Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144
                                                          www.elsevier.com/locate/na




          A dynamic boundary value problem arising in the
               ecology of mangroves
                      Gonzalo Galiano∗ , Julián Velasco
            Departamento of de Matemáticas, Universidad de Oviedo, c/ Calvo Sotelo, 33007-Oviedo, Spain

                       Received 23 June 2005; accepted 20 October 2005




Abstract
  We consider an evolution model describing the vertical movement of water and salt in a domain splitted in two parts: a water
reservoir and a saturated porous medium below it, in which a continuous extraction of fresh water takes place (by the roots of
mangroves). The problem is formulated in terms of a coupled system of partial differential equations for the salt concentration and
the water flow in the porous medium, with a dynamic boundary condition which connects both subdomains.
  We study the existence and uniqueness of solutions, the stability of the trivial steady state solution, and the conditions for the root
zone to reach, in finite time, the threshold value of salt concentration under which mangroves may live.
  2005 Elsevier Ltd. All rights reserved.

Keywords: Dynamic boundary condition; System of partial differential equations; Existence; Uniqueness; Dead core




1. Introduction

  Mangrove forests or swamps can be found on low, muddy, tropical coastal areas around the world. Mangroves are
woody plants that form the dominant vegetation of mangrove forests. They are characterized by their ability to tolerate
regular inundation by tidal water with salt concentration cw close to that of sea water (see, for example, [18]). The
mangrove roots take up fresh water from the saline soil and leave behind most of the salt, resulting in a net flow of
water downward from the soil surface, which carries salt with it. As pointed out by Passioura et al. [25], in the absence
of lateral flow, the steady state salinity profile in the root zone must be such that the salinity around the roots is higher
than cw , and that the concentration gradient is large enough so that the advective downward flow of salt is balanced
by the diffusive flow of salt back up to the surface. In [25] the authors presented steady state equations governing the
flow of salt and uptake of water in the root zone, assuming that there is an upper limit cc to the salt concentration at
which roots can take up water, and that the rate of uptake of water is proportional to the difference between the local
concentration c and the assumed upper limit cc . They also assumed that the root zone is unbounded, and that the constant
of proportionality for root water uptake is independent of depth through the soil. In [12], the model was extended in
two important ways. First, considering more general root water uptake functions and second, limiting the root zone to a
bounded domain. The authors proved mathematical properties such as the existence and uniqueness of solutions of the

  Supported by the Spanish DGI Project MTM2004-05417 and by the European RTN Contract HPRN-CT-2002-00274.
 ∗ Corresponding author. Tel.: +34 985182299; fax: +34 985103354.
  E-mail addresses: galiano@uniovi.es (G. Galiano), julian@orion.ciencias.uniovi.es (J. Velasco).

1468-1218/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.nonrwa.2005.10.005
1130          G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144


evolution and steady state problems, the conditions under which the threshold level of salt concentration is attained,
and others. In [12], it is assumed that tides, or other sources of fresh or not too saline water, renew the water on the
soil–water interface allowing to prescribe the salt concentration at this boundary (Dirichlet boundary data). Although
this is the usual situation in which mangroves live, in this article we shall focus in the situation in which the inflow
of fresh or sea water is impeded. In this situation, the continuous extraction of fresh water by the roots of mangroves
drives the ecosystem to a complete salinization and, henceforth, to death. This work is motivated by the occurrences
observed at Ciénaga Grande de Santa Marta, Colombia. As reported by Botero [8] (see also [28]), the construction
of a highway along the shore in the 1950s obstructed the natural circulation of water between both parts of the road
(Caribbean sea and lagoon). In addition, in the 1970s, inflow of fresh water from the river Magdalena was reduced
due to the construction of smaller roads and flooding control dikes. These changes caused a hypersalinization of water
and soil, which resulted in approximately 70% mangrove mortality (about 360 Km2 of mangrove forests), see [8,17].
Although other causes, like evaporation or sedimentation, may have had an important contribution to the salinization
of the Ciénaga, we shall keep our attention in the mechanisms of mangroves and their influence in this process.
  The main mathematical difficulty of this model when compared with that studied in [12] is that the closure of the
natural system, the lagoon, implies a new type of boundary condition in the water–soil interface, which is no longer of
Dirichlet type. Balance equations for salt and water content lead to a dynamical boundary condition at such interface,
i.e., a boundary condition involving the time derivative of the solution. Although not too widely considered in the
literature, dynamic boundary conditions date back at least to 1901 in the context of heat transfer [26]. Since then, they
have been studied in many applied investigations in several disciplines like Stefan problems [29,32], fluid dynamics
[16], diffusion in porous medium [27,15], mathematical biology [14] or semiconductor devices [30]. From a more
abstract point of view the reader is referenced to, among others [10,23,19,11,13,1,2,7].
  Apart from the mathematical technical details, one of the main features of the dynamic boundary condition when
compared to the Dirichlet boundary condition is the elimination of the boundary layer the latter creates in a neighborhood
of the water–soil interface, layer in which the salt concentration keeps well below the threshold salinity level. Thus,
this new model allows us to describe the situation in which a continuous increase of fresh water uptake by the roots of
mangroves drives the ecosystem to a complete salinization.
  The outline of the paper is the following: in Section 2 we formulate the mathematical model. We assume that
mangroves roots are situated in a porous medium in the top of which a water reservoir keeps the soil saturated. As
in [12], coupled partial differential equations for salt concentration and water discharge are considered in the porous
medium. Above it, in the water reservoir, balance laws for salt and water are formulated. The assumption of homogeneous
salt concentration in the water reservoir leads to a dynamic boundary condition in the water–soil interface. In Section
3 we state our hypothesis and formulate our main results on existence and uniqueness of solutions of the evolution
problem, as well as the convergence of this solution to the steady state solution. We also study the conditions under
which the complete salinization of the root zone is attained in finite time (dead core). Finally, in Section 4 we prove
our assertions.


2. The mathematical model

  In this section we formulate the mathematical model which describes the salt and water movement in the water–soil
system. We consider the case where the mangroves are present in the horizontal x, y plane, with an homogeneous
porous medium located below this plane and a water reservoir above it. The porous medium is characterized by a
constant porosity , indicating that we are assuming the mangroves roots to be homogenized throughout the porous
medium, without affecting its properties. Assuming further that the hydrodynamic dispersion tensor, D, is constant and
isotropic, i.e. neglecting the velocity dependence in the mechanical dispersion, we find for the salt concentration the
equation, see [6],

    jc
      + div(cq − D∇c) = 0,                                            (1)
    jt

where the vector q denotes the specific discharge of the fluid, D = DI, I is the identity matrix and t denotes time. We
also have a fluid balance in the porous medium. Disregarding density variations in the mass balance equation of the
            G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144  1131


fluid, we obtain a fluid volume balance expressed by

   divq + S = 0,                                                 (2)

where S is the volume of water taken up by the roots per unit volume of porous material per unit time. If the mangroves
are uniformly distributed throughout the x, y-plane and there is no lateral fluid flow, we may consider the problem
as one-dimensional in the vertical direction. If the z-axis is positive when pointing downwards, the flow domain is
characterized by the interval 0 < z < H < ∞. In the one-dimensional setting Eqs. (1) and (2) become

    ct + (cq − Dcz )z = 0,                                            (3)

   qz + S = 0,   in (0, H ) × (0, T ).                                     (4)

For S, we assume to have the form
                 p
             c
       s(z) 1 −         for 0 c cc ,
   S :=                                                      (5)
             cc
       0             for c > cc ,
where cc is the upper limit of salt concentration at which mangroves may uptake water, p > 0 and s(z) is determined
by the root distribution as a function of the depth z below the soil surface. This root distribution function will be
non-negative, and non-increasing with z. We shall keep in mind the following characteristic example: we assume that
the function s is a positive constant, s0 /z∗ , above a certain depth z∗ , and zero below that depth, i.e.

   s(z) = s0 /z∗             and s(z) = 0
           if 0 z z∗                  if z∗ < z H .                   (6)

The quantity s0 is the total amount of root water uptake in the profile with no salt present, in volume per unit surface
per unit time, i.e. the transpiration rate of the mangrove plants in the absence of salinity. On the bottom of the porous
medium domain, we assume no flux boundary conditions, resulting in

   q(H, t) = cz (H, t) = 0      for t ∈ (0, T ).                               (7)

On the water–soil interface we prescribe a boundary condition which is deduced from conservation laws for salt and
water in the whole system water–soil. We assume that salt concentration in the water domain, C, remains uniformly
distributed in space. This approximation is justified when assuming a much faster mixing of the salt in the reservoir
than in the porous medium. Then, the average height level of the water reservoir, W, and C are functions that only
depend on time. We further consider, based on a continuity assumption

   C(t) = c(0, t) for t ∈ (0, T ).                                        (8)

Then we have

• The salt balance. Assuming that the total amount of salt in the system water–soil remains constant, we have
               H
       d
         CW +        c =0     in (0, T ).
       dt      0

 Therefore, from Eq. (3) and the boundary condition (7),
       d(CW )
          = c(0, ·)q(0, ·) − Dcz (0, ·) in (0, T ).                            (9)
        dt
• The fluid balance, which asserts that the amount of water taken up from the soil by the roots of mangroves is replaced
 by water from the reservoir:
       dW
         = −q(0, ·)      in (0, T ).                                  (10)
       dt
1132           G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144


Combining (8)–(10) we deduce
    W (t)ct (0, t) = Dcz (0, t)     for t ∈ (0, T ),                                (11)
which is the dynamic boundary condition for the soil–water interface. Finally, we add to this formulation given initial
distributions of salt concentration, c(·, 0) = c0 in (0, H ), and of water reservoir height level, W (0) = W0 .
  We recast the above formulation in an appropriate dimensionless form introducing the following variables, unknowns
and parameters:
    ˜
    t := Dt/z∗ ,  x := z/z∗ ,     u := c/cc ,   q = qz∗ /D ,
                               ˜
         2


    w = W/ z∗ ,    s (x) := z∗ s(H x)/s0 ,
             ˜                d := H /z∗ ,    m := s0 z∗ /D
and we define f (x, u) := S(H x, cc u), with f : [0, d] × [0, 1] → R+ given by
                  p
    f (x, ) := s (x)(1 − )+ ,
          ˜                                                    (12)
with p > 0 and
    s (x) = 1
    ˜              and s (x) = 0
                     ˜
          if 0 x   1              if 1 < x                           (13)
                                     d.
                                           ¯        ¯
With the above changes we are led to the following problem (omitting tildes): find u : QT → [0, 1], q : QT → R and
W : [0, T ] → R such that
    ut + (uq − ux )x = 0,                                               (14)
    qx + mf (·, u) = 0   in QT = I × (0, T ), with I = (0, d),                           (15)
    w (t) + q(0, t) = 0   for t ∈ (0, T ),
subject to the boundary and initial conditions
    w(t)ut (0, t) = ux (0, t),                                             (16)
    ux (d, t) = q(d, t) = 0     for t ∈ (0, T ),                                  (17)
    u(·, 0) = u0          w(0) = w0 .
            in I,                                               (18)

Remark 1. In the recasting of our model there appeared a constant capturing all the important physical parameters,
the mangrove’s number:
    m := s0 z∗ /D .                                                  (19)
                                                           −5
Using [25] and [24] as a reference we find the following values for the physical constants: D = 7 × 10 m2 /day,
= 0.5, and s0 = 1 m−2 day−1 . Taking z∗ in the range 0.2–0.5 m, this implies a time scale in the range 2–10 yr and
m ∈ (6, 15).


3. Main results

 We shall refer to problem (14)–(18), as to Problem P, for which we assume the following hypothesis:

            ¯
H1 . The function f : I × [0, 1] → R, with I = (0, d) and d           1, satisfies
          ∞
       f ∈ L (I ; C([0, 1]),      |f | 1,
                      ¯
       f (·, s) is non-increasing in I and f (d, s) 0 for all s ∈ [0, 1],
       f (x, ·) is non-increasing in [0, 1] and f (x, 1) = 0 for a.e. x ∈ I .
                        ¯
                     0 in I × [0, 1].
    Note that, in particular, f
            G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144  1133


H2 . The initial data posses the regularity
      u0 ∈ H 1 (I )     with 0 u0    1 in I .
H3 . The function w is a positive constant. The number m is positive. We set w = m = 1.


Remark 2. The assumption w (or the dimensional W) constant in H3 has a reasonable range of validity. From (4), (5),
(10) and the mean value theorem we infer
                                  p
                                ¯
              t                 c
   W (t) = W0 −       q(0, ) d = W0 − ts 0 1 −        for some c ∈ (0, cc ).
                                        ¯
                               cc
            0
Set s0 as in Remark 1 and p = 1. A lower limit for c is sea water salt concentration cw ∼ 0.5cc . Then W0 must be
                            ¯
much greater than the 15 cm that the lagoon will decrease per year while keeping the sea water salt concentration. For
a value of c = 0.9cc the decrease of the height level is of about 3 cm per year.

Remark 3. Since the numbers m and w do not play any essential role in the results we prove in this work, we set
m = w = 1 for clarity.

 Under Hypothesis H1 –H3 we cannot expect the existence of classical solutions. We then introduce the notion of
solution we shall work with.

                                     ¯          ¯
Definition 1. We say that (u, q) is a strong solution of Problem P if u : QT → [0, 1] and q : QT → R satisfy the
following properties:

(1) For any r ∈ (0, ∞),
                                        ¯
      u ∈ W 1,r (0, T ; Lr (I )) ∩ Lr (0, T ; W 2,r (I )) ∩ C((0, T ]; C(I )),
      q ∈ C((0, T ]; W)
  with W := { ∈ W 1,∞ (I ) : (d) = 0}.
(2) The differential Eqs. (14) and (15) and the boundary conditions (16) and (17) are satisfied almost everywhere. The
  initial distribution is satisfied in the sense
      lim u(·, t) − u0          = 0.
                   L2 (I )
      t→0

 We prove the following result on existence and regularity of solutions.

Theorem 1. Assume H1 –H3 . Then there exists a strong solution of Problem P satisfying
   u um := min u0       a.e. in QT .                                   (20)
          ¯
          I

In addition, if for some p > 0
        ¯                  ¯
   f ∈ C p (I × [0, 1])     and u0 ∈ C 2+p (I )                               (21)
and if u0 satisfies the following compatibility condition:
              1
   u0 (0) + u0 (0)      f (x, u0 (x)) dx − u0 (0) = f (0, u0 (0))u0 (0),                  (22)
            0
          ¯            ¯
then u ∈ C 1+p,2+p (QT ) and q ∈ C 1+p,1+p (QT ).

  We prove uniqueness of solution for f (x, ·) being Lipschitz continuous in [0, 1]. For more general functions, we
show that uniqueness of solution holds true under an additional condition on the component u. In Proposition 1 we
give an example in which solutions of Problem P satisfy such condition.
1134             G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144


Theorem 2. Let (u1 , q1 ) and (u2 , q2 ) be two strong solutions of Problem P and let H1 –H3 be satisfied. If either

    f (x, ·) is Lipschitz continuous in [0, 1] for almost all x ∈ ,                          (23)

or anyone of the solutions satisfies
            x
              |ux (y, t)| dy  a.e. in QT ,                                 (24)
    u(x, t) >
          0

then (u1 , q1 ) = (u2 , q2 ) a.e. in QT .

Proposition 1. Assume H1 –H3 and (21)–(22), and let (u, q) be a solution of Problem P. Suppose that u0 satisfies
u0x L in I and

    f (·, um ) L < um −     1
                                                             (25)
                  2

for some positive constant, L, with um given by (20). Assume

    f (·, u) + uf u (·, u) < 0    in QT .                                     (26)

Then condition (24) is satisfied.

                                         ¯
Remark 4. In particular, if f (x, )=s(x)(1− )p , with s smooth, and u0 ∈ C 2+p (I ) satisfies u0x (1−um )p < um − 2 ,
                                                         1

then condition (24) is satisfied. Actually, the smoothness requirement on s may be dropped by using an approximation
argument.

  One important effect of the dynamic boundary condition when compared to the Dirichlet boundary condition at
the boundary water–soil is the elimination of the boundary layer the latter creates. It is straightforward to prove that
the unique solution of the steady state problem corresponding to Problem P, i.e., functions U ∈ H 1 (I ) and Q ∈ W
satisfying

    (QU − Ux )x = Qx + f (·, U ) = 0        in I ,

    Ux (0) = Ux (d) = 0,

is the trivial solution (U, Q) = (1, 0). Regarding the asymptotic convergence of solutions of Problem P to this trivial
solution when t → ∞, we have the following result.

Theorem 3. Assume H1 –H3 and um > 0, and let (u, q) be a strong solution of Problem P. Then

    (u, q) → (1, 0) in L2 (I ) and        u(0, t) → 1     pointwise as t → ∞.

  We finally state a result on the existence of a dead core for solutions of Problem P, i.e., sets where the threshold
salinization u = 1 is attained in finite time. The proof of this result, which is of local nature, i.e., independent of the
boundary data, can be found in [12]. First, we introduce some notation. For any t ∈ (0, T ) we consider the parabola of
vertex (x0 , t),

    P(t) := {(x, ) : |x − x0 | < ( − t) ,        ∈ (t, T )},

with 0 < < 1 and x0 ∈ I such that T < x0 < 1 − T , implying P(t) ⊂ QT for all t ∈ (0, T ). We define the local
energy functions

    E(t) :=       |ux |2 dx d   and C(t) :=        (1 − u)p+1 dx d .                 (27)
          P(t)                    P(t)

In [12] we proved the following theorem using the techniques introduced in [3,4].
                 G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144  1135


Theorem 4. Suppose there exist constants s0 and s1 such that

               f (·, 1 − ) s1               ∈ [0, 1],
        p+1                     p+1
   0 < s0                           for                             (28)

in P(t) for a.e. t ∈ (0, T ), with p ∈ (0, 1) and s0 > s1 /2, and let (u, q) be a strong solution of Problem P. Then there
exists a positive constant M such that if E(0) + C(0) M then u ≡ 1 in P(t ∗ ) for some t ∗ ∈ (0, T ).

 Let us finish this section with a remark on the assumptions of Theorem 4. First, if function f is given by f (x, ) =
s(x)(1 − )p , with s given by (13) then (28) is trivially satisfied in the region where s > 0 (root zone). Regarding the
bound of the initial energy, we have that testing the first equation of (14) with 1 − u and using the Eq. (15) we obtain

   2E(0) + C(0)           (1 − u0 )2 + (1 − u0 (0))2 1 +           f (·, u) .            (29)
                  I                         QT

Therefore, if the initial datum is close enough to one then the initial energy bound is satisfied. Combining Theorems 3
and 4 we deduce the following corollary.

Corollary 1. Let (u, q) be a strong solution of Problem P in QT , for T large enough. Under the conditions of Theorems
3 and 4 there exist T0 , t ∗ > 0 such that u ≡ 1 in P(t ∗ ), for some t ∗ ∈ (T0 , T ).

  Or, in other words, the threshold value of salt concentration is attained in any compact set contained in the root zone
in finite time.


4. Proofs


Proof of Theorem 1. We first prove the existence of weak solution of a time discretization of Problem P. Since, a
priori, the component u of solutions to approximated problems will not necessarily satisfy 0 u 1, we extend f by f¯
as f¯(x, ) = 0 if > 1, f¯(x, ) = f (x, ) if 0    1 and f¯(x, ) = f (x, 0) if < 0. We denote the corresponding
           ¯
problem by Problem P.

Lemma 1. For u ∈ H 1 (I ), and > 0 small enough, there exists a solution (u, q) ∈ W 2,r (I ) × W, with r < ∞, of
       ˜

   u + (uq − ux )x = u
            ˜           a.e. in I ,                                  (30)

   qx + f¯(·, u) = 0         a.e. in I ,                                      (31)

   u(0) = u(0) + ux (0),
       ˜                   ux (d) = 0.                                (32)

Proof. We introduce the set K = {v ∈ W, v W 1,∞     } for some > 0 to be fixed. It is clear that K is convex and
weakly compact in the star topology of W 1,∞ (I ). For q ∈ K, we define the map
                            ˆ
               d
                 f¯(s, u(s)) ds
   S(q)(x) :=
    ˆ
             x

with u ∈ H 1 (I ) solution of

      (u − u) +
         ˜                 +    (uq)x + (u(0) − u(0)) (0) = 0
                               ˆ       ˜                         (33)
                    ux   x
    I              I          I

for any ∈ H 1 (I ). The existence of a unique solution of (33) is guaranteed by the Theorem of Lax–Milgram (see, for
instance, [9]). In addition, we have
           1
                               f¯
               u−u
                ˜       +u           + ux       L∞ ,                 (34)
    uxx                                     q
                                 L∞
        L2             L2      L2          L2
1136             G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144

                      ¯
i.e., u ∈ H 2 (I ). Since u ∈ H 1 (I ) ⊂ C(I ), a boot-strap argument allows us to deduce u ∈ W 2,r (I ), for any r < ∞.
             ˜
                                         ˆ
A standard argument allows us to conclude that u satisfies (30) and (32) (with q replaced by q).
  Observe that a fixed point of S is a solution of (30)–(32). We prove the existence of such a fixed point using a theorem
by [5], for which we need to show: (i) S(K) ⊂ K and (ii) S is weakly–weakly continuous in the star topology of
W 1,∞ (I ). Showing S(K) ⊂ K is straightforward since for any q ∈ K, S(q) W 1,∞ 2 f¯ L∞ =: .
                                  ˆ      ˆ
  To prove the weak continuity, (ii), we consider a sequence qj and a function q in K such that qj → q weakly star in
                                 ˆ        ˆ        ˆ   ˆ
W 1,∞ (I ). Let uj and u be the corresponding solutions of problem (33). Taking = uj in (33) we obtain, after using
Schwarz’s inequality,

    uj (0)2 + (1 −     ˆ       − 2 qj x
                         ˆ               +           ˜    + u(0)2 .
                                                         ˜
                   2                   2         2      2
                qj              L∞ )   uj        uj x      u
                   L∞                  L2         L2     L2

For small enough and independent of j we get Ej +                      2   c, with c independent of and j, and with
                                           uj x  L2

    Ej = uj (0)2 + uj       2
                    .                                                   (35)
                   L2

Therefore, we obtained a uniform bound which allows us to extract a subsequence of uj (not relabelled) such that
                                            ¯
uj → v weakly in H 1 (I ), for some v ∈ H 1 (I ). Since the embedding H 1 (I ) ⊂ C(I ) is compact, extracting a new
                              ¯). Next we show that, actually, v = u. All the terms in (33)
subsequence if necessary we have uj → v uniformly in C(I
corresponding to (uj , qj ) are well defined in the limit j → ∞. For instance,
            ˆ

      (uj qj )x =
        ˆ            ˆ   +     ˆ      →       ˆ
                                        (v q)x ,
                 u j x qj      uj q j x
    I           I          I            I

                                   ¯
due to the convergences uj → u weakly in H 1 (I ) and uniformly in C(I ), and qj → q weakly star in W 1,∞ (I ) and
                                       ˆ  ˆ
        ¯) (by compact embedding, again). Then, by the uniqueness of solution of problem (33) we deduce
uniformly in C(I
v = u. Hence,
             d                   d
               f¯(s, uj (s)) ds →          f¯(s, u(s)) ds = S(q)(x),
    S(qj )(x) =
     ˆ                                       ˆ
           x                   x

          ¯
uniformly in C 1 (I ) and, in particular, weakly star in W 1,∞ (I ). Therefore, (ii) is proven and the existence of a fixed
point deduced.

                                                   ¯
  We now construct piecewise constant in time approximations of solutions of Problem P. Let (0, T ] = K        k=1
((k − 1) , k ], with = T /K and K ∈ N. For k = 1, . . . , K, define recursively (uk , qk ) as the solution of problem
                                                           ¯
(30)–(32) with u = uk−1 , u = uk and q = qk . Let the initialization of this recursion be the initial data of Problem P, u0 .
        ˜
We define the following piecewise constant in time functions: u( ) (x, t) = uk (x), q ( ) (x, t) = qk (x),
               uk (x) − uk−1 (x)
    ()
    jt u( ) (x, t) =                     E ( ) (t) =      |u( ) (0, t)|2 +     |u( ) |2
                                         1
                            ,            2
                                                      I

if x ∈ I , t ∈ ((k − 1) , k ], for k = 1, . . . , K.

Lemma 2. As → 0 there exist a subsequence of (u( ) , q ( ) ) (not relabelled) such that

           weakly star-weakly in L∞ (0, T ; H 1 (I )),
    u( ) → u                                                                (36)
    ()
    jt u( ) → jt u     weakly in L2 (QT ),                                              (37)
    ()
    jt u( ) (0, ·) → jt u(0, ·)       weakly in L2 (0, T ),                                     (38)

    u( ) → u   weakly in L2 (0, T ; H 2 (I )),                                             (39)
                        ¯
    u( ) → u   uniformly in C((0, T ]; C(I )),                                             (40)

    q( ) → q   uniformly-strongly in C((0, T ]; W).                                           (41)
                    G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144                1137


Proof. Replacing in (30) functions u, q and u by uk , qk and uk−1 , respectively, and using =uk in the weak formulation
                       ˜
(33) (with q = q), we obtain, after using the inequalities of Schwarz and x(x − y) (x 2 − y 2 )/2,
      ˆ
   Ek +                 Ek−1 + cf Ek
                 2
              ukx  L2

for

   Ek =          uk (0)2 +        u2
            1
                           k
            2
                         I

and with cf := f¯ 2 ∞ + f¯ L∞ . Then, from the Gronwall’s discrete inequality and k                           K, we deduce Ek  cE 0 , for
            L
k = 1, . . . , K, and for some constant, c, independent of . Therefore,
   Ek − Ek−1
                 + ukx     2
                             ccf E0 .
                         L2

Integrating in (0, t) for any t ∈ (0, T ), we obtain

   E ( ) (t) +           |u( ) |2    ccf E0 ,
                    x
                 Qt

which gives a uniform estimate for u( ) in the norm of L2 (0, T ; H 1 (I )) ∩ L∞ (0, T ; L2 (I )). On the other hand, from
              f¯ L∞ , which implies the uniform bound
(31) we obtain qk W 1,∞

                        f¯
    q(     )
                           L∞ .
            L∞ (W 1,∞ )

                 = (uk − uk−1 )/ in (33) (with q = q). We get
                                ˆ
We now choose
                    2                                                    2
          uk − uk−1                  uk − uk−1                  uk − uk−1    uk (0) − uk−1 (0)
                      +                     +                 +              = 0.
                           ukx                    (uk qk )x
    I                    I                     I
                                         x

Using again the inequality x(x − y) (x 2 − y 2 )/2, we obtain
              uk − uk−1           1
                                  (|ukx |2 − |u(k−1)x |2 )
        ukx
                             2
    I                            I
                         x

and therefore
                    2
            uk − uk−1                                          uk − uk−1
                         1
                      +         (|ukx |2 − |u(k−1)x |2 ) +        (uk qk )x
                         2
      I                       I                     I
                             2
              uk (0) − uk−1 (0)
        +                        0.

Integrating in ((k − 1) , k ) and adding from k = 1 to K leads to
                                         T
    1                           ()             ()
            |u( ) |2 (T , ·) +         |jt u( ) |2 +        |jt u( ) (0, ·)|2
             x
    2                                   0
          I                 QT
            1                         ()
                |u0x |2 −       (u( ) q ( ) )x jt u( ) .
            2  I          QT

Using Hölder’s inequality we deduce
                  ()          ()
                 + jt u( ) 2 2 (L2 ) + jt u( ) (0, ·) 2 2 (0,T )
            2
    u( )      L∞ (L2 )
     x                 L             L
            c( u0 H 1 + q
               2    () 2     () 2
                         u L2 (H 1 ) ),
                     W 1,∞

i.e., additional uniform bounds for
            in L∞ (0, T ; H 1 (I )),
   u(    )
1138               G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144

     ()
    j t u(    in L2 (QT ),
         )
                                                                  (42)
     ()
    jt u( ) (0, ·)      in L2 (0, T ).                                           (43)
Once we have the uniform bound on the time derivative, (42), we deduce from (34) a uniform bound for u( ) in
L2 (0, T ; H 2 (I )), i.e. (39). Therefore, there exist u ∈ L∞ (0, T ; H 1 (I )) ∩ H 1 (0, T ; L2 (I )) and q ∈ L∞ (0, T ; W)
such that (36) and (37) hold. In addition, the compactness result of [31] implies (40). Therefore, since f¯ ∈ L∞ (I ; C(R))
we have qx = f¯(·, u( ) ) → f¯(·, u) = qx uniformly-strongly in C((0, T ]; L∞ (I )), and then (41). Finally, from (43)
      ()

we deduce (38).

End of proof of Theorem 1. We are now ready to pass to the limit → 0. The pair (u( ) , q ( ) ) satisfies
                                             T
        ()                                       ()
        j t u(    +            +    (u( ) q ( ) )x +        jt u( ) (0, ·) (0, ·) = 0
            )
                     u( )                                           (44)
                         x
                      x
                                           0
     QT             QT          QT

    ∈ L2 (0, T ; H 1 (I )), and
for
    qx ) + f¯(·, u( ) ) = 0                    q ( ) (d, t) = 0     for all t ∈ (0, T ].
    (
                      a.e. in QT ,                                      (45)
Taking the limit → 0 in (44)–(45), and using (36)–(41) we obtain that (u, q) satisfies
                                     T
        ut +            +     (uq)x +        ut (0, ·) (0, ·) = 0                  (46)
                 ux  x
                                   0
     QT         QT          QT

and
    qx + f¯(·, u) = 0                  and q(d, t) = 0       for all t ∈ (0, T ].
                   a.e. in QT
Due to (39) we deduce uxx ∈ L2 (QT ). Integrating by parts in (46) and using f¯ ∈ L∞ (I, C(R)), we deduce that (u, q)
                                                ¯
satisfies the strong formulation (14)–(18) and it is, therefore, a strong solution of Problem P.
  Finally, using := min{0, u − m}, with m = minI¯ u0 , and := max{0, u − 1} as test functions in (46) one easily
shows that m u 1 in QT . We note at this point that this property implies f¯(·, u) = f (·, u) in QT and therefore the
              ¯                                      ¯
pair (u, q) is also a strong solution of Problem P.
  Finally, if function f and the initial condition satisfy the additional regularity and compatibility conditions stated in
              ¯                     ¯                  ¯
Theorem 1 then u ∈ C(QT ) which implies uf (·, u), qx ∈ C p (QT ) and, therefore, ut − uxx ∈ C p (QT ), implying the
additional regularity assertion.

Proof of Theorem 2. Let (u1 , q1 ) and (u2 , q2 ) be solutions of Problem P and set (u, q) := (u1 − u2 , q1 − q2 ). Then
(u, q) satisfies Problem PD
    ut + (uq 1 + u2 q)x − uxx = 0            in Q ,
    qx + f (x, u1 ) − f (x, u2 ) = 0           in Q ,
    ut (0, ·) = ux (0, ·)       on (0, T ),
    ux (d, ·) = q(d, ·) = 0         on (0, T ),
    u0 = 0    on I .
We first discuss the case in which f is Lipschitz continuous. Multiplying the first equation of Problem PD by u and
integrating by parts we obtain
    1d                             1           1
         u(0, t)2 +        u2 +      |ux |2   q1 (0, t)u2 (0, t) +         f (·, u1 )u2
    2 dt                            2           2
                   I        I                        I

                                   + u2 (0, t)|q(0, t)||u(0, t)| +         u2 |q||ux |.  (47)
                                                         I
               G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144                       1139


Using the second equation of Problem PD , assumption (23) and the continuous embedding L2 (I ) ⊂ L1 (I ), we deduce
                                 2             2
               d
   |q(x, t)|         |f (·, u1 ) − f (·, u2 )|              |u|
         2                           2
                                                  cC 2      u2 ,                       (48)
                                    CL               L
             x                            I              I

with CL the constant of Lipschitz of f (·, s) and c > 0. Using Schwarz’s inequality and u2                              1 we then get from
                                                                        L∞
(47)
   1d                    1
        u(0, t)2 +        u2 +      |ux |2    c1 u2 (0, t) + c2       u2
   2 dt                   2
                   I       I                      I

with c1 = ( q1 L∞ + 1)/2 and c2 = f L∞ /2 + cC 2 . Applying the Lemma of Gronwall with u0 = 0 we deduce u = 0
                           L
a.e. in QT , i.e., u1 = u2 . Then, from (48) we also deduce q1 = q2 a.e. in QT .
  We now let f be a general function satisfying H1 –H3 and assume that condition (24) holds for u2 . Multiplying the
differential equations of Problem PD by smooth functions , satisfying

    t (0, t) +  x (0, t) = 0,       x (d, t) =    (0, t) = 0      for any t ∈ [0, T ],                             (49)

integrating in Q , with ∈ (0, T ), and adding the resulting integral identities we obtain

   u(0, ) (0, ) +         u(·, ) (·, ) =             + q1      +  xx ) −            + u2x )
                                 u(                         q(
                                     t       x                 x
                 I              Q                         Q

                              +      (f (x, u1 ) − f (x, u2 ))(u2 + )
                                 Q

                              +                                                    (50)
                                    u(0, t)q1 (0, t) (0, t).
                                 0

We consider the function defined in Q by
       f (·, u1 ) − f (·, u2 )
   h=                   if u = 0,        h=0     if u = 0                                     (51)
            u
which is non-positive because f (x, ·) is non-increasing and possibly unbounded, since f is not Lipschitz continuous.
For m ∈ N, m 1, we consider the functions hm = T (h) − 1/m, where T (s) = s if −m < s 0, and T (s) = −m if
s − m. We regularize these functions in such a way that we obtain a smooth sequence {hm } ⊂ C 2 (Q ) satisfying

                            − m,     hm → h
   hm+1   hm            0 > hm
            in Q ,                             a.e. in Q .

The regularity of solutions of Problem P allows us to introduce sequences {q1 }n                          {un }n    ⊂ C 2 (QT ) such that, as
                                      n
                                                               1,      1
                                                                   2
n → ∞,
                                                 ¯
   q1 → q1          u n → u2    strongly in L2 (0, T ; H 1 (I )) ∩ C((0, T ]; C(I ))
    n
           and                                                                      (52)
                  2

and un satisfying (24). Using these approximations we rewrite (50) as
   2


   u(0, ) (0, ) +         u(·, ) (·, ) =             + q1      +     + hm (un + ))
                                        n
                                 u(   t       x    xx      2
                 I              Q

                              −           + un ) +          u(h − hm )(u2 + )
                                    q(   x   2x
                                 Q                    Q

                              −      ux (q1 − q1 ) −            u(q1x − q1x )
                                         n                  n
                                 Q                  Q

                              +      uhm (u2 − un ) −              q(u2x − un )
                                         2                     2x
                                 Q                     Q

                              +                                                    (53)
                                    u(0, t)q1 (0, t) (0, t).
                                 0
1140                G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144


Next we select the functions             and , being the solutions of
       + q1    +        + hm (un + ) = 0
        n
                                         in Q ,                              (54)
     t      x      xx       2

     x + u2x     =0
       n
                     in Q ,                                                  (55)
     ( )=      on ,                                                            (56)
                                      ¯
                                  ∈ C ∞ (I ).
with ,    satisfying (49) and with

                                            ¯
Lemma 3. Assume (24). Then, for each n and m there exists a unique solution , ∈ C 2,1 (Q ) of (54)–(56) and (49)
such that  ∞ (Q ) and   ∞ (Q ) are uniformly bounded with respect to n and m.
     L       L

End of proof of Theorem 2. Using the functions provided by Lemma 3 we obtain from (53)

    u(0, ) (0) +          u(·, ) =            u(h − hm )(u2 + ) −          ux (q1 − q1 )
                                                            n
                 I               Q                    Q

                                −     u(q1x − q1x ) +       uhm (u2 − un ) −      q(u2x − un )
                                          n
                                                         2            2x
                                   Q              Q             Q

                                +                                          (57)
                                     u(0, t)q1 (0, t) (0, t).
                                   0
By the uniform estimates from Lemma 3 and (52), we can pass to the limit in (57) and obtain for n → ∞ and then
m→∞

    u(0, ) (0) +          u(·, ) =                                                   (58)
                                   u(0, t)q1 (0, t) (0, t).
                                0
                 I
              ¯
We choose = j ∈ C ∞ (I ) with j → sign(u(·, )) pointwise and in L2 (I ), as j → ∞, where sign(s) = 1 if s                       0
and sign(s) = −1 if s < 0. Then, by Lemma 3

    u(0, ) j (0) +                           |u(0, t)|,
                     u(·, )        c
                            j
                                   0
                   I
with c independent of j. Passing to the limit j → ∞ we deduce

    |u(0, )| +        |u(·, )| c           |u(0, t)|                                      (59)
                              0
               I
and Gronwall’s Lemma with u0 = 0 implies u(0, t) = 0 in (0, ). Hence, from (59) we also deduce u = 0 in Q , and
therefore u1 = u2 a.e. in Q for any ∈ (0, T ). Checking that this implies q1 = q2 is straightforward.

Proof of Lemma 3. Since problem (54)–(56) and (49) is linear with smooth coefficients and data, existence, unique-
ness and regularity of solutions is well known, see for instance [22]. We assert that the global maximum of | |, given by
max | | = max{max , − min }, is attained initially at t = . On the contrary, suppose that max | | = max (positive)
              ¯
is attained at (x0 , 0 ) ∈ I × (0, ). Then t (x0 , 0 ) = 0. If x0 ∈ I then we also have x (x0 , 0 ) = 0. And, in fact,
this is also the case at the boundaries x = 0 and x = d due to the boundary conditions (49) satisfied by . Then, the
 -equation at (x0 , 0 ) yields

     xx (x0 , 0 ) = −h                                0) +
                     m
                                                                          (60)
                       (x0 ,  0 )(u2n (x0 , 0 )     (x0 ,     (x0 ,  0 )).

We assert that the right-hand side of (60) is positive, leading then to a contradiction to the assumption of (x0 ,               0 ) being
a maximum of . Suppose the contrary. Then using hm < 0,

                   0) +
    un (x0 ,                              0.                                    (61)
          0)  (x0 ,            (x0 ,   0)
    2

Integrating the -equation of (55) in (0, x) gives
               x
     (x, t) =        (−un (y, t)) (y, t) dy.                                              (62)
                  2x
             0
                G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144              1141


 Therefore, from (61), (62) and assumption (24) we obtain
                        x0
   un (x0 ,                    un (y,        0 ) dy
          0)  (x0 ,  0)              0)  (y,
    2                       2x
                      0
                        x0
                           |un (y,             | (y,
                                0 )| dy    sup       0 )|
                            2x
                      0               y∈(0,x0 )
                        x0
                    =      |un (y,                n
                                0 )| dy                                        (63)
                                      (x0 ,   0 ) < u2 (x0 , 0 )  (x0 ,  0 ),
                            2x
                      0

a contradiction. Therefore, (60) is non-negative and a maximum of cannot be attained at 0 > . The case of max | |=
− min at 0 > is treated in a similar way. Since max | | = 0 is overridden by the initial condition, we deduce that
the global maximum of must be attained at t = , i.e.,      L∞ =    L∞ , which is independent of m and n. To
finish the proof we use (62) and (52) to find                 L∞ (0, ;W 1,1 (I )) , which is also independent
                                 ∞ (Q ) u2
                           ∞ (Q )
                          L      L
of m and n.

Proof of Proposition 1. We first show that ux L in QT . Due to the regularity and compatibility assumptions
(21)–(22), we may differentiate the first equation of Problem P with respect to x to obtain the following problem
for v := ux

   vt + qv x − vxx − (2f + uf u )v = fx u             in QT ,                                    (64)

with v(0, ·) = ut (0, ·), v(d, ·) = 0 on (0, T ), and v(·, 0) = u0x on I. By assumption H1 and (26), we may apply
the maximum principle to (64) to deduce that the maximum of v must be non-negative and located on the parabolic
boundary. If the maximum is at x = d then we finished, since v(d, t) = 0. Let us examine the cases in which the
maximum is at x = 0 or at t = 0. Since, by assumption, u0x L in I, we have v(x, t) max{maxt∈(0,T ) v(0, t), L}
in QT . Suppose that the maximum of v is attained at (0, t0 ). Particularizing the differential equation satisfied by u at
(0, t0 ) and using the dynamic boundary condition we deduce

   ux (0, t0 )(1 + q(0, t0 )) = u(0, t0 )f (0, u(0, t0 )) + uxx (0, t0 ).

Since at a maximum of v, uxx (0, t0 ) = vx (0, t0 ) 0, we obtain

   v(0, t0 )(1 + q(0, t0 )) u(0, t0 )f (0, u(0, t0 ))

and then v(0, t0 ) f (0, um ) L, by (25). Therefore, ux = v                  L in QT .
 Define w(x, t) := u(x, t) − Lx for (x, t) ∈ QT . Then wx                   0 in QT and |ux |     − wx + L. Using (20) and (25) we
obtain
      x
        |ux | u(0, t) − u(x, t) + 2L 1 − um + 2L < um                 u(x, t) in QT .
    0

Proof of Theorem 3. First, we analyze the case in which q(0, t0 ) = 0 for some t0 0. From Eq. (15) and the boundary
condition (17) we deduce I f (·, u(·, t0 )) = 0, which implies u(·, t0 ) ≡ 1 in [0, 1] by assumption H1 and the continuity
of u. If the interval I = (0, d) has d = 1 then we finished. Otherwise, in (1, d) × (t0 , ∞) function u satisfies the equation
ut − uxx = 0, the boundary conditions u(1, t) = 1 and ux (d, t) = 0, for t t0 and the initial data u(·, t0 ) 0, and function
q is identically zero. It is then a standard result that u → 1 in L2 (1, d) as t → ∞.
  Let us now assume that q(0, t) > 0 for all t 0, and let : [0, T ] → [0, 2] be given by (t) = u(0, t) + I u(x, t) dx.
Integrating the u-equation (14) in I and using (20), we find
                  T                     T
    (T ) − (0) =           q(0, t)u(0, t) dt > um                                            (65)
                                         q(0, t)
                0                     0

and therefore
      ∞
        q(0, t) is bounded.                                                         (66)
    0
1142                G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144


We now obtain uniform estimates in time for ux and ut in the norms of L∞ ((0, ∞); L2 (I )) and L2 (Q∞ ). Multiplying
the u-equation (14) by u, integrating in QT and using (15) we get

    u2 (0, T ) +       u2 (·, T ) + 2          |ux |2 = u2 (0) +      u2 +        f (·, u)(u2 + u2 (0, t))
                                      0          0
              I              QT                I         QT
                                                                T
                                       2+2        f (·, u) = 2 + 2          q(0, t).
                                                              0
                                           QT

Therefore, estimate (66) implies
                is bounded.                                                           (67)
    ux   L2 (Q∞ )

Multiplying the u-equation by ut , integrating in QT and defining
(·, s) = sf (·, s) with
(·, 0) = 0, we get
                  T             1
        |ut |2 +        |ut (0, ·)|2 +         |ux (·, T )|2 +    
(·, u0 )
                               2
                0
     QT                            I            I
        1                             1                 1
            |u0x |2 +       
(·, u(·, T )) +                     +       |ut |2 .
                                         2       2
                                      q     ux
                                         L∞       L2
        2                             2                 2
          I           I                                  QT

Therefore, using (67) we deduce
                                        ut (0, ·)
                                    and                    are bounded.                 (68)
    ux   L∞ (0,∞;L2 (I )) ,       ut  L2 (Q∞ )                 L2 (0,∞)

                                                                        t ∗ , given by
Now we argue as in [21]. Let (u0 ) be the -limit set of the semi-orbits u(·, t), for t
    (u0 ) = {U ∈ H 1 (I ) : ∃tn → ∞ such that u(·, tn ) → U in L2 as n → ∞}.
Due to the bound of the gradient (68) the -limit set is well defined and non-empty. Let U = limn→∞ u(·, tn ) in L2 (I )
and a.e. in I. By the dominated convergence theorem, function Q given by
                                 d                d
    Q(x) = lim q(x, tn ) = lim                   f (·, u(·, tn )) =                                    (69)
                                                    f (·, U (·))
          n→∞             n→∞ x                     x

is well defined for a.e. x ∈ I . Consider the function Un (x, s) = u(x, tn + s) for x ∈ I , and s ∈ (−1, 1). We have
                               tn +1                                  tn +1
      |Un (·, s) − u(·, tn )|2                  |ut |2      |Un (0, s) − u(0, tn )|               |ut (0, ·)|2 .
                                        and
                             tn −1                                   tn −1
    I                      I

Hence, both
    Un − u(·, tn )                         Un (0, ·) − u(0, tn )
                  2                                   2
                             and
                  L2 (I ×(−1,1))                            L2 (−1,1)

tend to zero as n → ∞, due to the time derivative bounds (68). Define
                             d                     d
    Qn (x, s) = q(x, tn + s) =               f (·, u(·, tn + s)) =           f (·, Un (·, s)).
                           x                     x

By the dominated convergence theorem, we have Qn → Q in L2 (I × (−1, 1)) and a.e. in I × (−1, 1). Finally, showing
that the limit is (U, Q) = (1, 0) is standard. Let ∈ C 2 (I ) with (0) = (d) = 0, and ∈ C0 (−1, 1) such that
                                             2           0
    1
and −1 = 1. Multiplying the u-equation (14) by (t, x) = (t − tn ) (x), integrating in Qtn +1 and changing to the
variable s = t − tn leads to
      1
        ( (s) (0) + (s) (0)Qn (0, s))Un (0, s) +                         +      +            = 0.
                                                (              x Qn )Un
                                                        xx
    −1                                         I

Passing to the limit n → ∞ and using the properties of function                         we find

            +  x U Q) +      (0)U (0)Q(0) = 0.                                                (70)
      (  xx U
    I
               G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144               1143


Choosing (x) = 1, we deduce U (0)Q(0) = 0. Since for all t > 0 we have u(0, t) um > 0, we deduce U (0) > 0, and
then Q(0) = 0, implying Q ≡ 0 in I and U = 1 in [0, 1], due to (69) and the properties of f. Finally, since U ∈ H 1 (I )
we may integrate by parts in (70) to get, for any ∈ C 2 (I ) with (0) = (d) = 0,

           = 0,
       x Ux
     I

so U is constant in I, i.e., U ≡ 1 in I.

Proof of Corollary 1. Let (u, q) be a solution of Problem P corresponding to the initial data u0 . By Theorem 3 we
have that u(·, t) → 1 in L2 (I ) and u(0, t) → 1 pointwise as t → ∞. Therefore, for all M > 0 there exists T0 < ∞
such that

       (1 − u(·, T0 ))2 + (1 − u(0, T0 ))2 1 +         f (·, u) < M.
     I                            QT

We finish using (29) and Theorem 4 for Problem P with u0 := u(·, T0 ).


Acknowledgements

 The essentials of the modelling which gave rise to this article were worked out during a stay of the first author at the
University of Eindhoven. We thank Professors Hans van Duijn and Peter Raats for their valuable help.


References

[1] H. Amann, J. Escher, Strongly continuous dual semigroups, Ann. Mater. Pura et Appl. CLXXI (1996) 41–62.
[2] J.M. Arrieta, P. Quittner, A. Rodríguez-Bernal, Parabolic problems with nonlinear dynamical boundary conditions and singular initial data,
   Differential Integral Equations 14 (2001) 1487–1510.
[3] S.N. Antontsev, J.I. Díaz, S.I. Shmarev, The support shrinking properties for local solutions of quasilinear parabolic equations with strong
   adsorption terms, Ann. Fac. Sci. Toulouse 4 (1) (1995) 5–30.
[4] S.N. Antontsev, J.I. Díaz, S.I. Shmarev, Energy Methods for Free Boundary Problems, Birkhäuser, Boston, 2002.
[5] O. Arino, S. Gauthier, J.P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations,
   Funkcial Ekvac. 24 (1987) 273–279.
[6] J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.
[7] I. Bejenerau, J.I. Díaz, I.I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic
   boundary conditions, Election. J. Differential Equations 50 (2001) 1–19.
[8] L. Botero, Massive mangrove mortality of the Caribbean coast of Colombia, Vida Silve. Neot. 2 (1990) 77–78.
[9] H. Brezis, Analise Fonctionnelle, Théorie et Applications, Masson, Paris, 1987.
[10] R. Courant, D. Hilbert, Methoden der Mathesmatischen Physik II, Springer, Berlin, 1965.
[11] J.I. Díaz, R. Jiménez, Aplicación de la teoría no lineal de semigrupos a un operador pseudodiferencial, Actas VII CEDYA, 1984, pp. 137–182.
[12] C.J. van Duijn, G. Galiano, M.A. Peletier, A diffusion–convection problem with drainage arising in the ecology of mangroves, Interfaces Free
   Boundaries 3 (2001) 15–44.
[13] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Commun. Partial Differential Equations 18 (1993) 1309–1364.
[14] W. Feller, The parabolic differential equations and the associated semi-groups of transforms, Ann. Math. 55 (1952) 468–519.
[15] J. Filo, S. Luckhaus, Modelling surface runoff and infiltration of rain by an elliptic–parabolic equation coupled with a first order equation on
   the boundary, Arch. Ration. Mech. Anal. 146 (1999) 157–182.
[16] A. Friedman, M. Shinbrot, The initial value problem for the linearized equations of water waves, J. Math. Mech. 17 (1968) 107–180.
[17] E. González, El manglar de la Ciénaga Grande de Santa Marta, ecosistema en peligro de extinción, Colombia, sus gentes y regiones, vol. 21,
   1991, pp. 2–21.
[18] P. Hutchings, P. Saenger, Ecology of Mangroves, University of Queensland Press, Queensland, 1987.
[19] J. Kaˇ ur, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions, Math. Slovaca 30 (1980) 213–237.
      c
[21] M. Langlais, D. Phillips, Stabilization of solutions of nonlinear and degenerate evolution equations, Nonlinear Anal. Theoret. Methods Appl.
   9 (4) (1985) 321–333.
[22] G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.
[23] J.L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Gauthiers-Villars, Paris, 1969.
[24] C. Paliyavuth, B. Clough, P. Patanaponpaiboon, Salt uptake and shoot water relations in mangroves, Aquatic Botany 78 (2004) 349–360.
[25] J.B. Passioura, M.C. Ball, J.H. Knight, Mangroves may salinize the soil and in so doing limit their transpiration rate, Funct. Ecol. 6 (1992)
   476–481.
1144           G. Galiano, J. Velasco / Nonlinear Analysis: Real World Applications 7 (2006) 1129 – 1144


[26] W. Peddie, Note on the cooling of a sphere in a mass of well stirred liquid, Proc. Edinburgh Math. Soc. 19 (1901) 34–35.
[27] R.L. Peek, Solutions to a problem in diffusion employing a non-orthogonal sine series, Ann. Math. 30 (1929) 265–269.
[28] L. Perdomo, I. Ensminger, L.F. Espinosa, C. Elster, M. Wallner-Kersanach, M.L. Schnetter, The mangrove ecosystem of the Ciénaga Grande
   de Santa Marta (Colombia): observations on regeneration and trace metals in sediment, Marine Pollution Bull. 37 (1998) 393–403.
[29] M. Primicerio, J.F. Rodrigues, The Hele–Shaw problem with nonlocal injection condition, in: Nonlinear Mathematical Problems in Industry,
   Gatuko, Tokyo, 1993.
[30] S. Selberher, Analysis and Simulation of Semiconductor Devices, Springer, Wien, 1984.
[31] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Math. Pures et Appl. 146 (1987) 65–96.
[32] V.A. Solonnikov, E.V. Frolova, Lp -theory for the Stefan problem, J. Math. Sci. (2000) 989–1006.