Sobolev approximation for two-phase solutions of forward-backward parabolic problems

We discuss some properties of a forward-backward parabolic problem 
that arises in models of phase transition in which two stable phases are separated by an interface. 
Here we consider a formulation of the problem that comes from a Sobolev approximation of it. 
In particular we prove uniqueness of the previous problem extending to nonlinear diffusion function a result obtained 
 in [21] in the piecewise linear case. Moreover, we analyze the third order partial differential problem that approximates the forward-backward parabolic one. In particular, for some classes of initial data, we obtain a priori estimates that generalize that proved in [22]. Using these results we study the singular limit of the Sobolev approximation, as a consequence we obtain existence of the forward-backward problem for a class of initial data.

1. Introduction. In this paper we consider the Neumann initial-boundary value problem for the equation where T > 0, Ω ⊂ R is a bounded interval Ω := (ω 1 , ω 2 ), ω 1 < 0 < ω 2 , and the function φ ∈ C ∞ (R) satisfies the following assumption: In what follows, we will also denote by a ∈ (−∞, b) and d ∈ (c, ∞) the roots of the equation φ(u) = A, respectively φ(u) = B (see Figure 1). The conditions b < 0 < c and φ(0) = 0 are only technical and are introduced in order to simplify some proofs.
Since the response function φ is in general nonmonotone, equation (1) is of forward-backward parabolic type; it is well-posed forward in time at the points where φ > 0 and it is ill-posed where φ < 0. In this connection, we denote by and the stable branches of the equation v = φ(u), whereas is referred to as the unstable branch. Equation (1) arises in the theory of phase transitions (see e.g. [4], [5], [9], [11]). In that sense the function u represents the phase field and its values characterize the different phases; for diffusion functions as in (H 1 ), the half-lines (−∞, b) and (c, ∞) correspond to stable phases and the interval (b, c) to an unstable phase (see e.g [5]).
Initial-boundary value problems for equation (1) are in general ill-posed (see e.g. [14], [15], [33]). Then, it is important to introduce a proper regularization. In fact, a widely accepted idea is that ill-posedness of forward-backward equations derives from neglecting some relevant information in the modelling of the physical phenomenon. Therefore, a general strategy is to restore such an information by introducing some physically sensible regularization, that leads to well-posed approximating problems. Then, a natural question is whether, in the limit as the regularization parameter goes to zero, we can define solutions in some suitable sense to the original unperturbed problem. In this general framework, different types of regularizations for equation (1) have been introduced and investigated (e.g., see [2], [3], [4], [12], [32]). Here, for any given ε > 0, we consider the pseudoparabolic regularization, described by the Sobolev equation that takes nonequilibrium effects into account ( [4], [10], [11]). Equation (5) has been studied in [22], while the singular limit as ε → 0 + has been analyzed in [24,26]; see also [23] and [27] for other types of nonlinearities. In analogy with the theory of hyperbolic conservation laws, assuming that the physical solutions are obtained as limiting points of the solutions to equations (5) as ε → 0, it is possible to give an entropic formulation of the original ill-posed equation (1) (together with initial and boundary conditions). In fact, the solutions to (1), defined by this limiting process and hereafter called weak entropy solutions to (1), satisfy a class of suitable entropy inequalities (e.g., see [26], [8], [20]).
In this general framework, we consider the evolution problem   where φ is as in (H 1 ) and the initial data function u 0 ∈ L ∞ (Ω) satisfies   In view of the above assumption (H 2 ), we look for a solution to problem (6) with a particular structure. More precisely, since the initial datum u 0 takes values only in the stable phases S 1 and S 2 of the graph of φ, we consider solutions to problem (6) which describe evolution between such phases. In other words, we require that the unstable phase S 0 does not influence the dynamics. Moreover, we assume that there exists a regular interface separating the rectangle Q T into two different regions where the solution lies in each stable phase S i (i = 1, 2). Taking the above requirements into account, it is possible to introduce a notion of solution to (6) that is in accordance with the general entropy formulation given in [26]. We call this kind of solution to problem (6) a two-phase entropy solution, whose formal definition is given below.
Remark 1. Let us denote by C 2,1 (Q T ) the set of functions f ∈ C(Q T ) such that f t , f x ,f xx ∈ C(Q T ). Let (u, v, ξ) be any two-phase solution to problem (6) (in the sense of Definition 1.1). Then, it is easily seen that v(·, 0) = φ(u 0 ) in Ω and u ∈ C(V i ) (i = 1, 2); moreover, v = φ(u) in Q T and u is a weak solution of the equation 2). This implies that ( [17,31]).
In what follows, when we consider the class of the two-phase solutions introduced in Definition 1.1, both the initial-boundary value problem in a bounded spatial domain Ω ⊂ R and the Cauchy problem in R associated with equation (1) will be denoted by two-phase problems (TP-problems) for equation (1). Local existence and uniqueness of (regular) two-phase solutions to the Cauchy TP-problem in R × (0, T ) has been proven in [21] in the case of piecewise response functions φ. It is worth observing that local existence is obtained by using two auxiliary problems, namely a moving boundary problem (MB-problem) and a steady boundary problem (SB-problem). The former appears when the interface evolves in time and it can be regarded as a free boundary problem. Otherwise, the SB-problem arises when the interface does not move. An improvement of such results is obtained in [30], where it is shown that the solution to the Cauchy TP-problem can be extended in time by a suitable alternance of the above two auxiliary problems.
When assumption (H 1 ) holds (namely, φ is cubic-like), and the initial datum u 0 satisfies the additional constraint a ≤ u 0 ≤ d in Ω (see Figure 1), global existence of two-phase solutions to problem (6) was obtained in [28]. In particular, in [21,28] it was also proven that the condition a ≤ u 0 ≤ d in Ω ensures that any two-phase solution (u, v, ξ) to (6) verifies the condition ξ (t) = 0 for every t ≥ 0: in other words, following the terminoly in [20,21], in the case of such initial data functions u 0 , the triple (u, v, ξ) is a steady boundary two-phase solution to problem (6).
Finally, the long-time behaviour of two-phase solutions to the Cauchy-Dirichlet TP-problem has been studied in [28]; see also [13] and [29] for analogous results concerning general weak entropy solutions respectively to the Cauchy and to the Neumann initial-boundary value problem associated with equation (1). For numerical results regarding initial-boundary problems associated to equations (1) and (5) see [16].
In this paper we study the TP-problem and its pseudoparabolic regularization (5) for nonlinear diffusion functions φ satisfying assumption (H 1 ). In this connection, let us observe that the class of two-phase solutions introduced in Definition 1.1 is a slightly different from that considered in [21]. In fact, by the choice of the nonlinear term φ (see (H 1 )), in general we expect less regularity for the solution u to problem (6), since the parabolic equation (1) degenerates whenever u takes the values u = b or u = c, where the derivative φ vanishes (see Figure 1).
In Section 2 we analyze some qualitative properties of two-phase entropy solutions. In particular, using the entropy inequalities, we show that the interface ξ(t) moves only at the points where v(ξ(t), t) = A or v(ξ(t), t) = B. This is completely equivalent to the condition proven in [21] for the piecewise linear case.
Then, in Theorem 2.3 one of the main results of the paper is given: namely, uniqueness of two-phase solutions to problem (6) (in the sense of Definition 1.1). It is worth observing that the proof we outline includes and in some sense generalizes the case of regular two-phase solutions considered in [21]. In addition, we also prove some stability results with respect to the initial data function u 0 (see Remark 5), and in Proposition 3 we give a first description of the regions where the evolution equation (1) can not degenerate.
Next, in order to provide a local existence result of two-phase solutions to problem (6) for every u 0 satisfying (H 2 ) and less restrictive assumptions with respect to those formulated in [28] (see (A 1 )-(A 2 ) below), our strategy consists in regularizing the equation u t = [φ(u)] xx -hence the boundary conditions on ∂Ω × (0, T )-by means of the Sobolev regularization (5) which leads to well-posed approximating problems. Then, our aim is to recover local existence of two-phase solutions to the original problem (6) by characterizing the limiting points of the solutions to the above regularized problems.
The investigation of the Neumann initial-boundary value problems associated with (5) is the content of Section 3. As already recalled, this analysis has begun in [22]. Here we obtain more accurate informations about the regularity of their solutions and we establish some estimates that are useful to treat the case of initial data satisfying (H 2 ) (in particular, see Proposition 5). Moreover, introducing a family of viscous entropy inequalities, we also prove some comparison results that assure, under suitable initial and boundary conditions, the existence of subdomains in the rectangle Q T where the approximating solutions take values in one of the two stable phases S 1 and S 2 of the graph of φ (see Proposition 6). In some sense, the conclusions we draw seem to indicate that for a special class of initial data functions u 0 (see Section 4 for more details) the solutions to the psedoparabolic problems associated with (5) have the same structure of the corresponding two-phase solutions to the original unperturbed problem (6). Let us also remark that, although in our opinion the results obtained in Section 3 are interesting by themselves, their role will be crucial in the investigation of the singular limit as ε → 0 in (5), since they allow to prove local existence of two-phase solutions to (6) for a suitable choice of initial data functions u 0 .
For every u 0 satisfying (H 2 ), the analysis of the vanishing viscosity limit as ε → 0 is performed in Sections 4 and 5. As already mentioned, such singular limit has been studied in [26] in the general case of initial data u 0 ∈ L ∞ (Ω). Here the main result is the existence of two functions u and v, satisfying the equation u t = v xx in the weak sense and related in a suitable way, which are obtained respectively as limiting points of the families {u ε } and {φ(u ε )} of solutions to the pseudoparabolic problems for (5). The crucial point is that in general we have v = φ(u): in particular, in view of the nonmonotone character of φ, the function u can be regarded as a superposition of different phases, related to the different branches of the graph of φ (see Subsequence 3.2 for more details). In this general framework, we wonder whether under the more restrictive assumption (H 2 ) we can draw additional informations when studying the vanishing viscosity limit in the approximating problems. Specifically, a natural question is whether such a requirement allows to deduce (in some special cases) existence of two-phase solutions to problem (6) obtained by such a limiting process.
The affirmative answer is the content of Section 4, where we show that for every initial datum u 0 satisfying (H 2 ) and one between there exists a local (steady-boundary) two-phase solution to problem (6) (see Theorem 4.3). Let us observe that this is probably the main result of the paper. Moreover, it is interesting to note that we obtain local existence of two-phase solutions by taking the vanishing viscosity limit in (5) (see Theorem 4.1, 4.2 and Remark 9). A similar result was obtained in [28] for a more restrictive class of initial data functions u 0 than those considered in (A 1 )-(A 2 ). Moreover, the limiting solution we exhibit satisfies a steady-boundary problem that corresponds to a parabolic problem with a different diffusion function in the half lines (−∞, 0), (0, ∞), and a connection condition at the point x = 0. Againg, we think this is an interesting problem by itself. Finally, in Section 5 we study the singular limit as ε → 0 in (5) for initial data functions u 0 satisfying (H 2 ) and one between In this situation, we state again some results on the structure of the limiting points of the approximated solutions to (5) (see Propositions 9 and 10). In particular, we give sufficient conditions which imply the existence of domains where the limiting function u takes values only in one of the two stable phases S 1 and S 2 . Obviously, it would be very interesting to prove that u is a two-phase solution to the TP-problem (6) with a moving interface ξ. However, although we can not prove such a result, we think that in some sense the conclusions we draw go in this direction.

2.
Uniqueness. In this section we analyze some properties of the two-phase entropy solutions to problem (6) introduced in the previous section. In particular, we prove a uniqueness result that generalizes the uniqueness theorem proved in [21] in the case of a piecewise linear diffusion function φ.
We assume the following technical condition for the initial datum u 0 : Observe that condition (H 3 ) is a little different from that considered in [21], where the claim is that the functions φ(u 0 )(·) − A, φ(u 0 )(·) − B change sign at most a finite number of time in [ω 1 , ω 2 ], here we consider a condition that controls the number of zeros of the functions φ(u 0 )(·) − A, φ(u 0 )(·) − B that is a little stronger, this is done only to avoid further technical difficulties. On the other hand in (H 3 ) the condition on the number of zeros of the function φ(u 0 )(·)−A is only in [ω 1 , 0] and that of the function φ(u 0 )(·) − B is only in [0, ω 2 ]. Following the proof of uniqueness given in [21], we see that there is no technical reason to check the number of change of sign of the function φ( As observed in [20], there is an analogy between the two-phase entropy solutions given in Definition 1.1 and piecewise regular entropy solutions of scalar conservation laws. In some sense, the weak formulation (11) gives a Rankine-Hugoniot condition characterizing the evolution of the interface ξ, whereas the weak entropy inequalities (13) assure local admissibility conditions along the interface. More precisely, assuming regularity of the solution up to the boundary, it is proved in [8], [20], that for every entropy solution (u, v, ξ) to problem (6) the interface moves (ξ (t) = 0) only if v(ξ(t), t) = A or v(ξ(t), t) = B. Unfortunately, since the parabolic equation degenerates when u = b or u = c, in general we can not define the trace of the function v x along the boundary of the domains V 1 and V 2 , as observed in Remark 1. To overcome such a difficulty, we will establish a "Rankine-Hugoniot" and a local entropy condition along the interface in a weaker sense. These results could be obtained by the general theory of divergence-measure vector fields. Anyway, in this simple case we prefer to prove them directly.
It is clear that inequality (19) (respectively, (20)) gives informations about the jump of "the trace" of the original vector field (w, h) on the interface γ.
Next, let us consider a two-phase solution (u, v, ξ) of problem (6), hence the corresponding vector field (w, h) = (u, −v x ). Since in this case we haveh(y, t) = −ξ (t)u(y + ξ(t)) − v x (y + ξ(t), t), by (11) and (19)- (20) we deduce the following condition Remark 3. Suppose that there exists the trace of the function v x on the interface ξ(t) in a standard way. Then we obtain the classical Rankine-Hugoniot condition considered in [8,21], more precisely For this reason we call (21) the generalized Rankine-Hugoniot condition for problem (6).
Regarding admissibility conditions for the evolution of the interface ξ(t), we have to consider the entropy inequalities (13) with the vector field (w, h) = (G(u), − [F (v)] x ); here for every g ∈ C 1 (R), g ≥ 0, G(λ) is defined in (12) and F (λ) is any primitive of g(λ). In this case we haveh(y, , and by (20) for any interval (t 1 , t 2 ) ∈ (0, T ) there holds In some sense we can think that (22) is an entropy condition on the interface γ. By (22) we can prove the following result, that is analogous to that obtained in [20], [21], [8] when the trace of v x exists in the classical sense.
Proposition 1. Let (u, v, ξ) be a two-phase solution of problem (6). Then Proof. It is enough to prove that Let us prove i). Suppose that at some time t we have v(ξ(t), t) = B * < B and ξ (t) < 0. It is not restrictive to assume ξ(t) = 0. Let ρ > 0 be such that 0 < B * +ρ < B. By the continuity of ξ and v, there exists an interval (t 1 , t 2 ) containing t, and r > 0 such that ξ (t) < 0 for every t ∈ (t 1 , t 2 ) and v(x, t) ≤ B * + ρ for every (x, t) ∈ (−r, r) × (t 1 , t 2 ). Let us consider a nondecreasing function g ρ such that Since v ∈ C(Q) and u = s i (v) in V i for i = 1, 2 (see Definition 1.1-(ii)), there exist the traces of the function u along the interface ξ(t) (from the right and left side). Let us set u 1 (t) := u(ξ(t) − , t) and u 2 (t) := u(ξ(t) + , t). Then, inequality (22) Observe that by construction (24) gives a contradiction. Analogously we can prove ii).
Let us also state another consequence of the entropy inequalities (13).
the constant A and B being defined in assumption (H 1 ).
Proof. Let M ∞ be the constant defined in (25). Consider any Let G ∞ denote the function in (12) associated with g ∞ . Then, by such a choice of g ∞ and G ∞ , for every t ∈ (0, T ) The first estimate is obtained using the entropy inequalities (13), for example using a family of test functions s ≥ t and letting n goes to +∞. The last equality in the above estimate following by the definition of M ∞ and G ∞ .
Thus, the claim follows by the arbitrariness of t ∈ (0, T ).
Proof. The claim easily follows by Proposition 1. In fact, suppose that there exists t such that v(ξ(t),t) ∈ (A, B). Then, by the continuity of v and (23), there would exist an interval (t 1 , t 2 ) such that ξ (t) = 0 for every t ∈ (t 1 , t 2 ). This contradicts the hypothesis and concludes the proof. Lemma 2.2. Let (u, v, ξ) be a two-phase solution of problem (6). Suppose that there exists an interval (0, t) such that for any subinterval (t 1 , Proof. Suppose that v(0, 0) = B. Then, using Lemma 2.1 and the admissibility conditions in (23), we deduce that ξ (t) ≤ 0, v(ξ(t), t)) = B, u(ξ(t) − , t)) = b and u(ξ(t) + , t)) = d, for every t ∈ (0, t). Moreover, by hypothesis (H 3 ), there exists c > 0 such that φ(u 0 ) − B has a given sign in (0, c) ⊂ (0, ω 2 ). Arguing by contradiction, suppose that φ(u 0 ) < B in (0, c). Then, by the continuity of v there existst ≤t such that v(c, t) = φ(u(c, t)) < B and u(c, t) < d for every t ∈ (0,t). In particular, by the maximum principle we deduce that u ≤ d, Let us prove that this contradicts the generalized Rankine-Hugoniot condition (21). In fact, given any interval (t 1 , t 2 ) ⊂ (0,t) such that ξ (t) < 0 for every t ∈ (t 1 , t 2 ), by (21) and since v(ξ(t), t) = B for every such t, we takes values in the first stable phase S 1 of the graph of φ. The above contradiction ensures that φ(u 0 ) > B in (0, c). Hence, using again the maximum principle, the result follows (we omit the details). The case v(0, 0) = A is analogous. Now we can state the following uniqueness theorem. Proof. Let (u 1 , v 1 , ξ 1 ) and (u 2 , v 2 , ξ 2 ) be two two-phase solutions of problem (6). Then, by (11) we have equation (26) reads Concerning the right-hand side of (28) we have Let us analyze the left-hand side of (28). To this aim, we will distinguish between two different cases. Case 1. First, suppose that ξ 1 (t) = ξ 2 (t) for every t ∈ (0, T ). Let V 1 and V 2 be the two regions of Q T separated by the interface ξ(t) (see Definition 1.1-(ii)). Observe that by (7) there holds in both V 1 and V 2 . Then, by (28), (29), and (30), we have By Case 1, we have uniqueness if τ = T . Suppose τ < T , using standard methods of the maximum principle for parabolic equation (see [18]) u(·, τ ) satisfies again condition (H3), then it is not restrictive to assume Then, in view of Proposition 1 we have ξ 1 (t) = ξ 2 (t) = 0 for every t ∈ (0, τ * ), namely a contradiction since we have supposed τ = 0. Therefore, let us assume that v(0, 0) = B, the case v(0, 0) = A being analogous. Set For every i = 1, 2, let τ i ∈ (0, T ) be such that Here we set τ i = 0 if there not exists any positive time such that (32) is satisfied.
Since we have assumed τ = 0, necessarily at least one between τ 1 and τ 2 is positive.
In particular, in view of Lemma 2.2 we can suppose that τ 1 and τ 2 are small enough so that Using the test function the right-hand side in (28) (written in Qτ := Ω × (0,τ )) is nonpositive. Therefore, we have As in Case 1, we observe that ( In the first case, by Lemma 2.2 and (34), we obtain The second case is symmetric. The above arguments combined with (37) imply that for every k = 0, 1, 2 there holds Arguing by contradiction, assumeṼ 0 = ∅ and u 1 = u 2 , v 1 = v 2 inṼ 0 . Observe that by (33)-(34) only the possibility v 1 = v 2 = B inṼ 0 is allowed. Then, there would exist i ∈ {1, 2} and r > 0 such that u i ≡ d in the nonempty set V i 2 ∩Ṽ 0 , namely an absurd by the maximum principle (see the proof of Lemma 2.2). Therefore, we haveṼ 0 = ∅ and the conclusion follows.
Remark 4. Arguing as in Case 1 of the previous proof, it can be easily seen that, whenever ξ 1 (t) = ξ 2 (t) for every t ∈ (0, T ), in order to have uniqueness of two-phase solutions to problem (6), we do not need neither the condition v ∈ C(Q T ) nor the condition (H3). Such an information can be useful when we expect that ξ (·) ≡ 0.
Remark 5. By the same arguments used in the proof of Theorem 2.3, we can also obtain stability results with respect to the initial data function u 0 . More precisely, given two different initial data u 01 , u 02 , and denoting by (u i , v i , ξ i ), i = 1, 2, the corresponding two-phase solutions to problem (6), we have for every ψ ∈ C 1 (Q T ), ψ(·, T ) = 0 in Ω. Using the same test function defined in (27) and inequality (29) we obtain Defining as in (31) . Then, by (25) and (40) we deduce where |φ(u 02) |}, |u 02 |}.
By (41) we obtain the following stability estimate where Moreover, arguing as in Case 2 in the proof of Theorem 2.3, there exists τ ∈ (0, T ) such that FLAVIA SMARRAZZO AND ANDREA TERRACINA Then, definingṼ k , k ∈ {0, 1, 2} as in (38), we have In order to conclude the section, let us show that the parabolic equation (1) can not degenerate in some specific regions depending on the initial data function u 0 . More precisely, the following result holds.
Moreover, by the condition u 0 (x 0 ) < b and the continuity of u 0 in x 0 , it is possible to find 0 < r 1 < d 0 and ε > 0 such that (46) In particular this means that u satisfies the following problem   Since by assumption (H 1 ) we have φ (b) = 0 but φ (b) = 0, by a standard comparison principle there holds u(x 0 , t) < b (e.g., see [7] and Section 7 in [31]). Next, fix any t such that (x 0 , t) ∈ V 1 but assume that there existst ∈ (0, t) such that (x 0 ,t) ∈ V 1 . By such an assumption there also exists t 1 ∈ (0,t) such that ξ(t 1 ) ≤ x 0 . In this situation the set S = {t ∈ [0, t] : ξ(t) = x 0 } is nonempty and also compact by the continuity of ξ. Set t * := max S: observe that obviously we have 0 < t * < t, since both (x 0 , 0) and (x 0 , t) belong to V 1 . Moreover, Claim ξ (t * ) ≥ 0 and there exists η > 0 such that ξ (s) ≥ 0 and v(ξ(s), s) ≡ A for every s ∈ (t * , t * + η). In fact, assuming that ξ (t * ) < 0, we would have ξ(s * ) < x 0 for some s * ∈ (t * , t). Therefore, since (x 0 , t) ∈ V 1 (hence x 0 < ξ(t)) there would exists ∈ (s * , t) such that ξ(s) = x 0 , namely a contradiction by the definition of t * . Cleraly, if we suppose that ξ (t * ) > 0 we obtain easily the proof of the claim. Finally, let consider the case ξ (t * ) = 0. Then, by Definition 1.1-(i) we have two possibilities. In the first case there exists η > 0 such that ξ(·) ≡ x 0 in (t * , t * + η), which contradicts the definition of t * . Otherwise, there exists η > 0 such that ξ (·) ≥ 0 in (t * , t * + η) with ξ not identically zero in any interval (s 1 , s 2 ) ⊂ (t * , t * + η). It can be easily seen that in this case the claim is a consequence of Lemma 2.1. Now we can conclude the proof. In fact, if u(ξ(t), t) = b, then by the first part of the proof we have necessarily u(ξ(t), t) = b for every t ∈ [t * , t]. In particular, by the continuity of v we obtain v(ξ(t * ), t * ) = B, which contradicts the Claim.
3.1. The case ε > 0. For any ε > 0 let us consider the pseudoparabolic regularization of the Neumann initial-boundary value problem associated with equation where the chemical potential v is defined by setting In this section, concerning the initial data function u 0 we will only assume that u 0 ∈ L ∞ (Ω).  49) is a function u ε ∈ C 1 ([0, T ]; L ∞ (Ω)) such that: Remark 6. Let us observe that since v ε ∈ C([0, T ]; W 2,∞ (Ω)), then v ε ∈ C(Q T ) and there exists the partial derivative v ε x in Q T . In fact, for every (x, t) ∈ Q T we have The following well-posedness result was proven in [22]. Theorem 3.2. For any u 0 ∈ L ∞ (Ω) and ε > 0 there exists a unique solution u ε of (48)-(49). Moreover, for every t ∈ [0, T ] the function v ε ( . , t) solves the problem: In the following theorem we investigate the regularity properties of the chemical potential v ε , in particular the existence of the time partial derivative v ε t . Theorem 3.3. For any u 0 ∈ L ∞ (Ω) and ε > 0 let u ε be the solution of problem (48)-(49). Then, v ε ∈ C 1 (Q T ) and there holds: (ii) for every (x, t) ∈ Q T there exists the partial derivative v ε xt (x, t). Moreover, denoting by v ε tx the partial derivative of v ε t with respect to x -whose existence is ensured by (see also Remark 6). In addition, let us prove that there exists the partial derivative v ε t in Q T and that v ε t ∈ C(Q T ). This will ensure that v ε ∈ C 1 (Q T ). To this purpose, fix any t ∈ (0, T ) (the cases t = 0 and t = T being analogous). Then, in view of (51) for any h such that 0 < t + h < T the function is the unique solution of the problem where Multiplying the first equation in (55) by w h and by −w hxx , and using the boundary condition w hx = 0 on ∂Ω we obtain respectively From the above inequalities we easily deduce for any ε < 1. Since u ε ∈ C 1 ([0, T ]; L ∞ (Ω)) -hence also φ(u ε ) ∈ C 1 ([0, T ]; L ∞ (Ω)) -it can be easily proved that As a consequence of the above convergence and inequality (56), the family {w h } is uniformly bounded in H 2 (Ω), hence possibly up to a sequence {h n }, h n → 0, there holds: for some w ∈ H 2 (Ω). Moreover, in the light of the above convergences and (57), taking the limit as n → ∞ in problems (55) we obtain that the limiting function w solves (in the strong sense) the problem Since the above problem has a unique solutionw ∈ H 2 (Ω), it follows that w =w in Ω -hence also w =w ∈ W 2,∞ (Ω) since [φ(u ε )] t (·, t) ∈ L ∞ (Ω) -and the whole family {w h } converges tow in the weak sense of H 2 (Ω). In particular this implies that there exists for every x ∈ Ω. Therefore, for every such x there exists the partial derivative v ε t of v ε with respect to t at the point ( and v ε t (·, t) is the unique solution of problem (52) (see (58) above). Moreover, since φ(u ε ) t ∈ C([0, T ]; L ∞ (Ω)), by means of (52) and standard results on elliptic equations, arguing as above it can be easily proved that v ε t ∈ C([0, T ]; W 2,∞ (Ω)), thus v ε t ∈ C(Q T ) (see also (53)). (ii) To begin with, observe that the condition v ε t ∈ C([0, T ]; W 2,∞ (Ω)) implies that for every (x, t) ∈ Q T there exists the partial derivative v ε tx (x, t) = [v ε t (·, t)] x (x) and v ε tx ∈ C(Q T ) (see also Remark 6). Next, fix any t ∈ (0, T ) (the cases t = 0 and t = T can be treated in a similar way) and observe that for every x ∈ Ω and for any h such that where w h is the function in (54). Then, since we have proved in (i) above the strong convergence w hx → [v ε t (·, t)] x ≡w x in C(Ω) (w ∈ W 2,∞ (Ω) being the unique solution of problem (58)), it follows that for every x ∈ Ω there exists the partial derivative v ε xt (x, t) at the point (x, t) and This concludes the proof.

Remark 7.
Observe that in the light of the equality εu ε t = v ε − φ(u ε ), and since both the functions φ(u ε ) and v ε admit weak partial derivative with respect to time belonging to the space C([0, T ]; L ∞ (Ω)) (see Definition 3.1 and Theorem 3.3), we also have For any ε > 0 the solution (u ε , v ε ) to problem (48)-(49) satisfies a family of viscous entropy inequalities, this parlance being suggested by a formal analogy with the entropy inequalities for viscous conservation laws ( [8,21,22,24]).
Step (α). To begin with, fix any g ∈ C 1 (R), g ≥ 0 and let G be the function defined by (12) in correspondence to g. Then for every such g, for every ψ ∈ C 1 (Q T ), ψ ≥ 0, and for every t 1 , By the above inequality and the boundary condition v ε for every g, ψ as above.
Theorem 3.4. For any u 0 ∈ L ∞ (Ω) and ε > 0 let u ε be the solution to problem (48)-(49). There holds: Proof. Let us prove only (i), the proof of claim (ii) being analogous. In this direction, fix any t ∈ (0, t 0 ], any ϕ ∈ C 2 c ((ω 1 , x 0 )), ϕ ≥ 0, and observe that for a such ϕ the viscous entropy inequalities (60) give for any g ∈ C 1 (R), g ≥ 0 and G defined by (12). By standard regularization arguments the assumption g ∈ C 1 (R) can be dropped and inequalities (81) hold for any nondecreasing g. Following [24], for any ρ ≥ 0 we set: Let G ρ be the function defined by (12) in correspondence to g ρ and k = 0. Choosing g = g ρ in inequalities (81) we have for every ϕ as above. Let us study the different terms of the previous inequality in the limit as ρ → 0. First, since v ε ≤ B in (ω 1 , x 0 ) × (0, t) by (71), there holds Next, we consider the second term in the left-hand side of (83). To begin with, observe that by the assumption φ (b) = 0, there exists a constant C > 0 such that for any ρ > 0 there holds for any ρ < ρ * . On the other hand, for every x ∈ (ω 1 , x 0 ) such that u 0 (x) = b, there holds In the light of the above considerations and since lim ρ→0 (here use of the assumption u 0 ≤ b in (ω 1 , x 0 ) has been made). Finally, let us study the first term in the left-hand side of (83). In this direction, we decompose the function G ρ (u ε (x, t)) ϕ(x) in the following way: recall that in view of (70) we have φ(u ε (·, t)) ≤ B -hence u ε (·, t) ≤ d = s 2 (B)almost everywhere in (ω 1 , x 0 ). Arguing as above, passing to the limit as ρ → 0 in the first two terms of the right-hand side of (86) gives The third term in the right-hand side of (86) can be decomposed as follows: then by the Lebesgue theorem lim ρ→0+ I ρ 1 = 0. Moreover then, using the assumption s 2 (B) = 1/φ (d) < ∞, we have lim ρ→0 I ρ 2 = 0. Therefore, in the light of the above estimate and (90), we obtain Using (87), (88) and (91) we deduce Combining (84), (85) and (92), the limit as ρ → 0 in inequalities (83) gives for every ϕ ∈ C 2 c ((ω 1 , x 0 )), ϕ ≥ 0, and for every t ∈ (0, t 0 ]. To conclude the proof, fix any t ∈ (0, t 0 ] and, arguing by contradiction, assume that the set has a nonzero Lebesgue measure. Let K ⊂ {x ∈ (ω 1 , x 0 ) | u ε (x, t) > b} be any compact set with a strictly positive Lebesgue measure, |K| > 0, and let {ϕ n } ⊆ C ∞ c (R) be any sequence of smooth functions such that: (i) 0 ≤ ϕ n (x) ≤ 1 for every x ∈ R, n ∈ N; (ii) ϕ n (x) = 1 for every x ∈ K, n ∈ N; (iii) ϕ n (x) = 0 for every x ∈ R such that dist (x, K) ≥ 2/n. Write inequality (93) for ϕ = ϕ n and then take the limit as n → ∞. Since ϕ n → χ K in L 1 (R) as n → ∞, using the assumption u 0 ≤ b in (ω 1 , x 0 ) we obtain The above contradiction gives the claim.
3.2. The vanishing viscosity limit: preliminary results. In this section we give some preliminary results concerning the vanishing viscosity limit ε → 0 in the approximating problems (48)-(49), when u 0 is any initial datum to problem (6) that belongs to the space L ∞ (Ω). The case of initial data functions satisfying assumption (H 2 ) will be considered in more detail in Sections 4 and 5. In this direction, as a first step we have the following theorem, which is a consequence of (62)-(64) and of a characterization of the weak limit proved in [24] (see [20] for a more detailed proof).
In view of the nonmonotone character of φ, in general the relation v = φ(u) need not hold in the general case of arbitrary initial data u 0 ∈ L ∞ (Ω). In this regard, we recall the result given in [24]: precisely there exist λ 0 , λ 1 , for almost every (x, t) ∈ Q T , where s i (v) denote the three roots of the equation v = φ(u), i = 0, 1, 2 (see (2)-(4)); moreover, In other words, for arbitrary initial data u 0 ∈ L ∞ (Ω), the limiting function u is a superposition of different phases and the coefficients λ i (i = 0, 1, 2) can be regarded as phase fractions (see also [8,20,27,29]). Anyway, a natural question is whether under suitable assumptions on the initial datum u 0 we can recover or not the condition v = φ(u): observe that in this case the couple (u, v) would be a weak solution of equation (1). As already remarked, the answer is affirmative for initial data u 0 satisfying assumption (H 2 ) and the additional constraint a ≤ u 0 ≤ d in Ω ( [21,28,30]). In what follows, we will improve such a result, proving local existence of two-phase solutions to problem (6) in some specific cases, without the additional condition a ≤ u 0 ≤ d in Ω. To this aim, we will consider assumptions (A 1 )-(A 2 ) and under these conditions we will prove such an existence result (see Section 4 below).
4. Assumptions (A 1 ) and (A 2 ): Vanishing viscosity limit and local existence of two-phase solutions. To begin with let us suppose that, besides assumption (H 2 ), the initial data function u 0 satisfies also the condition (A 1 ). The case of assumption (A 2 ) will be taken under consideration at the end of this section.
Let x 0 ∈ [0, δ) be as in assumption (A 1 ), and let {ε k } be the vanishing sequence given in Theorem 3.5. Then, in view of (69) there exist k 0 ∈ N, A 0 , B 0 ∈ R such that for every k ≥ k 0 there holds Then, we define here we set τ k . By the continuity of v ε k in the cylinder Q T and in the light of (100)-(101), for every k sufficiently large we deduce For every k ∈ N set τ k := min{τ k A , τ k B }.
Next, we proceed to estimate from below the value τ k with a constant τ > 0 independent of k. In this direction, we begin by the following proposition.
As a consequence of the above proposition, we can now prove that the sequence {τ k } is uniformly bounded from below by a constant τ > 0. In particular, this ensures that the limiting functions u and v in Theorem 3.5 are related by the condition v = φ(u), that the function v is smooth in the cylinder Ω × (0, τ ), and that the inequalities in (111)-(112) carry over to the function u in Ω × (0, τ ). This is the content of the following main theorem.
Theorem 4.1. Let assumptions (H 2 ) and (A 1 ) hold. Let u, v be the limiting functions in Theorem 3.5 and let τ k be the defined in (106) for every k ∈ N. Then, there exists τ > 0 such that for every k sufficiently large. Moreover, v ∈ L ∞ ((0, τ ); Finally, there holds s 1 (v) and s 2 (v) being defined in (2) and (3), respectively.
To conclude the proof, observe that combining (111) and (112) with (129) we get for every k large enough, where s 1 and s 2 are defined in (2) and (3), respectively. Therefore characterization (125) follows from the previous equality and the convergence in (96).

Remark 9.
Observe that time τ defined in (129) does not depend on the sequence ε k , this is consequence of Assumption (A 1 ) and the uniform convergence (69) Let us also give a brief description of the results obtained in the case when assumption (A 2 ) holds. In this direction let {ε k } be the vanishing sequence given in Theorem 3.5, and fix any y 0 ∈ (−δ, 0] such that [φ(u 0 )](y 0 ) > A. Thus, in view of (69), for every k sufficiently large we have Therefore we have For every k ∈ N set the value η k being strictly positive for every k large enough in view of the above remarks. Proving the analogous of both Proposition 7 and Theorem 8, we can argue as in the first part of this section and we obtain the following result.
Finally, observe that in the light of Theorems 4.1-4.2 the following existence result holds.
Proof. Let us suppose that assumptions (H 2 ) and (A 1 ) hold, the case of assumption (A 2 ) following by similar arguments. Let (u, v) be the limiting functions given in Theorem 3.5 and let τ > 0 be the value considered in Theorem 4.1. Then, in the light of the relation (125), the triple (u, v, ξ), ξ(t) = 0, satisfies the properties asserted in (i)-(iii) of Definition 1.1 in Q τ := Ω × (0, τ ). Therefore let us show that (u, v) satisfies the entropy inequalities (13). To this aim, fix any g ∈ C 1 (R), g ≥ 0, and let G be the function defined in (12). Let {ε k } be the vanishing sequence given in Theorem 3.5 and observe that, possibly extracting another subsequence denoted again {ε k } for simplicity, we can also assume that almost everywhere in Q τ (see (96)). Next, for any ε k write the viscous entropy inequalities (60) in correspondence to g, and then consider the limit as k → ∞ in such inequalities. By (130) and (141) it easily follows that almost everywhere in Q τ (see also (125)). Moreover, in view of (97) and (124) there holds As a consequence of (142)-(144), taking the limit as ε k → 0 in the viscous entropy inequalities (60) gives the claim.

Remark 10.
Observe that from the uniqueness Theorem 2.3, Remark 4 and Remark 9 the two-phase solution (u, v, ξ) given by the above theorem in Ω × (0, τ ) is also the unique two-phase solution to problem (1). Moreover, by uniqueness, given any sequence ε k → 0 + , we have that u ε k converges to u and φ(u ε k ), v ε k converge to v in the sense of Theorem 3.5.
5. The general case: Some improved results about the vanishing viscosity limit. In this section, besides (H 2 ), we will assume that either the assumption holds. Let us begin by (A 3 ), the case of (A 4 ) being treated at the end of this section. To this aim, let {ε k } be the vanishing sequence given in Theorem 3.5 and let δ > 0 be the value whose existence is ensured by assumption (A 3 ), so that φ(u 0 ) ∈ (A, B) in (−δ, 0) and φ(u 0 ) ≥ B in [0, δ). For every k ∈ N and for every x ∈ [ω 1 , 0] set where Again it is not restrictive to assume τ k A , τ k B (x) < T . Clearly, for every such x and for every k such that τ k and (148) Moreover, the following proposition is a consequence of Proposition 6, Theorem 3.4, the condition φ(u 0 )(0) = B (hence φ(u 0 )(0) > A), and the convergence in (68).
(153) (iii) There exists k 0 ∈ N such that for every k ≥ k 0 there holds (iv) For every x ∈ (−δ, 0) there exists k x ∈ N such that for every k ≥ k x there holds Proof. Claims (i) and (ii) are a consequence of Proposition 6 and Theorem 3.4. To prove claim (iii) it suffices to combine the uniform convergence of v ε k (·, 0) to φ(u 0 ) in Ω (see (69)) and the condition φ(u 0 )(0) = B > A in assumption (A 3 ). Finally, fix any x ∈ (−δ, 0). Then, since φ(u 0 )(x) < B by assumption (A 3 ), and in the light of the uniform convergence (69), there exists k x ∈ N -in general depending on x -such that v ε k (x, 0) < B for every k ≥ k x . Hence, by the very definition of τ k B (x) in (145) claim (iv) follows. This concludes the proof.
Observe that in the light of (151) and Proposition 9-(iv), for every x ∈ (ω 1 , 0) we can find an index k x ∈ N such that τ k B (x) > 0 for every k ≥ k x . To prove it, we can fix any x * ∈ (−δ, 0), x * > x, and observe that the claim is a consequence of the inequality τ k B (x) ≥ τ k B (x * ), ensured by (151). By the above proposition, for every k ∈ N the map x → τ k B (x) is monotone nonincreasing. Then, claim (iv) of Proposition 9 guarantees that the function τ k B is not identically zero whenever k is large enough. In fact, it suffices to choose any x ∈ (−δ, 0), fix the corresponding k x ∈ N -so that τ k B (x) > 0 for every k ≥ k x -and observe that (151) ensures that τ k B (y) > 0 for every k ≥ k x and for every y ≤ x. Moreover, in view of such a monotone character, and since |τ k B (ω 1 ) − τ k B (0)| ≤ T for every k, the sequence {τ k B } is bounded in the space BV (ω 1 , 0) of functions with bounded total variation in the interval (ω 1 , 0). Therefore, eventually up to a subsequence, denoted again {τ k B } for simplicity, there exists a nonicreasing function τ B ∈ BV (ω 1 , 0), τ B L ∞ (ω1,0) ≤ T , such that for almost every x ∈ (ω 1 , 0) there holds as k → ∞. In what follows we will always identify the BV function τ B with its leftcontinuous representative. In an analogous way, we can also apply similar arguments to the sequence {τ k A }. Then, eventually up to a subsequence -denoted again {τ k A } for simplicity -there exists τ A ∈ [0, T ] such that as k → ∞.
Once introduced the function τ B ∈ BV (ω 1 , 0) in (157) and the value τ A in (158), it is reasonable to expect that, when τ A = 0 and τ B is not identically zero in (ω 1 , 0), the properties asserted in (149)-(150) and in (152)-(153) carry over to the limiting functions u and v given in Theorem 3.5, with the sequences {τ k B (x)} and {τ k A } replaced by τ B (x) and τ A , respectively. In this direction, we begin by the following proposition. Then, for every x ∈ [ω 1 , x 0 ) there holds Moreover, we also obtain τ A > 0.
As a consequence of (157) and (160) it follows that for every x ∈ (ω 1 , x 0 ), where x 0 ∈ (−δ, 0] has the property asserted in (159). In fact (174) holds for every x ∈ (ω 1 , x 0 ) \ S * , where S * is a null set in (ω 1 , 0) whose elements are the points where τ B is discontinuous or the points that we have to exclude in the pointwise convergence (157). Thus, since we have chosen the leftcontinuous representative of the nonicreasing function τ B ∈ BV (ω 1 , 0), for every x ∈ S * ∩ (ω 1 , x 0 ) we can fix any sequence {y n } ⊆ (ω 1 , x) \ S * , y n → x − , and get τ B (x) = lim n→∞ τ B (y n ) ≥ τ * , namely inequality (174). Then, assume that (159) holds and consider the sets In view of Proposition 10 we can now give a finer description of the structure of the limiting functions u and v in Theorem 3.5.
Proof. Let us first prove that u(x, t) = s 1 (v(x, t)) for almost every (x, t) ∈ U 1 . To this aim, let us recall that there exists a null set E * ⊆ Q such that for every (x, t) ∈ Q T \ E * there holds as k → ∞ (see (96)). Moreover, by (157) there exists a null set S * ⊆ (ω 1 , 0) such that for every x ∈ (ω 1 , 0) \ S * there holds as k → ∞. Finally, for every k ∈ N, let F k ⊆ Q T be a null set such that every (x, t) ∈ Q T \ F k is a Lebesgue point for both the functions u ε k and φ(u ε k ). Set N * := E * ∪ (S * × (0, T )) ∪ F * , F * := ∪ ∞ k=1 F k . Then, fix any (x, t) ∈ U 1 \ N * and observe that for every σ ∈ (0, τ B (x) − t) there exists k x,σ ∈ N such that for every k ≥ k x,σ we have Using (150) and the condition that (x, t) is a Lebesgue point for u ε k , the above inequality implies u ε k (x, t) = s 1 (φ(u ε k (x, t))) for every such k. Therefore, in view of (179) there exists lim k→∞ u ε k (x, t) = s 1 (v(x, t)). Namely by the arbitrariness of (x, t) ∈ U 1 \ N * and since the sequence {u ε k } is uniformly bounded in L ∞ (Q T ) (also observe that the set N * ⊆ Q T has zero Lebesgue measure). The above convergence and (94) give (178). By similar arguments, we can also prove that u(x, t) = s 2 (v(x, t)) for almost every (x, t) ∈ U 2 , hence characterization in (177) follows. Let us also observe that in the light of (160), for every x ∈ (ω 1 , x 0 ) and for every τ < lim inf k→∞ τ k B (x) we obtain τ k B (x) > τ for every k large enough. Such a consideration allows us to conclude that the estimate in (162) carries over to the limiting function v in the set (ω 1 , x) × (0, τ ), and this implies claim (i). Finally, as a consequence of (161), for every τ < τ A and for every k sufficiently large there holds τ k A > τ , namely claim (ii) by (164).
Observe that the above result is an improvement of the characterization (99) of u showed in [24]. Unfortunately, Theorem 5.2 does not give any further information (with respect to the equality in (99)) about the relation between u and v in the region U 0 := Q T \ {U 1 ∪ U 2 }. In other words, we can not prove that u = s 2 (v) (or u = s 1 (v)) in U 0 , and this inconvenience enables us to conclude that the triple (u, v, τ B (x)) (or (u, v, τ A )) is a local two-phase solution to problem (6).
Finally, in the case of assumptions (H 2 ) and (A 4 ), invoking arguments analogous to those used above in the proof of Theorem 5.2, we obtain the following theorem.
Finally, let us prove Lemma 5.1.
Proof of Lemma 5.1 Let us prove only claim (i), the proof of (ii) being formally analogous.