)๎‚„๐‘ˆ๐‘ก(๐‘–)๎€ท๐œ‰(๐‘–)๎€ธ+ฬ‡๐œ‰,๐‘ก(๐‘–)๐‘‘๐‘–โˆ’1๐‘‘๐‘ก๐‘–โˆ’1๎‚ƒ๐‘ˆ๐‘ก(๐‘–)๎€ท๐œ‰(๐‘–)๎€ธ๎‚„โˆ’ฬ‡๐œ‰,๐‘ก(๐‘–)๐‘‰๐‘œโˆ’๐‘‰๐‘œ๐‘ก๐‘–โˆ’1(๐‘‘๐‘–โˆ’1)!๐‘–โˆ’1๐‘‘๐‘ก๐‘–โˆ’1๎‚ƒฬ‡๐œ‰(๐‘–)๎‚„=๐‘ก๐›ฝ2๐™ฑ๐š—,(4.16)

which, in the limit ๐‘กโ†’0, reduces toโˆ’ฬ‡๐œ‰๐‘œ(๐‘–)๐‘‰๐‘œฬ‡๐œ‰=0โŸน๐‘œ(๐‘–)=0.(4.17) We therefore conclude that, assuming enough regularity for each problem ๐‘–=0,1,2,โ€ฆ, (4.10)โ€“(4.12) posses a unique local solution with ๐œ‰(๐‘–)(0)=๐œ‰๐‘œ and ฬ‡๐œ‰(๐‘–)(0)=0.

Before proceeding further we suppose that ๐‘ˆ(๐‘ฅ) has the following properties:

(H1)โ€ƒ ๐‘ˆ๐‘œ(๐‘ฅ)โˆˆ๐ถโˆž([0,๐œ‰๐‘œ]),

(H2)โ€ƒ0<๐‘ˆ๐‘œ(๐‘ฅ)<๐‘‰๐‘œforall๐‘ฅโˆˆ(0,๐œ‰๐‘œ), ๐‘ˆ๐‘œ(0)=0,๐‘ˆ๐‘œ(๐œ‰๐‘œ)=๐‘‰๐‘œ,

(H3)โ€ƒ๐‘ˆ๎…ž๐‘œ(๐‘ฅ)>0forall๐‘ฅโˆˆ[0,๐œ‰๐‘œ] and ๐‘ˆ๎…ž(๐œ‰๐‘œ)=๐›ฝ2๐™ฑ๐š—,

(H4)โ€ƒ๐‘ˆ๐‘œ(๐‘ฅ) satisfies all the compatibility conditions up to any order in the corner (๐œ‰๐‘œ,0).

4.1. Zero-Order Approximation

We introduce the new variable ๐œ™(๐‘œ)=๐œ‰(๐‘œ)โˆ’๐‘ก, so that (4.10) can be rewritten asฬ‡๐œ™(๐‘œ)๎€บ๐›ฝ2๐‘กโˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ™(๐‘œ)๎€ธ๎€ป=๐›ฝ2๐™ฑ๐š—,(4.18)

with ๐œ™(๐‘œ)(0)=๐œ‰๐‘œ. Then we look for the solution ๐‘ก=๐‘ก(๐œ™(๐‘œ)) which fulfills the following Cauchy problem:๐›ฝ2๐™ฑ๐š—๐‘‘๐‘ก๐‘‘๐œ™(๐‘œ)=๎€บ๐›ฝ2๐‘กโˆ’๐‘ˆ๎…ž๐‘œ๎€ท๐œ™(๐‘œ),๐‘ก๎€ท๐œ‰๎€ธ๎€ป๐‘œ๎€ธ=0,(4.19) that is,๐‘ก๎€ท๐œ™(๐‘œ)๎€ธ1=โˆ’๐›ฝ2๎€œ๐™ฑ๐š—๐œ™(๐‘œ)๐œ‰๐‘œ๐‘ˆ๎…ž๐‘œ๎‚ต๐œ™(๐‘ง)exp(๐‘œ)โˆ’๐‘ง๎‚ถ๐™ฑ๐š—๐‘‘๐‘ง.(4.20) Recalling that |ฬ‡๐œ‰(๐‘œ)|<1, that is,|||ฬ‡๐œ™(๐‘œ)|||ฬ‡๐œ™+1<1,โŸนโˆ’2<(๐‘œ)<0,โŸน๐‘‘๐‘ก๐‘‘๐œ™(๐‘œ)1<โˆ’2,(4.21) from (4.19)(1) we realize that (4.21) is fulfilled if๐œ‰๐‘œ+๐™ฑ๐š—2<1๐›ฝ2inf๎€บ๐‘งโˆˆ0,๐œ‰๐‘œ๎€ป๐‘ˆ๎…ž๐‘œ(๐‘ง).(4.22) Therefore, under hypothesis (4.22), local existence of a classical solution is guaranteed. Such a solution is given by ๐œ‰(๐‘œ)(๐‘ก)=๐œ™(๐‘œ)(๐‘ก)+๐‘ก, where ๐œ™(๐‘œ)(๐‘ก) is determined inverting (4.20).

4.2. First-Order Approximation

We now have to solve the problem๎‚€ฬ‡๐œ‰(1)๎‚๎‚ธ๐‘ˆโˆ’1๐‘œ๎€ท๐œ‰(1)๎€ธโˆ’๐‘กโˆ’๐‘‰๐‘œ+๐›ฝ2๐‘ก2๎‚น2!=๐‘ก๐›ฝ2๐™ฑ๐š—,(4.23) with ๐œ‰(1)(0)=๐œ‰๐‘œ. Proceeding as in Section 4.1 we introduce the new variable ๐œ™(1)=๐œ‰(1)โˆ’๐‘ก, so that (4.23) becomesฬ‡๐œ™(1)๎‚ธ๐‘ˆ๐‘œ๎€ท๐œ™(1)๎€ธโˆ’๐‘‰๐‘œ+๐›ฝ2๐‘ก2๎‚น2!=๐‘ก๐›ฝ2๐™ฑ๐š—,(4.24)

and we have to solve the following Cauchy problem:๐‘‘๐‘ก๐‘‘๐œ™(1)=1๐‘ก๐›ฝ2๎‚ธ๐‘ˆ๐™ฑ๐š—๐‘œ๎€ท๐œ™(1)๎€ธโˆ’๐‘‰๐‘œ+๐›ฝ2๐‘ก2๎‚น,๐‘ก๎€ท๐œ‰2!๐‘œ๎€ธ=0.(4.25) We notice that (4.25)(1) is a Bernoulli equation. Therefore, setting ๐‘ค=๐‘ก2, problem (4.25) becomes๐‘‘๐‘ค๐‘‘๐œ™(1)=๐‘ค๎ƒฌ๐‘ˆ๐™ฑ๐š—+2๐‘œ๎€ท๐œ™(1)๎€ธโˆ’๐‘‰๐‘œ๐›ฝ2๎ƒญ,๐‘ค๎€ท๐œ‰๐™ฑ๐š—๐‘œ๎€ธ=0.(4.26) whose solution is given by๐‘ก2๎€ท๐œ™(1)๎€ธ=๎€œ๐œ™(1)๐œ‰๐‘œ๎‚ป๐œ™2exp(1)โˆ’๐‘ ๐‘ˆ๐™ฑ๐š—๎‚ผ๎‚ธ๐‘œ(๐‘ )โˆ’๐‘‰๐‘œ๐›ฝ2๎‚น๐™ฑ๐š—๐‘‘๐‘ ,(4.27) which make sense only if ๐œ™(1)โฉฝ๐œ‰๐‘œ. Integrating (4.27) by parts we get๐‘ก2๎€ท๐œ™(1)๎€ธ๎€œ=2๐œ™(1)๐œ‰๐‘œ๎‚ป๐œ™exp(1)โˆ’๐‘ ๐‘ˆ๐™ฑ๐š—๎‚ผ๎‚ธ๎…ž๐‘œ(๐‘ )๐›ฝ2๎‚น2๐‘‘๐‘ +๐›ฝ2๎€บ๐‘‰๐‘œโˆ’๐‘ˆ๐‘œ๎€ท๐œ™(1).๎€ธ๎€ป(4.28) We recall from the previous section that the condition |ฬ‡๐œ‰(1)(๐‘ก)|<1 is guaranteed if๐‘‘๐‘ก๐‘‘๐œ™(1)1<โˆ’2,(4.29)

which, by virtue of(4.25)(1), is equivalent to require that๐‘ก22+๐™ฑ๐š—๐‘ก+๐›ฝ2๎€ท๐‘ˆ๎€ท๐œ™(1)๎€ธโˆ’๐‘‰๐‘œ๎€ธ<0.(4.30) Hence, under assumption (๐ป2), the discriminant ฮ”=๐™ฑ๐š—2+8๐›ฝโˆ’2