WELL-POSEDNESS AND LONGTIME BEHAVIOR FOR A SINGULAR PHASE FIELD SYSTEM WITH PERTURBED PHASE DYNAMICS

. We consider a singular phase ﬁeld system located in a smooth bounded domain. In the entropy balance equation appears a logarithmic non- linearity. The second equation of the system, deduced from a balance law for the microscopic forces that are responsible for the phase transition process, is perturbed by an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. We prove existence and uniqueness of the solution for this resulting highly nonlinear system. Moreover, under further assumptions, the longtime behavior of the solution is investigated.


1.
Introduction. This paper is devoted to the mathematical analysis of a system of partial differential equations (PDE) arising from a thermodynamic model describing phase transitions. The system is written in terms of a rescaled balance of energy and of a balance law for the microforces that govern the phase transition. Moreover, the second equation of the system is perturbed by the presence of an additional maximal monotone nonlinearity. This paper will focus only on analytical aspects and, in particular, will investigate existence, uniqueness and longtime behavior of the solution. In order to make the presentation clear from the beginning, let us briefly introduce the main ingredients of the PDE system and give some comments on the physical meaning.
We consider a two-phase system located in a smooth bounded domain Ω ⊆ R 3 and let T > 0 denote a final time. The unknowns of the problem are the absolute temperature ϑ and a phase parameter χ which may represent the local proportion of one of the two phases. To ensure thermomechanical consistency, suitable physical constraints on χ are introduced: if it is assumed, e.g., that the two phases may coexist at each point with different proportions, it turns out to be reasonable to require that χ lies between 0 and 1, with 1 − χ representing the proportion of the second phase. In particular, the values χ = 0 and χ = 1 may correspond to the pure phases, while χ is between 0 and 1 in the regions when both phases are present. Clearly, the model should provide an evolution for χ that complies with the previous physical constraint. Now, let us state precisely the equations as well as the initial and boundary conditions. The two equations governing the evolution of ϑ and χ are recovered as balance laws. The first equation is obtained as a reduction of the energy balance equation divided by the absolute temperature ϑ (see [7, formulas (2.33)-(2.35)]). Hence, the so-called entropy balance can be written in Ω × (0, T ) as follows: where k 0 > 0 is a thermal coefficient for the entropy flux, is a positive parameter and F stands for an external entropy source. We point out that in the previous equation one finds the entropy flux Q, related to the heat flux vector q by Q = q/ϑ, and specified by Q(t) = −k 0 ∇ϑ(t), t ∈ (0, T ). Moreover, due to the presence of the logarithm of the temperature in the entropy equation (1), the positivity of the variable representing the absolute temperature follows directly from solving the problem, i.e., from finding a solution component ϑ to which the logarithm applies. This is important, since we can avoid the use of other methods, or the setting of special assumptions, in order to guarantee the positivity of ϑ in the space-time domain.
The second equation of the system under study accounts for the phase dynamics and is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. According to [21,23], this balance reads where β + π represents the derivative, or the subdifferential, of a double-well potential W defined as W =β +π, whereβ : R −→ [0, +∞] is proper, l.s.c. and convex withβ(0) = 0, π ∈ C 1 (R) and π =π is Lipschitz continuous in R.
Due to (3), the subdifferential β := ∂β is well defined and turns out to be a maximal monotone graph. Moreover, asβ takes on its minimum in 0, we have that 0 ∈ β(0). Note that in (2) the inclusion is used in place of the equality in order to allow for the presence of a multivalued β. We recall that many different choices ofβ and π have been introduced in the literature (see, e.g., [5,8,20,26]). In the case of a solid-liquid phase transition, W may be chosen in such a way that the full potential (cf. (2)) χ →β(χ) +π(χ) − ϑχ exhibits one of the two minima χ = 0 and χ = 1 as global minimum for equilibrium, depending on whether ϑ is below or above a critical value ϑ c , which may represent a phase change temperature. A sample case is given byπ(χ) = ϑ c χ and by theβ that coincides with the indicator function I [0,1] of the interval [0, 1], that is, β(ρ) = I [0,1] (ρ) = 0 if 0 ≤ ρ ≤ 1 +∞ elsewhere so that β = ∂I [0,1] is specified by r ∈ β(ρ) if and only if r Of course, this yields a singular case for the potential W, in whichβ is not differentiable, and it is known in the literature as the double obstacle case (cf. [5,8,21]) In the present contribution, we assume that the second equation (2) of the system is perturbed by the presence of an additional maximal monotone nonlinearity, i.e., where ζ(t) ∈ A(χ(t) − χ * ) for a.e. t ∈ (0, T ).
Here, χ * is a positive and smooth function (χ * ∈ H 2 (Ω) with null outward normal derivative on the boundary) and A : L 2 (Ω) → L 2 (Ω) is a maximal monotone operator satisfying some conditions, namely: A is the subdifferential of a proper, convex and lower semicontinuous (l.s.c.) function Φ : L 2 (Ω) → R which takes its minimum in 0, and A is linearly bounded in L 2 (Ω). As widely described in [3], the role of this further nonlinearity is physically meaningful in the framework of phase transition processes.
In the last decades phase field models have attracted a number of mathematicians and applied scientists to describe many different physical phenomena. Let us just recall some results in the literature that are related to our system. Some key references are the papers [6][7][8]. Besides, we quote [10], where a first simplified version of the entropy system is considered, and [9,11] for related analyses and results. Besides, let us mention the contributions [18,19], where standard phase field systems of Caginalp type, perturbed by the presence of nonlinearities similar to (6), are considered and the existence of strong solutions, the global well-posedness of the system and the sliding mode property are proved. We also refer to [14], where the author prove the existence of solutions for a system characterized by the contemporary presence of two nonlinearities in the entropy balance equation: the resulting system is highly nonlinear and the main difficulties lie in the treatment of the doubly nonlinear equation for a.e. t ∈ (0, T ).
In the first part of the present contribution we prove existence and uniqueness of the solution for the system consisting of equations (1), (5)-(6) coupled with suitable boundary and initial conditions. In particular, we prescribe a no-flux condition on the boundary for both variables: where ∂ ν denotes the outward normal derivative on the boundary Γ of Ω. Besides, in the light of (6), initial conditions are stated for ln ϑ and χ: ln ϑ(0) = ln ϑ 0 , χ(0) = χ 0 in Ω.
The second part of the paper is concerned with the asymptotic behavior of the solution to (1), (5)-(8) as t goes to +∞. Let us point out that the longtime behavior has been already investigated for related equations with memory terms in, e.g., [4]. In our framework, we assume that A = ρ Sign, where ρ is a positive coefficient, Sign :−→ 2 H is defined as and B 1 (0) is the closed unit ball of H (it is straightforward to check that Sign satisfies the properties required for the operator A). Then, we show that the ωlimit, defined as is nonempty and consists only of stationary solutions. In particular, ϑ ∞ is a constant, while χ ∞ satisfies As far as the outline of the paper is concerned, we state precisely assumptions and main results in Section 2, then introduce the time-discrete problem (P τ ) in Section 3 and completely prove existence and uniqueness of the solution. Section 4 is devoted to the proof of several uniform estimates, independent of τ , involving the solution of (P τ ). Then, in Section 5 we pass to the limit as τ 0 by means of compactness and monotonicity arguments in order to find a solution to the problem (1), (5)- (8). Finally, in Section 6 and in Section 7, respectively, we prove the uniqueness of the solution and its longtime behavior.
2.1. Preliminary assumptions. We assume Ω ⊆ R 3 to be open, bounded, connected, of class C 1 and we write |Ω| for its Lebesgue measure. Moreover, Γ and ∂ ν still stand for the boundary of Ω and the outward normal derivative, respectively. Given a finite final time T > 0, for every t ∈ (0, T ] we set We also introduce the spaces with usual norms · H , · V and inner products (·, ·) H , (·, ·) V , respectively. We identify H with its dual space H , so that W ⊂ V ⊂ H ⊂ V ⊂ W with dense and compact embeddings. Let ·, · stand for the duality pairing between V and V . The notation · p , (1 ≤ p ≤ +∞) stands for the standard norm in L p (Ω). For short, in the notation of norms, we do not distinguish between a space (or its norm) and a power thereof.
From now on, we interpret the operator −∆ as the Laplacian operator from the space W to H, then including the Neumann homogeneous boundary condition. Moreover, we extend −∆ to an operator from V to V by setting Throughout the paper, we account for the well-known continuous embeddings V ⊂ L q (Ω) (1 ≤ q ≤ 6), W ⊂ C 0 (Ω) and for the related Sobolev inequalities: for v ∈ V and v ∈ W , respectively, where C s depends on Ω only, since sharpness is not needed. We will also use a variant of the Poincaré inequality, i.e., there exists a positive constant C p such that Furthermore, we make repeated use of Hölder inequality and of Young's inequalities, i.e., for every a, b > 0, z ∈ (0, 1) and δ > 0 Besides, for every a, b ∈ R we have that Finally, we also recall the discrete version of the Gronwall lemma (see, e.g., [24, Prop.
Finally, we state another useful result for the sequel.
Lemma 2.2. Assume that a, b ∈ R are strictly positive. Then Proof. We consider a > b (if b > a the technique of the proof is analogous) and obtain Then, dividing by b, we have that Letting x = a/b, we can rewrite (26) as Now, we observe that (24) is verified if and only if the function f (x) := 2(x + 1) ln x − x + 1 is nonnegative for every x ≥ 1.
Since f (1) = 0 and f (x) > 0 for every x ≥ 1, we conclude that the proof of the lemma is complete.
In the following, the small-case symbol c stands for different constants which depend only on Ω, on the final time T , on the shape of the nonlinearities and on the constants and the norms of the functions involved in the assumptions of our statements. On the contrary, we use different symbols to denote precise constants to which we could refer. The reader should keep in mind that the meaning of c might change from line to line and even in the same chain of inequalities.
2.2. Statement of the problem and results. As far as the data of our problem are concerned, let and k 0 > 0 be two real constants. We also consider the data F , χ * , ϑ 0 and χ 0 such that Moreover, we introduce the functionsβ andπ, satisfying the conditions listed below: β : R −→ [0, +∞] is lower semicontinuous and convex withβ(0) = 0, π ∈ C 1 (R) and π is Lipschitz continuous.
(37) Indeed, thanks to the definition of the subdifferential and to (33), we have that In the following, the same symbol β will be used for the maximal monotone operators induced on H ≡ L 2 (Ω) and L 2 (0, T ; H) ≡ L 2 (Q). In our problem a maximal monotone operator also appears. We assume that A is the subdifferential of a convex and l.s.c. function Φ : H −→ R which takes its minimum in 0 and has at most a quadratic growth.
These properties are related to our assumptions on A = ∂Φ, which read In the following, the same symbol A will be used for the maximal monotone operators induced on L 2 (0, T ; H). Examples of operators A. Now, we consider the operator and its nonlocal counterpart Sign : where B 1 (0) is the closed unit ball of H. It is straightforward to check that Sign satisfies (41) and turns out to be the subdifferential of the norm function v → v H . Moreover, let us recall that the subdifferential of the convex function v → Ω |v| is a maximal monotone operator from L ∞ (Ω) to 2 L ∞ (Ω) defined as in (45).
(55) In order to obtain a variational formulation of Problem (P ), from (50) and (54) we infer that Theorem -(Existence and uniqueness) 2.3. Assume (29)-(41). Then problem (P ) stated by (50)-(55) has a unique solution (ϑ, χ, ξ) satisfying (46)-(49) and the regularity properties Theorem -(Longtime behavior) 2.4. Assume (29)-(41). In addiction, if χ * is constant, ρ is a positive parameter, A = ρ Sign and then the ω-limit, defined as is nonempty and consists only of stationary solutions. In particular, ϑ ∞ is a constant, while χ ∞ satisfies 3. The approximating problem (P τ ). The following three sections are devoted to the proof of Theorem 2.3. In order to obtain this result, we introduce a backward finite differences scheme. Assume that N is a positive integer and let Z be any normed space. By setting the time step τ = T /N we introduce the interpolation , we define the piecewise constant functions z τ and the piecewise linear functions z τ , respectively: if 0 < s < 1 and i = 0, · · · , N − 1. We also define the operator for (z 0 , z 1 , · · · , z N ) ∈ Z N +1 and (w 0 , w 1 , · · · , w N −1 ) ∈ Z N , and we denote by By a direct computation, it is straightforward to prove that Then, we consider the approximating problem (P τ ). We set and we look for two vectors (ϑ 0 , satisfying, for i = 1, · · · , N , the system Now, we rewrite the equations (72) and (73) using the piecewise constant functions z τ and the piecewise linear functions z τ defined in (63)-(64), respectively, and obtain that In view of (29)-(32), we infer that for i = 1 the right-hand side of (72) is an element of H, and we have to find ϑ 1 along with ξ 1 fulfilling (71)-(72) and (74); in case we succeed, from a comparison in (72) it will turn out that ϑ 1 ∈ W . Then, we insert ϑ 1 in the right-hand side of (73) and seek χ 1 ∈ W and ξ 1 ∈ H satisfying (73) and (75). Once we recover them, we can start again our procedure, and so on. Then, it is important to show that, for a fixed i and known data Theorem 3.1. There exists some fixed value τ 1 ≤ min{1, T }, depending only on the data, such that for any time step 0 < τ < τ 1 the approximating problem (P τ ) stated by (71)-(77) has a unique solution Let us now rewrite the discrete equation (72)-(72) by using the piecewise constant and piecewise linear functions defined in (63), with obvious notation, and obtain that 3.1. The auxiliary approximating problem (AP ε ). In this subsection we consider the auxiliary approximating problem (AP ε ) obtained by considering the approximating problem (P τ ) in each interval of range τ and replacing the operators appearing in (72)-(77) with their Yosida regularizations. About general properties of maximal monotone operators and subdifferentials of convex functiions, we refer the reader to [1,12]. Yosida regularization of ln. We introduce the Yosida regularization of ln. For ε > 0 we set where I denotes the identity. We remark that ln ε is monotone, Lipschitz continuous (with Lipschitz constant 1/ε) and satisfies the following properties: denoting by L ε = (I + εln) −1 the resolvent operator, we have that We also introduce the nonnegative and convex functions Note that the graph x → ln x is nothing but the subdifferential of the convex function Λ extended by lower semicontinuity in 0 and with value +∞ for x < 0.
On the other hand, Λ ε coincides with the Moreau-Yosida regularization of Λ and, in particular, we have that Yosida regularization of A. We introduce the Yosida regularization of A. For ε > 0 we define Note that A ε is Lipschitz-continuous (with Lipschitz constant 1/ε) and maximal monotone in H. Moreover, A satisfies the following properties: denoting by J ε = (I + εA) −1 the resolvent operator, for all δ > 0 and for all x ∈ H, we have that where A 0 x is the element of the range of A having minimal norm. Let us point out a key property of A ε , which is a consequence of (41): indeed, there holds Notice that 0 ∈ A(0) and 0 ∈ I(0): consequently, for every ε > 0 we infer that J ε (0) = 0. Moreover, since A is maximal monotone, J ε is a contraction. Then, from (41) and (94) it follows that for every x ∈ H. Yosida regularization of β. We introduce the Yosida regularization of β. For ε > 0 we define We remark that β ε is Lipschitz continuous (with Lipschitz constant 1/ε) and satisfies the following properties: denoting by R ε = (I + εβ) −1 the resolvent operator, we have that where β 0 (x) is the element of the range of β(x) having minimal modulus. We also introduce the Moreau-Yosida regularization ofβ. For ε > 0 and x ∈ R we definẽ and we recall thatβ (98) We also observe that β ε is the derivative ofβ ε . Then, for every x 1 , x 2 ∈ R we have thatβ Definition of the auxiliary approximating problem (AP ε ). We fix τ and consider the auxiliary approximating problem (AP ε ) obtained by considering (72)-(77) in the interval of range τ and regularizing the operators appearing in (P τ ). We set  32) and (71)). We look for a pair (Θ ε , X ε ) such that in Ω, (102) where ln ε , A ε and β ε are the Yosida regularization of ln, A and β defined by (90), (93) and (97), respectively. Here, according to the extended meaning of −∆ (see (16)), we omit the specification of the boundary conditions as with (76).

3.2.
Existence of a solution for (AP ε ). In order to prove the existence of the solution for the auxiliary approximating problem (AP ε ) we intend to apply [1, Corollary 1.3, p. 48]. To this aim, we point out that, for τ small enough, the two operators both with domain W and range H, are monotone and coercive. Indeed, they are the sum of a monotone, Lipschitz continuous and coercive operator: and of a maximal monotone operator that is −∆ with a positive coefficient in front. We now check our first claim. Letting v 1 , v 2 ∈ H, we have that (105) Due to the monotonicity of ln ε , the last term on the right-hand side of (105) is nonnegative. Then we infer that i.e. the operator [τ 1/2 I + ln ε ] is strongly monotone, hence coercive in H. Next, for all v 1 , v 2 ∈ H we have that where C π denotes a Lipschitz constant for π. Since β ε and A ε are monotone, it turns out that and, choosing τ 2 ≤ 1/2C π , from (106) we infer that whence the operator [I + τ β ε + τ π + τ A ε (· − χ * )] is strongly monotone and coercive in H, for every τ ≤ τ 2 . Now, in order to prove Theorem 3.2, we divide the proof into two steps. In the first step, we fix Θ ε ∈ H on the right hand side of (102) and find a solution X ε for (102). In the second step, we insert on the right hand side of (101) the solution X ε obtained in the first step and find a solution Θ ε to (101). Now, let Θ 1,ε and Θ 2,ε be two different initial data. We denote by X 1,ε , X 2,ε the corresponding solutions for (102) obtained in the first step and by Θ 1,ε , Θ 2,ε the related solution of (101) founded in the second step.

3.3.
A priori estimates on AP ε . In this subsection we derive a series of a priori estimates, independent of ε, inferred from the equations (101)-(102) of the auxiliary approximating problem (AP ε ). First a priori estimate. We test (101) by τ (Θ ε − ϑ * ) and (102) by X ε , then we sum up. By exploiting the cancellation of the suitable corresponding terms and recalling the definition (91) of Λ ε , we obtain that Let us note that all the terms on the left-hand side are nonnegative. Due to (30) and the continuity of the positive function ϑ * , (92) helps us in estimating the second term on the right-hand side of (114): Due to the sub-linear growth of A ε and the Lipschitz continuity of π the first two terms on the right hand side of (114) can be estimated as Since g, h ∈ H and (100) holds, by applying the Young inequality (20) to the other therms on the right hand side of (114), we find that Then, due to (116)-(118), from (114) we infer that taing into account tat, e.g., τ ≤ 1. Second a priori estimate. We test (102) by β ε (X ε ) and obtain that (120) Thanks to the monotonicity of β ε and to the condition β ε (0) = 0, the terms on the left-hand side are nonnegative. As π is Lipschitz continuous and A ε has a linear growth, applying the Young inequality (20) to every term on the right hand side of (120) and using (119), for 0 < τ ≤ 1 we obtain that Then, owing to (121)-(124), from (120) it follows that Hence, by comparison in (102), we conclude that τ ∆X ε H ≤ c and, from (119) and standard elliptic regularity results, Finally, recalling (96), (30) and (119), we immediately deduce that Third a priori estimate. Next, (101) by ln ε Θ ε and obtain that (128) Then, by applying the Cauchy-Schwarz inequality to every term on the right-hand side and using (20) and (119), we infer that whence τ 1/4 ln ε Θ ε H ≤ c.
(130) Moreover, due to (130), by comparison in (101) it is straightforward to see that τ 5/4 ∆Θ ε H ≤ c and consequently 3.4. Passage to the limit as ε 0. In this subsection we pass to the limit as ε 0 and prove that the limit of subsequences of solutions (Θ ε , X ε ) for (AP ε ) (see (101)-(102)) yields a solution (ϑ i , χ i ) to (72)-(77); then we can conclude that the problem (P τ ) has a solution.

3.5.
Uniqueness of the solution of (P τ ). In this section we prove that the approximating problem (P τ ) stated by (72)-(77) has a unique solution. Then, the proof of Theorem 3.1 will be complete.
We write problem (P τ ) for two solutions (ϑ i 1 , χ i 1 ), (ϑ i 2 , χ i 2 ). and set ϑ i := ϑ i 1 − ϑ i 2 and χ i := χ i 1 −χ i 2 . Then, we multiply by τ ϑ i the difference between the corresponding equations (72) and by χ i the difference between the corresponding equations (73). Adding the resultant equations, we obtain that (142) Since ln, A and β are monotone, the second, the third and the seventh term on the left hand side of (142) are nonnegative. Besides, if τ ≤ 1/(2C π ), thanks to the Lipschitz continuity of π, the right hand side of (142) can be estimated as Then, due to (143), from (142) we conclude that whence we easily conclude that ϑ i = χ i = 0, i.e., ϑ i 1 = ϑ i 2 and χ i 1 = χ i 2 .
4. Uniform estimates on (AP τ ). In this section we deduce some uniform estimates, independent of τ and inferred from the equations (72)-(77) of the approximating problem (P τ ). First uniform estimate. We add (72) and (73) tested by ϑ i and (χ i − χ i−1 )/τ , respectively. Adding (χ i , χ i − χ i−1 ) to both side of the resultant equation and exploiting the cancellation of the suitable corresponding terms, we obtain that Due to (21), we can rewrite the first, the fifth and the sixth term on the left hand side of (145) as (147) Moreover, since the function u −→ e u is convex and e u turns out to be its subdifferential, by setting u i = ln ϑ i we obtain that Since β is the subdifferential ofβ, it follows that while, due (20) and the sub-linear growth of A stated by (41), we obtain that where the constant C 1 depends on C A , ϑ * H and C p . Due to the the boundedness of F i in L ∞ (Ω) and the Lipschitz continuity of π, we also infer that where C 2 depends on C π and |π(0)|. Now, we apply the estimates (146)-(151) to the corresponding terms of (145) and sum up for i = 1, · · · , n, as n ≤ N . We obtain that On account of (31)-(32) and (37), the first four terms on the right hand side of (152) are bounded. Now, recalling the definition of F i (see (70)), we have that Thanks to the absolute continuity of the integral, if τ is small enough (independent of n) we have that Then, on the basis of (153), from (152) we infer that Now, we easily deduce that Beside, according to (29), we have that Then, we can apply (23) and, recalling the notation (63), we conclude that Besides, in view of (74) and due to the sub-linear growth of A stated by (41) and to (30), we deduce that ζ τ L 2 (0,T ;H) ≤ c, (158) Since the third and the fourth term of the left hand side of (157) are bounded, using (18), we also infer that ϑ τ L 2 (0,T ;V ) ≤ c.
(159) Finally, by comparison in (72), we conclude that Second uniform estimate. We formally test (73) by ξ i and obtain We point out that the previous estimate (161) can be rigorously derived by testing (102) by β ε (X ε ) and then passing to the limit as ε 0. Since β is the subdifferential ofβ, we have that

MICHELE COLTURATO
Due to the Lipschitz continuity of π, applying the Young inequality (20) to the first term on the right hand side of (161), we deduce that Moreover, due to the sub-linear growth of A stated by (41), using (157), we have that Now, combining (161)-(164) and summing up for i = 1, · · · , n, n ≤ N , we infer that whence, due to (157)-(160), we obtain that Finally, by comparison in (79), we conclude that ∆χ τ L 2 (0,T ;H) ≤ c. Then, thanks to (157) and elliptic regularity, we find out that Third uniform estimate. We introduce the function ψ n : R −→ R obtained by truncating the logarithmic function in the following way: It is easy to see that ψ n is an increasing and Lipschitz continuous function. Then, defining j n (u) = u 1 ψ n (s) ds, u ∈ R and j(u) = u 1 ln s ds, u > 0 (168) and testing (72) by ψ n (ϑ i ), we obtain that Recalling that j n is a convex function with derivative ψ n , we have that whence, from (169) we infer that τ k 0 Due to the properties of the subdifferential we have that 0 ≤ j(ϑ k ) ≤ j(1) + (ln ϑ k , ϑ k − 1) for k = 0, 1, . . . , N.
Passage to the limit on the logarithmic nonlinearity. Due to the weak convergence of ϑ τ ensured by (181) and to the strong convergence of ln(ϑ τ ) stated by (191), we have that whence λ = ln ϑ and the equation (56) is also achieved.
Passage to the limit on the other nonlinearities. In this paragraph we check that ξ ∈ β(ϕ) a.e. in Q and that ζ(t) ∈ A(ϑ(t) − χ * ) for a.e. t ∈ [0, T ]. Denoting with the same symbol β the operator induced by β on L 2 (0, T ; H), we recall that Consequently, due to [2, Proposition 2.2, p. 38], we conclude that and this is equivalent to saying that ξ ∈ β(ϕ) a.e. in Q.
Now, denoting by the same symbol A the operator induced by A on L 2 (0, T ; H), we recall that as τ 0, whence, setting and this is equivalent to saying that 6. Uniqueness. In this section we prove the uniqueness of the solution of Problem (P ) (see (50)-(55) and Theorem 2.3).
We integrate (50) over (0, t) and we obtain Then, we couple (215) with (51)-(55). We assume that F , χ * , ϑ 0 and χ 0 are given as in (29)-(32) and (ϑ i , χ i ), i = 1, 2, are the corresponding solutions of problem (P ) (see (50)-(55)). Then we write both (215) and (51) for such solutions and multiply the difference of the first equations by ϑ := ϑ 1 − ϑ 2 and the difference of the second ones by χ := χ 1 − χ 2 . Finally, we sum the equalities that we have obtained to each other and integrate over Q t . We have that Due to the Lipschitz continuity of π, the right hand side of (216) can be estimated as follows while the third and the last term on the left hand side of (216), due to the monotonicity of A and β, respectively, can be treated in this way: Finally, using (217)-(219) and applying the Gronwall Lemma to (216), we infer that (220) Consequently, since ln is strictly monotone, we conclude that 7. Long time behavior. In this section we prove Theorem 2.4. Our procedure is the following. First we perform a number of a priori estimates that provide some compactness and ensure, in particular, that the ω-limit is nonempty and fulfills the basic properties stated in Theorem 2.4. Then, we pick any element (ϑ ∞ , χ ∞ ) ∈ ω and prove its relationship with the limit problem − k 0 ∆ϑ ∞ = 0 for a.e. in Ω, in Ω.
(224) Our argument is the following. We choose a sequence t n such that t n +∞, as → +∞, according to definition (60) and introduce the auxiliary functions and which solve problems close to (50)-(55). We show that the a priori estimates derived in the previous steps yield a number of estimates for such functions which allow us to take a weak limit point (ϑ ∞ , χ ∞ ) of the sequence {(ϑ n , χ n )} . We infer that (ϑ ∞ , χ ∞ ) solves a system close to (222)-(224), and the last step of the proof is to show that (ϑ ∞ , χ ∞ ) does not depend on time and coincides with the original pair (ϑ ∞ , χ ∞ ) of the ω-limit.
Our proof relies on a number a priori estimates. However, the regularity of the solution is not sufficient to completely justify the calculation we would like to perform. Therefore, we should come back to the procedure used in [6], where an analogous problem has been solved by passing to the limit as ε 0 in an approximating problem depending on the positive parameter ε, and prove a priori estimates which are uniform with respect to ε. However, in order not to make the exposition too heavy, we prefer to proceed formally on the solution of problem (50)-(55). Of course, we think of a more regular structure and of smoother initial data for a while, but it is understood that we cannot use constants related to such a further regularity. We remark that the source term F and the boundary datum are smooth enough by assumption and that no new property of the initial data ϑ 0 and χ 0 is needed (i.e., more regularity is assumed just for the approximating initial data) since we use weighted test functions, if necessary. Now, we recall the main feature of the approximating problem, which has the following form where ln ε , A ε and β ε are the Yosida regularization of ln, A and β defined by (90), (93) and (97), respectively. It has been proved that problem (226)-(231) has a solution (ϑ ε , χ ε , ξ ε ) and that such a solution tends to (ϑ, χ, ξ) in some appropriate topology as ε 0, at least for a subsequence.
7.1. A priori estimates. Under the assumptions of Theorem 2.4 the solution of problem (50)-(55) will satisfy new a priori estimates, provided that corresponding uniform estimates are fulfilled by the solution of problem (226)-(231) and that just norms related either to reflexive Banach spaces or to dual spaces of separable Banach spaces are involved. Hence, everything would work if we were dealing with the approximating problem. In order to clarify this point, we write a remark after each formal estimate. First a priori estimate We take the difference between (50) and (222) and test it by ϑ. Then, we multiply (51) by ∂ t χ, sum the obtained equality to each other and integrate over (0, t). We obtain that and treat each term that need some manipulation, separately. Since Sign is the subdifferential of the map · : H → R, we have that Now, we deal with the π term. It is easy to see that (33)-(35) imply that r 2 ≤ δβ(r) + c δ and |π(r)| ≤ c(r 2 + 1) for every r in R, where δ denotes an arbitrary positive parameter, whose value is chosen whenever it is convenient to do it. Hence, we deduce that − Ωπ (χ(t)) ≤ 1 2 Ωβ (χ(t)) + c.
Moreover, due to (59), by comparison in (50), we have that Remark. As said before, the above estimates (238)-(239) should be performed on the approximating problems. Doing that, we would obtain a uniform bound for both ϑ ε and ln ε (ϑ ε ) in the space L ∞ (0, +∞; L 1 (Ω)). Moreover, we note that the main trouble in our formal procedure relies on the fact that the time derivative ∂ t ln ϑ belongs just to L 2 (0, T ; V ). On the contrary, as the graph of the logarithm is replaced by a bi-Lipschitz relation (see (226)), the corresponding term of the approximating problem is a function. Second a priori estimate We set for convenience α(t) = tanh(t) for t ≥ 0 and note that bot α and α are bounded by 1. Now, we take the difference between (50) and (222). and test it by α∂ t ϑ = α∂ t (ϑ − ϑ ∞ ). Next, we differentiate (51) with respect to time and test it by α∂ t χ. Finally, we add the equalities we get to each other, integrate over (0, t) and obtain that We treat each term that need some manipulation, separately. First, integrating by parts the second term on the left hand side of (240) and using (238), we obtain Now, we deal with the third term on the left hand side of (240). With an analogous strategy, we infer that Since β and Sign are monotone, the last two terms on the left hand side of (240) are nonnegative. Besides, due to (59) and (238) the first term on the right hand side of (240) can be treated as follows: Indeed, from (59) we easily deduce that ∂ t F L 1 (0,+∞;L ∞ (Ω)) ≤ c.