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Well-posedness for a class of phase-field systems modeling prostate cancer growth with fractional operators and general nonlinearities

This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy that has been recently introduced and analyzed (see [P. Colli et al., Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. 30 (2020), 1253-1295], preprint in arXiv:1907.11618 [math.AP]). The related system includes two evolutionary operator equations involving fractional powers of selfadjoint, nonnegative, unbounded linear operators having compact resolvents. Both equations contain nonlinearities and in particular the equation describing the dynamics of the tumor phase variable has the structure of a Allen-Cahn equation with double-well potential and additional nonlinearity depending also on the other variable, which represents the nutrient concentration. The equation for the nutrient concentration is nonlinear as well, with a term coupling both variables. For this system we design an existence, uniqueness and continuous dependence theory by setting up a careful analysis which allows the consideration of nonsmooth potentials and the treatment of continuous nonlinearities with general growth properties.

Here Ω is a bounded domain in R 3 with a smooth boundary Γ, ∂ ν denotes the normal derivative on Γ, and T > 0 is some final time.
Moreover, the nonlinearity f is defined by 3) denote positive diffusion coefficients, and γ i , S i , and α i , with i = c or i = h, are given constants as well, while u and s are prescribed functions. Finally, ϕ 0 , σ 0 and p 0 are given initial data.
The above system models a prostate cancer growth with chemotherapy, where the physical variables ϕ, σ and p denote the relative amount of tumor and the concentrations of nutrient and of the PSA released by the cells, respectively. In fact, the model describes the tumor dynamics using the phase field ϕ, whose evolution is ruled by (1.1): this equation accounts for the transitions from the value ϕ ≈ 0 in the host tissue to ϕ ≈ 1 in the tumor. The dynamics of the nutrient concentration σ is governed by the reactiondiffusion equation (1.2), while the concentration p of PSA in the prostatic tissue obeys the diffusive equation (1.3) with right-hand side depending linearly on ϕ.
By looking at (1.1) and (1.6), about the term 2ϕ(1 − ϕ)f (ϕ, σ, u) we note that the common factor 2ϕ(1 − ϕ) is on one hand multiplied by M(1 − 2ϕ) to render the derivative of the double-well potential ϕ → Mϕ 2 (1 − ϕ) 2 , and on the other by −3M m(σ) − m ref u , where the term m(σ) describes the tumor net proliferation rate depending on the nutrient. In the definition of m(σ), the values ρ and A stand for constant proliferation and apoptosis (i.e., programmed cell death) indices, and σ r and σ l are a reference and a threshold value for the nutrient concentration, respectively. The positive constant m ref scales the function u that represents the tumor-inhibiting effect of a cytotoxic drug. When |m(σ) − m ref u| is sufficiently small, the function 2ϕ(1 − ϕ)f (ϕ, σ, u) is a double-well potential with local minima at ϕ = 0 and ϕ = 1. Within this range, low values of the nutrient concentration (or large values of u) produce a lower energy level in the healthy tissue (ϕ = 0) than in the tumoral tissue (ϕ = 1). The opposite is true for high values of the nutrient concentration (or low values of u).
As for (1.2), we point out that γ h , γ c are positive constants that represent the nutrient uptake rate in the healthy and cancerous tissue, respectively; the coefficients S h and S c are the nutrient supply rates in the respective tissues; s is a given function yielding the reduction in nutrient supply caused by antiangiogenic therapy. In the model, S h , S c , and s are all nonnegative and s satisfies the constraint s ≤ S c . Both healthy and cancerous prostatic cells release PSA, although tumor cells do so at a much larger rate: by (1.3) the PSA is assumed to diffuse through the prostatic tissue and to decay naturally at rate γ p . The constants α h and α c in (1.3) denote, respectively, the tissue PSA production rate of healthy and malignant cells. About the boundary conditions (1.4) we emphasize that they are very natural, since we assume that the domain Ω is large enough in order that the prostatic tumor has not yet reached the boundary, whence ϕ = 0 on Γ × (0, T ). On the other hand, for the concentrations σ and p no flux is assumed across the boundary, whence Neumann homogeneous boundary conditions are prescribed for them.
More details on the model and a large list of concerned references can be found in [23] and also in the twin paper [24], dedicated to the study of optimal control problems in which the functions u and s, related to cytotoxic and antiangiogenic therapies, act as controls in the system. We point out that the complete system (1.1)-(1.5) is designed to describe the growth of a prostatic tumor under the influence of therapies and it turns out to be a phase-field model. In recent years the phase-field (or diffuse interface) method has been extensively used to describe tumor growth in the computational and mathematical literature (see, e.g., [8,9,11,13,15,26,32,34,37,39,40,44,46,47,49,[54][55][56]). Indeed, tumor growth has become an important issue in applied mathematics and a significant number of models has been introduced and discussed, with numerical simulations as well, in connection and comparison with the behavior of other special materials: one may also see [5,7,12,16,20,25,27,28,31,33,35,36,42,48,50,52,53].
The basic reference [23] contains, in particular, a mathematical study of the wellposedness of the problem (1.1)-(1.5) that is based on a fixed-point approach to equations (1.1) and (1.2). The argumentation makes use of the boundedness property for ϕ, namely this phase variable is assumed (and then shown) to remain between the values 0 and 1, i.e., in the right physical range, during the evolution. In this paper, we aim to significantly generalize system (1.1)-(1.5) by replacing the elliptic operators v → −λ∆v in (1.1) and v → −η∆v+γ h v in (1.2) by operators of fractional type and introducing nonlinear variants of the structural functions appearing in the system, especially of the double-well potential hidden in (1.1) and (1.6). For our purposes, it is convenient to replace the variable ϕ acting in (1.1)-(1.5) by (1 + ϕ)/2, in order to let the 'new' ϕ take the admissible values in the interval [−1, 1]. Note that this change does not affect the structure, since it implies only a rescaling in the equations. Moreover, we decide that in the sequel the third equation (1.3) can be forgotten: indeed, since our aim is essentially to provide a general theory for well-posedness, the (even generalized) third equation can be immediately solved once that ϕ is known. Thus, the system we are interested in is the following (with different notation with respect to the above one) where A 2ρ and B 2τ , with ρ > 0 and τ > 0, denote fractional powers of the selfadjoint, monotone, and unbounded linear operators A and B, respectively, which are densely defined in the Hilbert space H := L 2 (Ω) and have compact resolvents. Notice that the boundary conditions are implicit in the definition of the operators. Moreover, F is a potential of double-well type; h, m, γ, and κ, are real functions defined in the whole of R; m 0 is a constant. Finally, u and S are given functions on Ω × (0, T ), and ϕ 0 and σ 0 are prescribed initial data as before. Well-known and important examples of F are the so-called classical double-well potential and the logarithmic potential, defined by the formulas respectively, where c 0 and c 1 are positive constants. Other significant potentials are the following where I [0,1] is the indicator function of the interval [0, 1] and c 2 and c 3 are positive constants. We recall that the indicator function I X : R → (−∞, +∞] of the generic subset X ⊂ R is defined by I X (r) := 0 if r ∈ X and I X (r) := +∞ otherwise. Note that the potential F sing (r) blows up as r approaches 1 (tumorous phase) while may become largely negative for negative values of r (no problem to go down to the healthy phase).
In cases like (1.13), F is not differentiable in the endpoints of its domain, so that the derivative F ′ appearing in (1.7) is meaningless and has to be suitably replaced. Namely, we split F as F = β + π, where β : R → [0, +∞] is convex and lower semicontinuous (e.g., the indicator function I [0,1] for F 2obs ) and π is a smooth perturbation ( π(r) = −c 3 r 2 , r ∈ R, in (1.13)). Accordingly, we replace F ′ by β +π, where β := ∂ β is the subdifferential of β and π is the derivative of π, and read (1.7) as a differential inclusion. In general, we can rewrite the equation or inclusion (1.7) as a variational inequality involving β rather than β. Actually, we will do this in the following. We analyze the system (1.7)-(1.9) by proving the existence of solutions in a large set of assumptions for the data of system and using a double approximation based on the regularization of nonlinearities and a Faedo-Galerkin discretization. Then, in a more focused framework for nonlinearities we prove a continuous dependence result by dealing with very sharp estimates in the proofs. Of course, our analysis takes advantage of wellestablished approaches for the study of parabolic systems and, in this respect, we would like to recall the pioneering and important contribution given by Claudio Baiocchi [1,2], a master of mathematics and excellent teacher for at least two of the authors of this manuscript.
All in all, we point out that the theory developed in this paper offers a different approach to well-posedness with respect to the one in [23], since in our general setting with fractional operators no L ∞ -estimate is proved for the components of the solution (in particular, not for ϕ), but we are able to show anyway existence and uniqueness of the solution, by exploiting in a very careful way the shape of nonlinearities in the system.
Let us spend some words on the use of fractional operators, which in recent years provided a challenging subject for mathematicians: they have been successfully utilized in many different situations, and a wide literature already exists about equations and systems with fractional terms. For an overview of recent contributions, we refer to the papers [14,18] and [10], which offer to the interested reader a number of suggestions to expand the knowledge of the field. Moreover, we underline that the authors of the present paper already investigated systems with fractional operators in the papers [15][16][17][19][20][21][22], in particular studying another class of tumor growth models [15,20] inspired by [40] and the related contributions [11,13,34,50]. In our approach here, we adopt the same setting for fractional operators, that are defined through the spectral theory. This framework includes, in particular, powers of a second-order elliptic operator, and other operators like, e.g., fourth-order ones or systems involving the Stokes operator. A precise definition for our fractional operators A 2ρ , B 2τ along with their properties follows in Section 2 below. About the use of fractional operators in a physiological framework, we notice that some components in tumor development, such as immune cells, exhibit an anomalous diffusion dynamics (as it observed in experiments [29]), and other components, like nutrient concentration, are possibly governed by different fractional or non-fractional flows. We conclude by arguing that fractional operators are becoming more and more implemented in the field of biological applications and related reaction-diffusion equations (cf, e.g., [3,8,29,30,38,41,43,52,53,57]).
The paper is organized as follows. As for Section 2, we state precisely the problem as well as the assumptions and the well-posedness results. Section 3 contains the proof of the continuous dependence result. The approximation of the problem via regularization of nonlinearities and introduction of the discrete problem is carried out in Section 4, while the existence of solutions is shown in Section 5 by performing a limit procedure on the regularized problem.

Statement of the problem and results
In this section, we state precise assumptions and notations and present our results. First of all, the set Ω ⊂ R 3 is assumed to be bounded, connected and smooth, with outward unit normal vector field ν on Γ := ∂Ω. Moreover, ∂ ν stands for the corresponding normal derivative. We use the notation H := L 2 (Ω) (2.1) and denote by · and ( · , · ) without indices the standard norm and inner product of H. On the contrary, for a generic Banach space X, its norm is denoted by · X , with the following exceptions: a special notation is used for the norms in the spaces V ρ A and V τ B introduced below, and the norm in any L p space is denoted by · p for 1 ≤ p ≤ +∞. Now, we start introducing our assumptions. As for the operators, we postulate that This assumption implies that there are sequences {λ j } and {λ ′ j } of eigenvalues and orthonormal sequences {e j } and {e ′ j } of corresponding eigenvectors, that is, By the same assumption, the powers of A and B for an arbitrary positive real exponent are well defined. Indeed, we can set the series being convergent in the strong topology of H, due to the properties (2.6) of the coefficients. We endow V ρ A with the (graph) norm and inner product v 2 This makes V ρ A a Hilbert space. In the same way, starting from (2.2)-(2.5) for B, we can set , with the norm · B,τ associated to the inner product From now on, we assume: π : R → R is of class C 1 with a Lipschitz continuous first derivative.
(2.12) m : R → R is continuous and bounded, and m 0 ∈ R .
However, it is clear that the present framework is much more general. Nontrivial situations are given in the examples below.
Example 2.4. We modify the previous example by assuming that V ρ A and V τ B are embedded in L 5 (Ω) and L 6 (Ω), respectively, and take α h = α γ = α κ = 5/4. Then, an admissible choice of the exponents of the L p spaces is given by p h = 2 and p γ = q γ = 4, as one immediately sees. We notice that this example is compatible with the further assumptions we introduce later on (see the forthcoming Remark 2.6).
At this point, we can state the problem under investigation. We give a weak formulation of the equations (1.7)-(1.8) and present (1.7) as a variational inequality. For the data, we make the following assumptions: (2.20) Then, we set and look for a pair (ϕ, σ) satisfying and solving the system for every v ∈ V ρ A and for a.a. t ∈ (0, T ), The last integral in (2.26) has to be read as +∞ if β(v) ∈ L 1 (Ω), of course. We also remark that all the other integrals involving nonlinearities are meaningful. Indeed, π is Lipschitz continuous and, by combining (2.23) and (2.24) with our assumptions on the structure and the data, one can show that (similarly as in the proof of the forthcoming estimates (5.8)-(5.10)) with the same warning as above for the last term. Also in this inequality and in the analogous equation for σ, all the integral are meaningful due to ( with a constant C 1 that depends only on the structure of the system, the norms of the data corresponding to (2.20)-(2.21), and T .
(2.36) h, γ and κ are of class C 1 and satisfy where C ′ 0 and C ′ 1 are given constants. We notice that the inequalities (2.37) imply both (2.16) and the inequality as well as its analogues for γ and κ.
Remark 2.6. The assumptions (2.34)-(2.35) look very complicated. However, they are satisfied in a number of situations. One is that of the system (1.1)-(1.5) described in the Introduction, as one immediately sees by also accounting for Remark 2.2. A nontrivial case is given by Example 2.4, where we recall that V ρ A ⊂ L 5 (Ω), V σ B ⊂ L 6 (Ω) and α h = α γ = α κ = 5/4. Indeed, an admissible choice of the new parameters is the following: p γ,1 = 15/2, q γ,1 = 5, r γ,1 = 6, and all of the other exponents are taken as 4.
Here is our result.
Theorem 2.7. Besides the hypotheses of Theorem 2.5, assume that (2.34)-(2.37) are satisfied. Then the solution to problem (2.26)-(2.28) is unique. Moreover, given a constant M, let u i , S i and ϕ 0,i , i = 1, 2, , be two choices of u, S and ϕ 0 , respectively, and (ϕ i , σ i ) be corresponding solutions, and assume that for i = 1, 2. Then the estimate holds true with a constant C 2 that only depends on the structure of our system, T , and M.
The remainder of the paper is organized as follows. The uniqueness and continuous dependence result is proved in Section 3, while the existence of a solution is established in the last Section 5 and is prepared by the study of the approximating problem introduced in Section 4.
Throughout the paper, we make a wide use of the Hölder inequality and of the elementary Young inequality Moreover, we use the notation so that Q = Q T . Finally, we state a general rule concerning the constants we are going to follow. We always use a small-case italic c without subscripts for different constants that may only depend the structure of our system (i.e., the operators A ρ and B τ , the shape of the nonlinearities, the parameters that appear in our assumptions), the final time T and the properties of the data involved in the statements at hand. In particular, the values of such constants do not depend on the regularization parameter ε we introduce in Section 4. Symbols like c δ (e.g., with δ = ε) denote constants that depend on the parameter δ, in addition. It has to be clear that the values of c and c δ might change from line to line and even within the same formula or chain of inequalities. In contrast, we use different symbols (e.g., capital letters like C i in (2.16)) for precise values of constants we want to refer to.

Continuous dependence and uniqueness
This section is devoted to the proof of the uniqueness and the continuous dependence stated in Theorem 2.7. More precisely, we prove just the continuous dependence, since uniqueness follows as a consequence. We pick two choices of the data as in the statement, the corresponding solutions and a constant M satisfying (2.39). We set for convenience We also set for brevity ϕ * := max{|ϕ 1 |, |ϕ 2 |} pointwise (3.2) and denote by D the maximum of the norms of the embeddings (2.15) and (2.35). We assume that D ≥ 1 without loss of generality. At this point, we are ready to start. According to our general rule, we use the same symbols c and c δ (where δ > 0 is chosen later on) for (possibly) different constants, as explained at the end of the previous section.
In this proof, the values of c and c δ are allowed to depend on M, in addition. We write (2.26) at the time s for (ϕ 1 , σ 1 ) and (ϕ 2 , σ 2 ) and test the inequality obtained by ϕ 2 (s) and ϕ 1 (s), respectively. Then, we sum up, integrate over (0, t) with respect to s and add the same quantity Qt |ϕ| 2 to both sides. Since the terms involving β cancel each other, we obtain that By the Young inequality and the Lipschitz continuity of π, we immediately see that For the next term, we owe to both the assumptions of Theorem 2.5 and the supplementary assumptions on the structure (in particular, to (2.38)). We have that and we estimate the last two integrals, separately. We have that The other integral is estimated this way (as for p h , recall (2.14)-(2.15)): By collecting (3.3)-(3.8), we conclude that and we observe that the function s → u 1 (s) 2 ∞ belongs to L 1 (0, T ) and that its norm is bounded by M 2 . Now, we write (2.27) at the time s for both solutions, test the difference by σ(s), integrate over (0, t) and add the same quantity Qt |σ| 2 to both sides. We obtain that We estimate the last three terms, separately, by accounting for the analogues of (2.38) for γ and κ. As for the first one, we have that We estimate the next term in this way: Finally, we observe that By collecting (3.10)-(3.13) and rearranging, we conclude that where we observe that the function s → S 2 (s) 2 ∞ belongs to L 1 (0, T ) and that its norm is bounded by M 2 . At this point, we add (3.9) and (3.14) to each other, choose δ small enough and apply the Gronwall lemma. This yields (2.40) with a constant that has the same dependence as the constant C 2 in the statement. This completes the proof.

Approximation
In this section, we deal with an approximation of problem (2.26)-(2.28) depending on the parameter ε ∈ (0, 1) and solve it by a Faedo-Galerkin scheme. We first introduce the Moreau-Yosida regularizations β ε and β ε of β and β at the level ε > 0 (see, e.g., [6, p. 28 and p. 39]), and we recall that β ε is the derivative of β ε and is Lipschitz continuous. We also remark that the following properties hold true (the first inequality being due to (2.11)): (4.1) The approximating problem, whose unknown is the pair (ϕ ε , σ ε ), is obtained by replacing β by β ε in (2.26)-(2.28) and approximating the other nonlinearities by smoother functions. Moreover, we also regularize the data u and S. However, since β ε is differentiable, the variational inequality (i.e., the analogue of (2.26)) can be replaced by an equation involving the derivative β ε of β ε . As for the other approximating nonlinearities, we require that they are bounded and Lipschitz continuous and converge to the original ones uniformly on every compact interval of R. Moreover, inequalities that are analogous to (2.16)-(2.17) should be satisfied by the regularized functions uniformly with respect to ε. A possible construction of such approximations is based on the lemma stated below. It is clear that the second assumption in (4.7) is empty if D( β) = R. In this case, K 3 can be any positive constant. Then, ψ ε is bounded and Lipschitz continuous on R, and it holds that ψ ε → ψ uniformly on every bounded interval of R. Moreover, if |ψ(r)| ≤ K 0 |r| α + K 1 for every r ∈ R and some positive constants α, K 0 and K 1 , (4.4) then there are constants K 0 and K 1 depending only on α, K 0 and K 1 such that |ψ ε (r)| ≤ K 0 |r| α + K 1 for every r ∈ R and ε ∈ (0, 1) . (4.5) Finally, also assume that |ψ(r)| 2 ≤ K 2 β(r) + K 3 for every r ∈ R, (4.6) where K 2 and K 3 are constants satisfying for every end-point r 0 of D( β) (if any) and some δ > 0. Then, the function ψ ε : R → R defined by setting is bounded and Lipschitz continuous on R and satisfies | ψ ε (r)| 2 ≤ K 2 β ε (r) + K 3 for every r ∈ R and ε ∈ (0, 1), (4.9) as well as the analogues of (4.3) and (4.5).
Proof. Clearly, ψ ε is of class C 1 and its derivative is given by (1/(2ε)(ψ ε (r + ε) −ψ ε (r −ε). In particular, if |r| > (1/ε) + ε, we have that ψ ′ ε (r) = 0, so that ψ ε is bounded and Lipschitz continuous on R. Take now any M > 0 and η > 0. Since ψ is uniformly continuous on every bounded interval, we have that |ψ(r) − ψ(s)| ≤ η whenever ε ∈ (0, 1) is small enough, |r|, |s| ≤ M + 1 and |r − s| ≤ ε. Then, if ε also satisfies M + 1 < 1/ε and r belongs to [−M, M], we conclude that All this proves (4.3). Assume now (4.4). Then, if |s| ≤ |r|, we have that |ψ(s)| ≤ K 0 |s| α + K 1 ≤ K 0 |r| α + K 1 . Therefore, sup |s|≤|r| |ψ(s)| ≤ K 0 |r| α + K 1 , whence also Set now M α := sup r∈R (|r| + 1) α /(|r| α + 1), which is obviously finite. Then, for every r ∈ R and ε ∈ (0, 1), we have that Let us come to the properties of ψ ε under the assumption (4.6). First of all, recall that K 2 and K 3 are positive, that β ε is nonnegative and locally Lipschitz continuous, and that it tends to +∞ as its argument tends to ±∞ (as a consequence of the last condition in (2.11)). Thus, it turns out that ψ ε (r) = ψ ε (r) for |r| large enough. Moreover, ψ ε is (globally) Lipschitz continuous. This yields that the function ψ ε it is well defined and Lipschitz continuous. Moreover, ψ ε is bounded in view of the boundedness of ψ ε . Furthermore, (4.9) trivially follows from the definition of ψ ε , and the analogue of (4.5) for ψ ε is a consequence of (4.5) itself, since | ψ ε | ≤ |ψ ε |. Let us prove the analogue of (4.3) by assuming that the interior of D( β) is nonempty (the opposite case is even easier since then D( β) is a singleton). Take a compact interval I contained in the interior of D( β). Then, the restriction of β to I is continuous. Therefore, by (4.1) and Dini's theorem on monotone convergence, β ε converges to β uniformly in I. By combining this with (4.3), we infer that ψ ε uniformly converges in I to the function the last equality being due to (4.6). If D( β) = R, then the convergence properties under investigation are completely proved. In the opposite case, we consider two situations regarding the compact interval I. In the first one, I = [a, b] is contained in the exterior of D( β). Then, β ε (r) tends to +∞ as ε tends to zero for every r ∈ I. Moreover, the convergence is monotone due to (4.1). By applying Dini's theorem to 1/ β ε (whose pointwise limit is obviously continuous), we deduce that β ε is uniformly divergent. Thus, by setting M := sup a−1≤r≤b+1 |ψ(r)|, we have that β ε (r) ≥ M 2 for every r ∈ I and ε > 0 small enough. On the other hand, |ψ ε (r)| ≤ M for every r ∈ I (see the first part of this proof). We thus have for such values of ε (since K 2 ≥ 1 and β ε is nonnegative) that Therefore, ψ ε coincides with ψ ε on I for ε > 0 small enough, and our assertion follows from (4.3). In the other situation, I = [r 0 − δ, r 0 + δ] is the δ-neighborhood of an endpoint r 0 of D(β) like in (4.7). Then, for every r ∈ I, we have that |ψ ε (r)| 2 ≤ K 3 ≤ K 2 β ε (r) + K 3 , whence also ψ ε (r) = ψ ε (r), so that our assertion follows from (4.3) also in this case. Since every compact interval of R is the union of n ≤ 3 intervals of the previous type, the uniform convergence property we have claimed is completely proved.
The remainder of the section is devoted to the proof of this theorem. As for uniqueness, we can apply Theorem 2.7, since β ε has the same properties as β, and the functions m ε , h ε , γ ε and κ ε are bounded and have bounded derivatives, so that all of the assumptions that are needed are satisfied by the approximating nonlinearities. Hence, we just have to prove the existence of a solution. To this end, we introduce a discrete problem depending on the parameter n ∈ N by means of a Faedo-Galerkin scheme. Then, we solve it and then take the limit of its solution as n tends to infinity.
Now that the discrete problem is solved, we can start estimating. According our general rule, the symbol c ε denotes (possibly different) constants that are allowed to depend on the structure, the data, T , and ε, but not on n.
First a priori estimate. We write (4.17) at the time s and choose v = ∂ t ϕ n (s) ∈ V ρ, n A . Then, we integrate with respect to s over the interval (0, t) with t ∈ (0, T ]. Moreover, we add the same quantity 1 2 ϕ n (t) 2 − 1 2 ϕ n (0) 2 = Qt ϕ n ∂ t ϕ n to both sides. After rearranging terms, we obtain the identity All of the terms on the left-hand side are nonnegative, and the whole right-hand side can be estimated by the quantity since β ε grows quadratically at infinity, π is Lipschitz continuous on R, and the functions h ε , m ε , and u ε , are bounded. Moreover, we can owe to (4.21). We thus obtain the estimate Second a priori estimate. Similarly, we test (4.18) written at the time s by ∂ t σ n (s) and integrate over (0, t). Also in this case, we add the same quantity to both sides. We thus obtain that Using (4.21) for σ n , we can estimate the first term on the right-hand side. Since γ ε and κ ε are bounded, as well as S ε , the whole right-hand side is thus bounded by By accounting for (4.25), and applying the Gronwall lemma, we conclude that Limit. By virtue of (4.25)-(4.27) we can find subsequences (still labeled with the index n for simplicity) that converge to some limits in the weak or weak star topologies associated with the estimates. Since both V ρ A and V τ B are compactly embedded in H (due to assumption (2.2)), by recalling, e.g., [51,Sect. 8,Cor. 4], we conclude that there is a pair (ϕ ε , σ ε ) satisfying the analogues of the regularity conditions (2.23)-(2.24) such that, at least for a subsequence of n, We now show that such a pair solves the approximating problem. Namely, we consider the time integrated version mentioned in Remark 4.2. Take any integern ≥ 1 and any v ∈ L 2 (0, T ; V ρ,n A ). Then, if the (selected) index n satisfies n ≥n, then v(t) ∈ V ρ, n A for a.a. t ∈ (0, T ) and we can use v(t) as a test function in (4.17). After integration over (0, T ), we obtain that and we can let n tend to infinity. Since all of the nonlinearities are Lipschitz continuous and h ε and m ε , as well as u ε , are even bounded, we conclude that Sincen is arbitrary, this equality holds for every step function v with values in the union , an easy density argument shows that (4.30) holds for every v ∈ L 2 (0, T ; V ρ A ). Concerning the equations for σ n and σ ε , we observe that γ ε (ϕ n ) converges to γ ε (ϕ) a.e. in Q, since γ ε is continuous. On the other hand, γ ε is bounded. Hence, we infer that γ ε (ϕ n ) also converges to γ ε (ϕ) in the weak star topology of L ∞ (Q), and combining this with the strong convergence given by (4.29), we conclude that γ ε (ϕ n )σ n converges to γ ε (ϕ)σ ε weakly in L 2 (0, T ; H). Finally, κ ε (ϕ n ) converges to κ ε (ϕ ε ) strongly in C 0 ([0, T ]; H) by Lipschitz continuity. By arguing as for the previous equation, and using a similar density property related to the spaces V τ, n B , we conclude that (ϕ ε , σ ε ) solves the integrated version of (4.13) with test functions taken in L 2 (0, T ; V τ B ) as well. Finally, ϕ n (0) and σ n (0) converge to ϕ ε (0) and σ ε (0) in H by (4.28)-(4.29). On the other hand, (4.20) implies that ϕ n (0) and σ n (0) converge to ϕ 0 and σ 0 , respectively, in the same topology. We infer that the initial conditions (4.14) are satisfied and conclude that (ϕ ε , σ ε ) actually solves problem (4.12)-(4.14).

Existence
In this section, we prove Theorem 2.5. We start from the solution (ϕ ε , σ ε ) to the approximate problem and take the limit as ε tends to zero. To perform this project, we have to prove some a priori estimates. In particular, we show that (ϕ ε , σ ε ) satisfies the analogue of (2.33) with a constant whose dependence is the same as that of C 1 in Theorem 2.5 (in particular, it is independent of ε), since the symbol c we use always stands for (possibly different) constants independent of ε, according to our general rule. It follows that this estimate is kept in the limit as ε goes to zero and that the last sentence of Theorem 2.5 is proved as well. Hence, we do not return to this point in the sequel.
First a priori estimate. We claim that from (4.24), by a limit procedure as n ր ∞, it is possible to derive the following inequality Indeed, by (4.28) and the weak lower semicontinuity of norms we infer that since, ϕ n , ϕ ε being weakly continuous from [0, T ] to V ρ A , for all t ∈ [0, T ] it occurs that ϕ n (t) weakly converges to ϕ ε (t) in V ρ A . Besides, note that if a sequence v n converges to v strongly in H, then This is due to the mean value theorem in the integral form, which gives and to the Lipschitz continuity of β ε . Hence, the terms Ω β ε (ϕ n (t)) and Ω β ε (ϕ n (0)) in (4.24) converge to the respective ones Ω β ε (ϕ ε (t)) and Ω β ε (ϕ 0 ) in (5.1). Moreover, we point out that thanks to (4.21), and that in the last term of (4.24) we can pass to the limit by strong-weak convergence in L 2 (0, T ; H) (cf. (4.28)-(4.29)). Thus, (5.1) is completely verified.
All of the terms on the left-hand side of (5.1) are nonnegative. As for the ones on the right-hand side, we recall (4.1) for the second one and use the Schwarz and Young inequality for the volume integral. Moreover, we notice that (4.2) immediately yields that ψ ε is uniformly bounded if ψ is bounded, so that m ε is uniformly bounded since m is so. Furthermore, by recalling (4.10), we account for the inequality (4.9) applied to h ε . Finally, we owe to the inequality |u ε | ≤ |u| a.e. in Q (see (4.11)). Then, the right-hand side of (5.1) is estimated from above by Since the function s → u(s) 2 ∞ belongs to L 1 (0, T ) by (2.20), we can apply the Gronwall lemma and conclude that Second a priori estimate. Similarly, as for the derivation of (5.1), from (4.26) and (4.28)-(4.29) it is straightforward to deduce the inequality We treat the nontrivial terms on the right-hand side, separately. To do this, we owe to our assumptions (2.14)-(2.16) and to their consequences given by Lemma 4.1 applied with ψ = γ and ψ = κ. Moreover, we also account for (5.2) already established. We have, by virtue of Hölder's and Young's inequalities, that where in the last estimate we have employed (2.15) and (5.2) to see that Now, we treat the next term on the right-hand side of (5.3). By applying the Young inequality, we immediately obtain that On the other hand, we have that Finally, by using the inequality |S ε | ≤ |S| a.e. in Q (see (4.11)), we can write and, due to assumption (2.20) for S and to (5.2) once more, By combining all the inequalities just obtained with (5.3) and applying the Gronwall lemma, we conclude that Estimates of the nonlinear terms. By recalling (5.5), and repeating the arguments that led to (5.6) without time integration, we see on account of (5.2) that γ ε (ϕ ε ) L ∞ (0,T ;L pγ (Ω)) ≤ c and κ ε (ϕ ε ) L ∞ (0,T ;H) ≤ c . (5.8) By combining with (5.7) and recalling (2.14), we deduce that Since the terms involving h ε are a little more complicated, we prepare an auxiliary estimate. We recall (2.14) for the definition of p 0 . Take any w ∈ V ρ A and notice that thanks to the assumption on h given by (2.16) and Lemma 4.1 applied with ψ = h. By accounting for (2.15), we deduce that By applying this with w = ϕ ε (t) for a.a. t ∈ (0, T ) and recalling (5.2), we deduce that Since m ε is uniformly bounded (as already observed) and u ε is bounded in L 2 (0, T ; L ∞ (Ω)) by (4.11) and (2.20), we conclude that Conclusion. We are ready to take the limit of (ϕ ε , σ ε ) as ε tends to zero. More precisely, in considering the approximating problem, we replace (4.30) (equivalent to (4.12)) and (4.13) with the equivalent variational inequality and integrated variational equation, respectively. We thus start from as well as the initial conditions (4.14). By (5.2) and (5.7), and accounting for standard weak, weak star and strong compactness results (see, e.g., [51,Sect. 8,Cor. 4] for the latter), we see that there exists a strictly decreasing subsequence ε n ց 0 such that the corresponding convergence holds true. However, we still write ε, at least for a while, to simplify the notation. Namely, we have that for some pair (ϕ, σ) satisfying (2.23)-(2.24). It follows that such a pair fulfills the initial conditions (2.28) and that π(ϕ ε ) → π(ϕ) strongly in C 0 ([0, T ]; H), (5.15) because π is Lipschitz continuous. Moreover, since p γ , p h and p 0 are larger than 1, m ε is uniformly bounded, and the estimates (5.8)-(5.10) hold true, we also have that γ ε (ϕ ε )σ ε → ζ 1 weakly star in L ∞ (0, T ; H), (5.16) κ ε (ϕ ε ) → ζ 2 weakly star in L ∞ (0, T ; H), (5.17) h ε (ϕ ε )(m ε (σ ε ) − m 0 u ε ) → ζ 3 weakly in L 2 (0, T ; L p h (Ω)), weakly in L 2 (0, T ; L p 0 (Ω)), (5.19) m ε (σ ε ) → ζ 5 weakly star in L ∞ (Q), (5.20) for a suitable subsequence and some limiting functions ζ i . On the other hand, γ ε converges to γ uniformly on every compact interval of R thanks to Lemma 4.1 applied with ψ = γ.