Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 08, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu A PARABOLIC SYSTEM WITH STRONG ABSORPTION MODELING DRY-LAND VEGETATION JESÚS ILDEFONSO DÍAZ, DANIELLE HILHORST, PARIS KYRIAZOPOULOS Abstract. We consider a variant of a nonlinear parabolic system, proposed by Gilad, von Hardenberg, Provenzale, Shachak and Meron, in desertification studies, in which there is a strong absorption. The system models the mutual interaction between the biomass, the soil-water content w and the surface- water height which is diffused by means of the degenerate operator ∆hm with m ≥ 2. The main novelty in this article is that the absorption is given in terms of an exponent α ∈ (0, 1), in contrast to the case α = 1 considered in the previous literature. Thanks to this, some new qualitative behavior of the dynamics of the solutions can be justified. After proving the existence of non-negative solutions for the system with Dirichlet and Neumann boundary conditions, we demonstrate the possible extinction in finite time and the finite speed of propagation for the surface- water height component h(t, x). Also, we prove, for the associate stationary problem, that if the precipitation datum p(x) grows near the boundary of the domain ∂Ω as d(x, ∂Ω) 2α m−α then hm(x) grows, at most, as d(x, ∂Ω) 2 m−α . This property also implies the infinite waiting time property when the initial datum h0(x) grows at fast as d(x, ∂S(h0)) 2m m−α near the boundary of its support S(h0). 1. Introduction We study a parabolic system which captures the interactions between vegetation and water in arid and semi-arid porous areas such as modeled in [16]. A slight variation in the modeling is introduced in order to get some new qualitative behavior by its solutions. We consider non-dimensionalized system ∂tb = db∆b+ wG1(b)(1− b)b− b, ∂tw = dw∆w − (L(b) +G2(b))w + I(b)hα, ∂th = dh∆hm − I(b)hα + p. (1.1) Here, b represents the concentration of the above ground biomass, w the soil water content and h the height of a thin surface water layer per unit area. The equation for the evolution of biomass involves a water dependent growth rate G1(b), a mortality term with constant loss rate and a linear diffusion term modeling growth due to seeds dispersal or clonal growth. In the equation for the soil water, we have a loss 2010 Mathematics Subject Classification. 35K55, 35K65, 35B05. Key words and phrases. Nonlinear parabolic system; dry-land vegetation; positive solution; free boundary problem. c©2021 Texas State University. Submitted October 12, 2020. Published February 20, 2021. 1 2 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 term which consists of the water up-take rate by the plant roots denoted by G2(b) and the biomass dependent evaporation rate L(b). Moreover, the equation contains the source term I(b)hα representing the infiltrated surface water, which is discussed in more detail below, and a linear diffusion term modeling the soil water transport. The main novelty in this paper is that the exponent α will be assumed such that α ∈ (0, 1), in contrast to the case α = 1 considered in the previous literature. The third equation models the surface water flow and how this infiltrates into the ground. The variable h corresponds to the dimensionless quantity H = ρd, where d(t̃, x̃) represents the depth in meters of the surface water where x̃ ∈ Ω ⊂ R2 and ρ is the constant density of the fluid. In fact, the third equation in dimensional quantities (and m ≥ 2) can be derived from the continuity equation ∂H ∂t̃ + div(H~u) = P − IH and the shallow water momentum equation D−→u ∂t̃ = −g∇(ζ + d) + 1 ρ F, where ~u is the horizontal velocity of the fluid, F represents ground surface friction, P stands for the precipitation rate and IH is the infiltration rate of water through the soil surface. Moreover, ζ denotes the height of the soil surface and g stands for the acceleration of gravity. For non-trivial land topographies ζ(x̃) is a nonnegative function of the space variable x̃ and it is convenient to set Z = ρζ. We consider a friction term of the form F = −k~u/dl, for l ≥ 0 and k > 0, a biomass and surface water dependent infiltration rate term of the form I(B,H) = IB(B)IH(H) and we let ζ = 0 which corresponds to a region with flat topography. Then ∂H ∂t̃ − c∆Hm = P − I(B,H)H, where m = l + 2 and c = g/(mkρl). The biomass dependent infiltration rate IB captures the infiltration contrast between vegetated regions and bare soil due to the formation of biogenic crusts in non-vegetated regions which reduce the infiltration of surface water. Therefore, this term is monotonically increasing with B approaching a constant infiltration for high biomass concentrations. The counterpart IH of the infiltration rate in this paper is chosen to be a decreasing function of H, taking the explicit form Hα−1 for α ∈ (0, 1). Other models related to desertification studies can be found, for instance, in [1, 21]. From the mathematical viewpoint, we mention the study of the corresponding dynamical system in the case α = 1 made in [18, 17]. Notice that, curiously for the associated stationary system (considered in [10, 9]) the assumption α ∈ (0, 1) does not introduce any big change in the problem, since the change of variables ĥ = hm leads to the stationary equation −dh∆ĥ+ Ib(ĥ)α/m = p which involves an exponent α/m < 1 even for α = 1. So, the modifications implied by the assumption α ∈ (0, 1) mainly affect the dynamics of solutions of (1.1). This article is organized as follows. In Section 2 we complete the mathematical formulation of the system (1.1). In particular, we consider two cases of boundary conditions: the Dirichlet and Neumann boundary conditions. For the first case, we define a regularized approximating system which possesses positive bounded EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 3 solutions. This allows us to pass to the limit of the approximating problem prov- ing the existence of solutions for the original problem. For the second case, we use a different approach, specifically, the existence of solutions is given by a fixed point argument employing a fixed point theorem for sequentially weakly continuous mappings in Banach spaces. Section 3 is devoted to the qualitative behavior of so- lutions. We examine the behavior, in time, of the vanishing set of the surface water component h in the absence of precipitation during sufficiently long time intervals. The spatial location of the vanishing set of h is also analyzed. 2. Existence of solutions In what follows, we denote by Ω a bounded domain in R2 with regular boundary ∂Ω and for T > 0 we let QT = Ω× (0, T ) and ST = ∂Ω× (0, T ). Our purpose is to prove the existence of a solution U = (b, w, h) of the system ∂tb = db∆b+ wG1(b)(1− b)b− b, in QT , ∂tw = dw∆w − (L(b) +G2(b))w + I(b)hα, in QT , ∂th = dh∆hm − I(b)hα + p, in QT , (2.1) with the initial conditions b(x, 0) = b0(x), w(x, 0) = w0(x), h(x, 0) = h0(x) for x ∈ Ω, (2.2) and boundary conditions which are either of Dirichlet typ b = w = h = 0, on ∂Ω× (0, T ), (2.3) or of Neumann type ∂b ∂n = ∂w ∂n = ∂hm ∂n = 0, on ∂Ω× (0, T ). (2.4) We shall assume b0, w0, h0 ∈ L∞(Ω), (2.5) and to obtain more regularity we will additionally assume that b0, w0, h0 ∈ C(Ω̄). (2.6) In any case, we are specifically interested in the case in which the initial data satisfy 0 ≤ b0 ≤ 1, w0 ≥ 0, h0 ≥ 0, on Ω. (2.7) Concerning the precipitation term p, we assume that p ∈ L∞(QT ) is nonnegative. Moreover, we suppose the structural conditions I(b) = θ b+ r/c b+ r , (2.8) L(b) = ν 1 + ρb , (2.9) G1(b) = ν(1 + ηb)2, (2.10) G2(b) = γb(1 + ηb)2, (2.11) and that db, dw, dh, η, ρ, r, ν, θ are given positive constants and that c ≥ 1. For later use we also note that for s ∈ [0, 1], I(s), G1(s), G2(s) are nondecreasing functions anL(s) is non-increasing function, so I(0) ≤ I(s) ≤ I(1), for s ∈ [0, 1] (2.12) 4 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 and L(1) ≤ L(s) +G2(s) ≤ L(0) +G2(1), for s ∈ [0, 1]. (2.13) In what follows we refer to the Dirichlet problem (2.1), (2.3) and (2.2) as Problem (PD), and to the Neumann problem (2.1), (2.2) and (2.4) as Problem (PN ). To begin with we define the notions of weak solutions of both problems. Definition 2.1. We call (b, w, h) a weak solution of Problem (PD) on [0, T ], if it satisfies (1) (b, w, h) ∈ C([0, T ] : L1(Ω)3) ∩ L∞(QT )3 and b, w, hm ∈ L∞(0, T ;H1 0 (Ω)) (2) for all ψ ∈ C1(QT ) ∩ L2(0, T ;H1 0 (Ω)) ∫ Ω b(t)ψ(t) + db ∫ t 0 ∫ Ω {∇b · ∇ψ − bψt} dx dτ = ∫ Ω b0ψ(0) + ∫ t 0 ∫ Ω {G1(b)w(1− b)b− b}ψ, (2.14) ∫ Ω w(t)ψ(t) + dw ∫ t 0 ∫ Ω {∇w · ∇ψ − wψt} dx dτ = ∫ Ω w0ψ(0) + ∫ t 0 ∫ Ω {−(L(b) +G2(b))w + I(b)hα}ψ, (2.15) ∫ Ω h(t)ψ(t) + dh ∫ t 0 ∫ Ω {∇hm · ∇ψ − hψt} dx dτ = ∫ Ω h0ψ(0) + ∫ t 0 ∫ Ω {p− I(b)hα}ψ dx dτ. (2.16) Definition 2.2. We call (b, w, h) a weak solution of Problem (PN ) on [0, T ], if it satisfies (1) U ∈ C([0, T ] : L1(Ω)3) ∩ L∞(QT )3 and b, w, hm ∈ L∞(0, T ;H1(Ω)), (2) for all ψ ∈ C1(QT ) ∩ L2(0, T ;H1(Ω)) b, w, h satisfy (2.14)–(2.16). In the case of Dirichlet boundary conditions we shall be able to prove, addition- ally, that the weak solutions are in fact continuous functions if (2.6) holds. Other regularity properties could be obtained by different techniques (see, e.g. [23]). 2.1. Regularized system for (PD). The main difficulty for the study of Problem (PD) is the fact that the equation for h is degenerate. Here, we overcome this difficulty by defining a sequence of approximating uniformly parabolic problems for which classical solutions exist. Finally, we prove existence of Problem (PD) by passing to the limit thanks to some a priori estimates. For ε ∈ (0, 1), κ ≥ 1 and 0 < α < 1, we let φε(s) := (s+ ε)m − εm, fε(s) := (s+ ε)α − εα. EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 5 We consider the regularized system ∂tbε − db∆bε + bε = G1(bε)wε(1− bε)bε, in QT , ∂twε − dw∆wε + (L(bε) +G2(bε))wε = I(bε)fε(hε), in QT , ∂thε − dh∆(φε(hε)) + I(bε)fε(hε) = pε, in QT , b = w = h = 0, on ST , bε(x, 0) = b0,ε(x), wε(x, 0) = w0,ε(x), hε(x, 0) = h0,ε(x), for x ∈ Ω, (2.17) where pε ∈ C∞ such that 0 ≤ pε ≤ ‖p‖L∞(QT ), (2.18) ‖pε − p‖L1(QT ) → 0 as ε→ 0, (2.19) for T > 0 arbitrary and the initial conditions b0,ε, w0,ε, h0,ε, with bε(0) = b0,ε, wε(0) = w0,ε, hε(0) = h0,ε ∈ C∞c (Ω), such that 0 ≤ b0,ε(x) ≤ ‖b0‖L∞(Ω), 0 ≤ w0,ε(x) ≤ ‖w0‖L∞(Ω), 0 ≤ h0,ε(x) ≤ ‖h0‖L∞(Ω) (2.20) for a.e. x ∈ Ω and (b0,ε, w0,ε, h0,ε)→ (b0, w0, h0) in L1(Ω)3 as ε→ 0. (2.21) We also note for later use that (2.18), (2.19) and (2.20), (2.21) imply pε → p in Lq(QT ) as ε→ 0, (2.22) and (b0,ε, w0,ε, h0,ε)→ (b0, w0, h0) in Lq(Ω)3 as ε→ 0, (2.23) for all q > 1. Under the above considerations the following result holds. Theorem 2.3. For every ε ∈ (0, 1), problem (2.17) possesses a unique classical solution (bε, wε, hε) such that 0 ≤ bε ≤ 1, in QT (2.24) and there exists a positive constant C̄ such that 0 ≤ wε, hε ≤ C̄, in QT , (2.25) where C̄ does not depend on ε. Proof. The existence of a classical solution of (2.17) for the non-negative initial data (b0,ε, w0,ε, h0,ε) follows from [20]. Moreover, from the classical maximum principle we have that for bε,0 ∈ [0, 1], 0 ≤ bε ≤ 1. Similarly, we can show that wε, hε ≥ 0. Next we prove that hε is bounded from above. We first recall the definition of the negative and positive parts of a function f , namely (f)+ = max{f, 0}, (f)− = max{−f, 0}. We set ĥ = hε − h̄, with h̄ an arbitrary positive constant to be determined later. We multiply the equation for hε in (2.17), by ĥ+ and integrate over Ω to obtain∫ Ω ∂hε ∂t ĥ+ dx− dh ∫ Ω ∆φε(hε)ĥ+ dx+ ∫ Ω I(bε)fε(hε)ĥ+ dx = ∫ Ω pεĥ+ dx, (2.26) 6 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 which implies 1 2 d dt ∫ Ω |ĥ+|2 dx+ dh ∫ Ω φ′ε(hε)|∇ĥ+|2 dx+ ∫ Ω I(bε)fε(hε)ĥ+ dx = ∫ Ω pεĥ+ dx. (2.27) From (2.12), (2.27) and the fact that, φ′ε(hε) > 0, 0 ≤ pε(t, x) ≤ ‖p‖L∞(QT ), we have 1 2 d dt ∫ Ω |ĥ+|2 + I(0) ∫ Ω fε(hε)ĥ+ ≤ ‖p‖L∞(QT ) ∫ Ω ĥ+ dx, (2.28) from which we infer that 1 2 d dt ∫ Ω |ĥ+|2 + I(0) ∫ Ω (fε(hε)− fε(h̄))(hε − h̄)+ ≤ ‖p‖L∞(QT ) ∫ Ω ĥ+ dx, (2.29) which we may write as 1 2 d dt ∫ Ω |ĥ+|2 + I(0) ∫ Ω (fε(hε)− fε(h̄))(hε − h̄)+ ≤ (‖p‖L∞(QT ) − I(0)fε(h̄)) ∫ Ω ĥ+ dx, (2.30) Thanks to the monotonicity of φε(·), the second term on the left-hand side of the above inequality is nonnegative. Next, we look for h̄ > ‖h0‖L∞(Ω) such that ‖p‖L∞(QT ) − I(0)fε(h̄) ≤ 0. (2.31) Since, −fε(h̄) ≤ (1− h̄α), we may choose h̄ := max {(‖p‖L∞(QT ) I(0) + 1 )1/α , ‖h0‖L∞(Ω) } , (2.32) so that d dt ∫ Ω |ĥ+|2(t) dx ≤ 0, (2.33) which in turn implies |ĥ+(t)|2L2(Ω) ≤ |ĥ+(0)|2L2(Ω) = |(h0 − h̄)+|2L2(Ω) = 0 , (2.34) for h̄ given by (2.32), and so hε ≤ h̄ in QT . (2.35) To obtain an upper bound for wε we work similarly. We set ŵ = wε − w̄, where w̄ is a positive constant to be determined later, we multiply the equation for wε in (2.17) by ŵ+ and integrate over Ω to obtain that 1 2 d dt ∫ Ω |ŵ+|2 dx+ δw ∫ Ω |∇ŵ+|2 dx+ + ∫ Ω (L(bε) +G2(bε))(|ŵ+|2 + w̄ŵ+) dx = ∫ Ω I(bε)fε(hε)ŵ+ dx. (2.36) EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 7 where we have used the equality wε = (wε − w̄)+ − (wε − w̄)− + w̄. Then, from (2.12), (2.35) and the fact that fε(s) ≤ sα for s ≥ 0, we have∫ Ω I(bε)fε(hε)ŵ+ dx ≤ I(1)h̄α ∫ Ω ŵ+ dx. (2.37) So, using (2.13) and dropping the appropriate non-negative terms on the left hand side of (2.36), we end up with 1 2 d dt ∫ Ω |ŵ+|2 dx+ (w̄L(1)− I(1)h̄α) ∫ Ω ŵ+ dx ≤ 0. (2.38) Therefore, arguing as before we may choose w̄ := max {I(1)h̄α L(1) , ‖w0‖L∞(Ω) } , so that wε ≤ w̄ in QT . � Next we remark that bε solves a problem of the form bεt = db∆bε + Fε in QT bε = 0 on ST bε(x, 0) = b0,ε(x) in Ω (2.39) where 0 ≤ b0,ε ≤ ‖b0‖L∞(Ω) ≤ 1 and Fε ∈ L∞(QT ). (2.40) Multiplying the equation by bε and integrating by parts, ‖bε‖L2(0,T ;H1 0 (Ω)) ≤ C, (2.41) Further taking the duality product 〈·, ·〉(H−1,H1 0 ) of bεt with an arbitrary test func- tion from L2(0, T ;H1 0 (Ω)), we deduce that ‖bεt‖L2(0,T ;H−1(Ω)) ≤ C. (2.42) Then (2.41) and (2.42) imply that {bε} is relatively compact in L2(QT ). cf. [22, Theorem 2.1 p. 27]. We deduce that there exist a function b ∈ L2(0, T ;H1 0 (Ω)) with bt ∈ L2(0, T ;H−1(Ω)) and a subsequence {bεj} of {bε} such that bεj → b strongly in L2(QT ) bεj t → bt weakly in L2(0, T ;H1 0 (Ω)). Moreover, it follows from [22, Lemma 1.2 p. 260] that b ∈ C([0, T ];L2(Ω)). Finally, it is clear that 0 ≤ b ≤ 1 for all t ∈ [0, T ] and a.e. x ∈ Ω. Since, wε satisfies the equation it follows in a similar way that wε converges along a subsequence to a limit w strongly in L2(QT ) and weakly in L2(0, T ;H1 0 (Ω)) as ε→ 0 where w ∈ C([0, T ];L2(Ω)) and 0 ≤ w ≤ w̄ for all t ∈ [0, T ] and a.e. x ∈ Ω. Next we consider the problem for hε, namely ∂thε = ∆φε(hε)− I(bε)fε(hε) + pε in QT , hε = 0 on ST , hε(x, 0) = h0,ε(x) for x ∈ Ω, (2.43) We first prove the following estimate. 8 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 Lemma 2.4. We have 1 2 ∫ Ω (hε) 2(t) dx+ ∫ T 0 ∫ Ω |∇ψε(hε)|2 dx dt ≤ C(T ) (2.44) where ψε(s) = ∫ s 0 √ φ′ε(s) ds, which in turn implies that∫ T 0 ∫ Ω |∇φε(hε(t))|2 ≤ C(T ). (2.45) Proof. The function hε solves the initial value problem hεt = dh∆φε(hε) +Gε (2.46) with zero Dirichlet boundary conditions where we have set Gε = −I(bε)fε(hε) + pε, and so ‖Gε‖L∞(QT ) ≤ C. We multiply the equation (2.46) by hε and integrate by parts to deduce that 1 2 d dt ∫ Ω (hε) 2(t) dx+ dh ∫ Ω ∇φε(hε) · ∇hε dx dt = ∫ Ω Gεhε dx (2.47) which implies ∫ Ω (hε) 2(t) dx+ 2dh ∫ T 0 ∫ Ω φ′ε(hε)|∇hε|2 dx dt ≤ ∫ Ω G2 ε dx dt+ ∫ T 0 ∫ Ω h2 ε dx dt+ ∫ Ω h2 0ε dx. (2.48) Since ∫ T 0 ∫ Ω φ′ε(hε)|∇hε|2 dx dt = ∫ T 0 ∫ Ω ( √ φ′ε(hε)∇hε)2 = ∫ T 0 ∫ Ω |∇ψε(hε)|2 dx dt, (2.49) we deduce that ∫ Ω (hε(T ))2 dx+ 2dh ∫ T 0 ∫ Ω |∇ψε(hε)|2 ≤ ∫ T 0 ∫ Ω G2 ε dx dt+ ∫ T 0 ∫ Ω h2 ε dx dt+ ∫ Ω h2 0ε dx, (2.50) which in turn yields inequality (2.44). To prove (2.45), we observe that∫ T 0 ∫ Ω |(∇φε(hε(t))|2 = ∫ T 0 ∫ Ω φ′ε(hε(t)) 2|∇hε(t)|2 dx dt ≤ sup |φ′ε(hε(x, t))| ∫ T 0 ∫ Ω φ′ε(hε(t))|∇hε(t)|2 dx dt ≤M ∫ T 0 ∫ Ω (√ φ′ε(hε(t))∇hε(t) )2 , (2.51) where M is independent of ε and so (2.45) follows from (2.49) and (2.44). � Next we set Uε = φε(hε) and βε(·) = φ−1 ε (·), to apply a result from [15, Theorem 6.2]. EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 9 Lemma 2.5. (i) For all τ > 0, the function Uε is equicontinuous in Q̄τT . Pre- cisely, there exists a continuous nondecreasing function ωτ (·) with ωτ (0) = 0, such that |Uε(x1, t1)− Uε(x2, t2)| ≤ ωτ (|x1 − x2|+ |t1 − t2|1/2) (2.52) for all (xi, ti) ∈ Q̄τT , i = 1, 2. The function ωτ does not depend on ε. (ii) If in addition U(0, x) = U0(x) ∈ C(Ω̄), then {Uε} is equicontinuous on Q̄T . We deduce from Lemma 2.5 (i) that for all τ > 0, Uεj is precompact in C(Q̄τT ) and thus there exists a subsequence that we denote again by Uεj and a function ζ ∈ C(Q̄τT ) such that Uεj → ζ, uniformly in Q̄τT as εj → 0. Then |hεj − ζ1/m| = |βε(Uεj )− ζ1/m| ≤ |βεj (Uεj )− (Uεj ) 1/m|+ |(Uεj )1/m − ζ1/m|, (2.53) Therefore, since for all ε > 0, |βε(Uε)− (Uε) 1/m| < 2ε, setting h = ζ1/m we have hεj → h, uniformly in Q̄τT , φε(hεj )→ hm uniformly in Q̄τT , (2.54) as εj → 0, for all τ > 0. Moreover, from Lemma 2.4 there exists a subsequence of {hεj} which we denote again by hεj and a function χ ∈ L2((0, T );H1 0 (Ω)) such that φεj (hεj ) ⇀ χ weakly in L2(0, T ;H1 0 (Ω)), (2.55) as εj → 0. Since, L2(Ω) ⊂ H−1(Ω) we further deduce that φεj (hεj ) ⇀ χ weakly in L2(QT ). On the other hand, φεj (hεj ) ≤ (hεj ) m ≤ h̄m and from (2.54) we have that φεj (hεj ) → hm a.e. in QT . Then, by the dominated convergence theorem we deduce that φεj (hεj ) → φ(h) strongly in L2(QT ). Therefore, φ(hεj ) ⇀ hm weakly in L2(QT ) and uniqueness of the weak limits implies that χ = hm. Hence, we conclude that φεj (hεj ) ⇀ hm weakly in L2(0, T ;H1 0 (Ω)), (2.56) as εj → 0. Next we prove that (b, w, h) is a weak solution of Problem (P ). We multiply the three partial differential equations in (2.17) by ψ ∈ C1(QT ) ∩ L2(0, T ;H1 0 (Ω)) and integrate by parts to obtain (here for simplicity ε = εj)∫ Ω bε(t)ψ(t) + db ∫ t 0 ∫ Ω {∇bε · ∇ψ − bεψt} dx dτ = ∫ Ω b0,εψ(0) + ∫ t 0 ∫ Ω {G1(bε)wε(1− bε)bε − bε}ψ, (2.57) ∫ Ω wε(t)ψ(t) + dw ∫ t 0 ∫ Ω {∇wε · ∇ψ − wεψt} dx dτ = ∫ Ω w0,εψ(0) + ∫ t 0 ∫ Ω {−(L(bε) +G2(bε))wε + I(bε)fε(hε)}ψ, (2.58) ∫ Ω hε(t)ψ(t) + dh ∫ t 0 ∫ Ω {∇φε(hε) · ∇ψ − hεψt} dx dτ = ∫ Ω h0,εψ(0) + ∫ t 0 ∫ Ω {pε − I(bε)fε(hε)}ψ dx dτ. (2.59) 10 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 In summary, bε, wε, hε are positive and bounded. Moreover, there exists a subse- quence of (bε, wε, hε), which converges strongly to (b, w, h) in L2(QT )3, and a.e. in QT , and (bε, wε, φε(hε)) converges weakly to (b, w, φ(h)) in L2(0, T ;H1 0 )3. To pass to the limit in the terms involving fε(hε) we notice that |fε(s) − sα| ≤ 2εα, and thus |I(bε)fε(hε)− I(bε)fε(hε)| ≤ |I(bε)||(fε(hε)− f(hε)|+ |I(bε)− I(b)||f(hε)|+ |I(b)||f(hε)− f(h)| ≤ 2I(1)εα + |I(bε)− I(b)||f(h̄)|+ |I(1)||f(hε)− f(h)| . (2.60) Moreover, recalling (2.22) and (2.23), we can let ε → 0 in (2.57)-(2.59), to obtain the integral identities (2.14)-(2.16). Finally, if h0 ∈ C(Ω̄) from Lemma 2.5 (ii), h ∈ C(Q̄T ). In fact, a weak solution of Problem (PD) exists even if h0 is just essentially bounded, since working as above we know that there exists h ∈ C((0, T ] : L1(Ω)). We would also like to know if ‖h(t)‖L1(Ω) is continuous at 0. To this end let h0,n be a sequence of smooth bounded functions which converges to h0 in L1(Ω). Working as above and using Lemma 2.5 (ii) there exists a solution of the system, denoted by hn, obtained as a limit of the approximating system such that hn ∈ C(Q̄T ). Next note that ‖h(t)− h0‖L1(Ω) ≤ ‖h(t)− hn(t)‖L1(Ω) + ‖hn(t)− h0,n‖L1(Ω) + ‖h0,n − h0‖L1(Ω), (2.61) where the second term on the right hand side goes to zero as t tends to 0, while the last term becomes arbitrarily small for n large enough. On the other hand, for any ε > 0 and h1(0) and h2(0) smooth initial data, the solutions h1,ε and h2,ε of the corresponding approximating problems satisfy ‖h1,ε(t)− h2,ε(t)‖L1(Ω) ≤ ‖h1,ε(0)− h2,ε(0)‖L1(Ω) + ∫ t 0 ‖b1,ε(s)− b2,ε(s)‖L1(Ω) ds ≤ ‖h1,ε(0)− h2,ε(0)‖L1(Ω) + tD. where D is a positive constant. Letting ε→ 0, this in turn implies that ‖h1(t)− h2(t)‖L1(Ω) ≤ ‖h1(0)− h2(0)‖L1(Ω) + tD. (2.62) Finally, by (2.61) and (2.62), ‖h(t)− h0‖L1(Ω) ≤ ‖h(0)− hn(0)‖L1(Ω) + tD + ‖hn(t)− h0,n‖L1(Ω) + ‖h0,n − h0‖L1(Ω) (2.63) Then t 7→ ‖h(t)‖L1(Ω) is continuous at zero and so h ∈ C([0, T ];L1(Ω)). We thus have the following result. Theorem 2.6. If the initial condition (h0, b0, w0) satisfies (2.5) and (2.7), then there exists a weak solution (b, w, h) of Problem (PD) such that 0 ≤ b ≤ 1, 0 ≤ w ≤ w̄ and 0 ≤ h ≤ h̄. If in addition h0, b0, w0 ∈ C(Ω̄), then h ∈ C(Q̄T ) and b, w ∈ C(Ω̄× [δ, T ]) for all δ > 0. Proof. To complete the proof we note that from (2.40) and [20, Theorem 9.1 p. 341] it follows that ‖bε‖W 2,1 q (QδT ) ≤ C(δ, T, q,Ω) (2.64) EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 11 for all δ ∈ (0, T ) and all q ∈ (1,∞), where QTδ = (δ, T ) × Ω and W 2, 1 q (QTδ ) = W 1, q(δ, T ;Lq(Ω)) ∩ Lq(δ, T ;W 2, q(Ω) ∩W 1, q 0 (Ω)). This in turn implies ‖bε‖Cα, α2 (Ω̄×[δ,T ]) ≤ C, (2.65) for α = 2− N+2 q and q 6= N+2 [6, Lemma 3.5]. Therefore, we can conclude (passing if necessary to another subsequence) that bεj → b uniformly in Q̄δT for all δ > 0 and so b ∈ C(Ω̄× [δ, T ]). Similarly, w ∈ C(Ω̄× [δ, T ]). � 2.2. Neumann boundary conditions (PN). Although the above strategy can also be adapted to obtain the existence of weak solutions for the Neumann problem (PN ), in this section, we use a different approach which is based on a fixed point argument. Theorem 2.7. There exists a weak solution of Problem (PN ). Before, giving the proof, it is useful to state a lemma related to the problem ∂tu−∆ϕ(u) = v in QT , ∂ϕ(u) ∂n = 0 on ST , u(x, 0) = u0(x) for x ∈ Ω. (2.66) where ϕ : R → R is nondecreasing continuous function with ϕ(0) = 0, u0 ∈ L1(Ω) and v ∈ L1(0, T ;L1(Ω)). It is known that problem (2.66) possesses a unique weak solution (see [5, 23]). For fixed u0, let us denote by uv the unique weak solution of (2.66) for some v ∈ L1(0, T ;L1(Ω)). Lemma 2.8. Suppose that ϕ : R → R is a strictly increasing continuous function with ϕ(0) = 0, then (i) for each u0 ∈ L1(Ω) and a weakly relatively compact set K in L1(0, T ;L1(Ω)), the set {uv : v ∈ K} is relatively compact in C([0, T ];L1(Ω)), (ii) for each fixed u0 ∈ L∞(Ω) and a bounded set K in L∞(0, T ;L∞(Ω)) the mapping v 7→ uv, is sequentially continuous from K endowed with the weak topology of L1(0, T ;L1(Ω)) into C([0, T ];Lp(Ω)) endowed with the strong topology, for all p ∈ [1,∞). For the proof of Lemma 2.8 (i) we refer to Diaz-Vrabie [12, 13]. It must be pointed out that although the compactness results of the above references concern the case of Dirichlet boundary conditions, the arguments are identical for the case of Neumann boundary conditions (see [13, Section 2] and [5]). On the other hand Lemma 2.8 (ii) is a consequence of the counterpart (i) thanks to the uniqueness of the weak solution (for details see [14, Corollary 1, Section 2] or [14, Corollary 3.1] for the case p = 1). For convenience, if u = (u1, u2, u3) is a vector function with ui ∈ X, where X a Banach space, we shall use the notation ‖u‖X := maxi=1,2,3{‖ui‖X}. Proof of Theorem 2.7. Let us start with the existence of a local (in time) weak solution of (PN ). We introduce the reaction functions R : R3 → R3 given by R(b, w, h) = (R1(b, w, h), R2(b, w, h), R3(b, w, h)) 12 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 with R1(b, w, h) = wG1(b)(1− b)b− b R2(b, w, h) = −(L̃(b) +G2(b))w + Ĩ(b)h|h|α−1 R3(b, w, h) = −Ĩ(b)h|h|α−1. (2.67) where Ĩ(b) (respectively L̃(b)) is a truncation of I(b) (respectively L(b)) extending it continuously by a constant equal to I(0) (respectively L(0)) for b < 0. We choose K > 0 such that max(‖b0‖L∞(Ω), ‖w0‖L∞(Ω), ‖h0‖L∞(Ω)) + 1 ≤ K. Since the functions Ri : R3 → R are continuous it is possible to find M > 0, such that max{|R1(b, w, h)|, |R2(b, w, h)|, |R3(b, w, h)|+ ‖p‖L∞(QT )} ≤M assumed that 0 ≤ b, w, h ≤ K. Now we define the “solution operator” S : L1(0, T : L1(Ω))3 → C([0, T ];L2(Ω))3 by S(f, g, v) = (b, w, h) where b, w, h are the unique weak solutions of the decoupled system ∂tb− db∆b = f in QT , ∂tw − dw∆w = g in QT , ∂th− dh∆hm = v in QT , ∂b ∂n = ∂w ∂n = ∂h ∂n = 0 on ST , b(x, 0) = b0(x), w(x, 0) = w0(x), h(x, 0) = h0(x) for x ∈ Ω. Next, to control a priori estimates it is useful to introduce the following convex set (adapted to the reaction terms R(b, w, h)): Kr,T0 = {(f, g, v) : f, g, v ∈ L1(0, T0 : L1(Ω)), ‖(f, g, v)‖L∞(QT0 ) ≤ r}, where QT0 := (0, T0)× Ω, r ≥M and T0 ∈ (0, T ] is such that S(Kr,T0 ) ⊂ BL∞(QT0 )(0,K) with BL∞(QT0 )(0,K) := {u ∈ L∞(QT0)3 : ‖u‖L∞(QT0 )) ≤ K}. Recall that M depends on K through the properties of R. Moreover, it is not difficult to see that Kr,T0 is nonempty and weakly compact in (L1(0, T0 : L1(Ω)))3. Next let us define the restriction of the solution operator on Kr,T0 : Ŝ = S ∣∣ Kr,T0 : Kr,T0 → L∞(QT0 )3. We also define the composition of the realization operator associated to R and Ŝ, namely, the operator R : Kr,T0 → C([0, T ];L2(Ω))3 defined by R(f, g, v) = (R1(Ŝ(f, g, v)), R2(Ŝ(f, g, v)), R3(Ŝ(f, g, v)) + p) i.e. R(f, g, v) = R(b, w, h) with (b, w, h) = Ŝ(f, g, v). Then, from the choice of the set Kr,T0 we know that R maps Kr,T0 into Kr,T0 . Next we prove that there exists at least one fixed point of R : Kr,T0 → Kr,T0 . This will be a consequence of a variant of the Schauder fixed point theorem given in [24, Theorem 1.2.11], which requires R to be weakly-weakly sequentially con- tinuous. It is actually enough to show that the graph of R, is weakly-weakly sequentially closed [24, Corollary 1.2.5]. To this end, let {(fn, gn, vn)}n∈N ∈ Kr,T0 EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 13 and {(Fn, Gn, Vn)}n∈N ∈ R(fn, gn, vn) be sequences which converge weakly in (L1((0, T );L1(Ω)))3 to (f, g, v) and (F,G, V ), respectively. Then from Lemma 2.8 (ii) Ŝ is weakly-strongly sequentially continuous from L1(0, T0;L1(Ω)) into C([0, T0];Lp(Ω)) and so we may assume without loss of generality (taking a subse- quence if necessary) that Ŝ(fn, gn, vn)→ Ŝ(f, g, v) a.e. in QT0 , (2.68) which, by continuity of Ri, implies Ri(Ŝ(fn, gn, vn))→ Ri(Ŝ(f, g, v)) a.e. in QT0 . (2.69) Moreover, Ri(Ŝ(fn, gn, vn)) is a.e. bounded in QT0 and therefore by the domi- nated convergence theorem we have that R(fn, gn, vn) → R(f, g, v) strongly in (L1(QT0 ))3. Consequently, by the uniqueness of weak limits, (F,G, V ) = R(f, g, v). Therefore, the graph of R is weakly-weakly sequentially closed and so R has at least one fixed point (f, g, v). Since (b, w, h) = Ŝ(f, g, v) we conclude that (b, w, h) is a weak solution of the problem (PN ) on the cylinder QT0 := (0, T0) × Ω, i.e. a local (in time) solution of (PN ) on QT0 . It only remains to prove that no possible blow-up of the norm in C([0, T ];L2(Ω))3 may arise to get the continuation of the local weak solution to the whole cylinder QT . But for the reaction terms R(b, w, h) given by (2.67) and for positive initial conditions satisfying (2.5), (2.7) this is an easy task: indeed, similar arguments to the ones of the proof of Theorem 2.3 show that the local weak solution satisfies 0 ≤ b ≤ 1, 0 ≤ w ≤ C a.e. in QT0 , where C > 0 is independent of T0 and, by well-known estimates for the porous medium with monotone absorption 0 ≤ h ≤ ‖h0‖L∞(Ω) + T‖p‖L∞(QT ) a.e. in QT0 which is also independent of T0. Therefore, the local weak solution can be extended, by taking T0 as initial time and the values of b, w, h, at t = T0 as new initial data, to the complete cylinder QT producing at least one global weak solution of (PN ) in view of the fact that b, w, h are nonnegative. � The above type of arguments can be applied to prove the convergence of some numerical algorithms that relies in suitable decoupling of the system and applying the Dı́az-Vrabie ([12]) ad hoc compactness argument. For instance we can consider the following iterative argument: we solve the uniformly parabolic equations by prescribing the h-component ∂tbn = db∆bn + wnG1(bn)(1− bn)bn − bn in QT , ∂twn = dw∆wn − (L(bn) +G2(bn))wn + I(bn)hαn−1 in QT , (2.70) with the initial conditions bn(x, 0) = b0(x), wn(x, 0) = w0(x), for x ∈ Ω, (2.71) and Neumann boundary conditions ∂bn ∂n = ∂wn ∂n = 0, on ∂Ω× (0, T ). (2.72) Then we solve the degenerate equation by prescribing the b-component (Ph,n): ∂thn = dh∆hmn − I(bn−1)hαn + p in QT , 14 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 ∂hn ∂n = 0 on ∂Ω× (0, T ), hn(x, 0) = h0(x) for x ∈ Ω. Obviously the iteration starts with the initial data. The existence of weak solutions for the decoupled problems are easy modifications of previous results in the litera- ture (or they can be obtained by following some ideas of the preceding section for the treatment of the Dirichlet case). The convergence of the algorithm is a small variant of the proof of Theorem 2.7. 3. Qualitative properties of the surface water component In this section, we focus on the qualitative properties of the surface water compo- nent h investigating the impact of dry periods on the zero set of h. More precisely, we start by assuming that precipitation is negligible for sufficiently long time, in the sense that p(t) = 0 for t ∈ (0, T ), (3.1) with T large enough. Then, we will show that h vanishes after a finite time for the Dirichlet boundary conditions. As a second qualitative property, we will consider a compactly supported initial condition h0 and we will show that h has a compact support (which defines a free boundary during a dry period in which p = 0). In what follows, without loss of generality we suppose that δh = 1. We let (b, w, h) be a solution of system (2.1) for a non-negative and bounded initial datum (b0, w0, h0), with 0 ≤ b0 ≤ 1. To determine the properties of h it suffices to study the scalar equation ∂th−∆hm + I(b(t, x))hα = 0 in QT , (3.2) which involves the bounded solution component b. We consider (3.2) subject to the homogeneous Dirichlet boundary conditions h(t, x) = 0, on (0, T )× ∂Ω, (3.3) and a given non-negative initial datum h0(x) = h(0, x), x ∈ Ω. (3.4) At the end of the section some remarks are given concerning the homogeneous Neumann boundary conditions as well as the non-homogeneous Dirichlet bound- ary conditions. We point out that these qualitative behavior properties can be proved by means of some energy methods (see, e.g. [4]) but here we shall use some comparison arguments because they are simpler and lead to sharper estimates. 3.1. Extinction in finite time. We recall that 0 ≤ b0 ≤ 1 implies 0 ≤ b ≤ 1, and thus I(0) ≤ I(b(t, x) ≤ I(1). Therefore, if h satisfies (3.2)–(3.4) and Ū is such that ∂Ū ∂t −∆Ūm + I(0)Ūα ≥ 0 in Ω Ū(t, x) ≥ h(t, x) on (0, T )× Ω Ū(0, x) ≥ h(0, x) in Ω (3.5) since ∂h ∂t −∆hm + I(0)hα ≤ ∂h ∂t −∆hm + I(b(t, x))hα = 0. (3.6) EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 15 by comparison we have that h ≤ Ū in QT [8]. This simple observation leads to the following result. Theorem 3.1. Let (3.1) hold and let (b, w, h) be a solution of problem (2.1)–(2.3) in the time interval (0, T ). Then, if T > 0 is large enough, there exists T ∗ ∈ (0, T ) such that h(t, x) = 0 for all t > T ∗. Proof. Let U be uniform in space satisfying the non-linear ODE: ∂U ∂t + λUα = 0, U(0) = ‖h0‖L∞(Ω), (3.7) for λ > 0. Then, for α < 1, (3.7) possesses the explicit solution U(t;λ) = ( max{0, ‖h0‖1−αL∞(Ω) − λ(1− α)t} )1/(1−α) . (3.8) Obviously, U(t; I(0)) = Ū(t) satisfies (3.5). As a result, letting T ∗(‖h0‖L∞(Ω), I(0), α) = (‖h0‖L∞(Ω)) (1−α) I(0)(1− α) , by comparison 0 ≤ h(t) ≤ Ū(t), h(t) = 0 for all t ≥ T ∗(‖h0‖L∞(Ω), I(0), α). � 3.2. Estimates on the support of h(t, ·). First, let us introduce the following notation. If f is a real-valued function defined on Ω, the support of f in Ω is supp(f) := {x ∈ Ω : f(x) 6= 0}, and by N(f) the complement of the support, namely, N(f) := Ω̄− supp(f). Next we estimate the location of the support of h(t, ·) in Ω, which is equivalent to study the location of the set N(h(t, ·)). Theorem 3.2. Let σ = α/m < 1 and suppose that h0 ∈ L∞(Ω), h0 ≥ 0 and with compact support. Then N(h(t, ·)) ⊂ N(h0(·)) for all t ∈ (0, T ). In particular, if we set M = ‖h0‖L∞(Ω), L ∗ = ( Mm K ) 2 1−σ and K = ( I(0)(1−σ)2 2(2σ+N(1−σ)) ) 1 1−σ , we have N(h(t, ·)) ⊂ {x ∈ (Ω− supp(h0)) such that dist(x, supp(h0)) ≥ L∗}. Proof. We look for local supersolutions which may vanish at points of the zero set of the initial datum h0. Letting σ = α/m, we have that for 0 < σ < 1 and λ > 0, the function V (x) = K(λ)|x−x0| 2 1−σ , with K(λ) = ( λ(1−σ)2 2(2σ+N(1−σ)) ) 1 1−σ satisfies the equation −∆V + λV σ = 0 (see [7]). Now, let x0 ∈ Ω − supp(h0), R := dist{x0, supp(h0)} and Ω̃ := (BR(x0) ∩ Ω). Then for ū(t, x) = (V (x))1/m we have ∂tū−∆ūm + I(0)ūσ = 0 in (0, T )× Ω̃, ū(x) ≥ 0 = h0(x) on Ω̃, (3.9) and ū ≥ 0 on (0, T )× (∂Ω ∩ Ω̃). (3.10) By (3.6), ū is a local super solution of (3.2) as long as the inequality h(t, x) ≤ ū(t, x) is also satisfied for all x in ∂BR(x0) ∩ int (Ω) and t ∈ (0, T ). In fact, since |x− x0| = R on ∂BR(x0), if R ≥ ( Mm h K(I(0)) ) 1−σ 2 (3.11) 16 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 and ‖h‖L∞(0,T ;Ω) ≤Mh, we have ( K(I(0))R 2 1−σ )1/m ≥Mh which in turn implies ū ≥ h, on (0, T )× ∂Ω̃− ∂Ω. (3.12) Therefore, when (3.11) holds, ū is a local supersolution thanks to (3.6), (3.9), (3.10) and (3.12). Finally, ‖h‖L∞(0,t;Ω) ≤ ‖h0‖L∞(Ω), so we may set Mh = ‖h0‖L∞(Ω) and since 0 ≤ h(x0) ≤ ū(x0) = 0 the result follows. � Remark 3.3. In the case of homogeneous Neumann boundary conditions, a similar result is true due to the local nature of the supersolutions. In particular, in the proof above we may take R := dist{x0, supp(h0) ∪ ∂Ω} so that the ball BR(x0) for x0 ∈ Ω− supp(h0) is entirely contained in Ω, then if (3.11) is satisfied, (3.6), along with (3.9) and (3.12) ensure that the super-solution is appropriately defined. Remark 3.4. The same result holds for the problem with compactly supported inhomogeneous Dirichlet boundary conditions, i.e. when h(t, x) = g(t, x) ≥ 0 on (0, T )× (∂Ω ∩ Ω̃) with g(t, ·) > 0 on a compact subset of ∂Ω. In this case, we may take R := dist{x0, supp(h0) ∪ (∪τ>0 supp(g(τ, ·)))}. Remark 3.5. It seems possible to extend most of the results of this paper to the case in which α ∈ (−1, 0]. See, e.g., the treatment made in [11] for a scalar equation. We shall end with a result which implies an infinite waiting time (see [4, 23] for some general expositions on the subject). More precisely, as in [9] it is enough to consider the stationary problem this time for the condition α ∈ (0,m). Moreover, we shall not assume that p = 0 but that p(x) vanishes outside a closed subset ω of R2 (the study could be extended to Rn for any n ≥ 1). The case ω ⊂⊂ Ω and p(x) = pχω(x) on Ω, where χω denotes the characteristic function of ω (as well as with Neumann boundary conditions on ∂Ω) was considered in [19]. In this paper we will extend the mentioned study to the case in which ω = Ω, i.e. p(x) > 0 in Ω and p = 0 on ∂Ω. It is easy to see that in the stationary problem there exists a positive constant cb such that cbd(x) ≤ b(x) ≤ 1 for any x ∈ Ω, where d(x) = d(x, ∂Ω). Indeed, it suffices to apply the strong maximum principle to the stationary equation satisfied by b(x). We set Λ(x) := θ b(x) + r/c b(x) + r in Ω. Then θ (cbd(x) + r/c) 1 + r ≤ Λ(x) ≤ θ (1 + r/c) cbd(x) + r in Ω, and the stationary version of the third equation of (1.1) can be written for ĥ = hm as the stationary problem −dh∆ĥ+ Ib(ĥ)α/m = p, −∆ĥ+ Λ(x) dh ĥα/m = φ(x) in Ω, ĥ = 0 on ∂Ω, (3.13) with φ(x) := p(x) dh . For b fixed (i.e., for a given Λ(x)) it is well-known that there is a unique solution ĥ of (3.13). The following result gives a sufficient condition on EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 17 p(x) in order to get that ĥ is a flat solution (in the sense that also ∂ĥ ∂n = 0 on ∂Ω). In fact, the following result holds for α ∈ (0,m). (3.14) Theorem 3.6. Assume (3.14), let σ = α/m and suppose that p(x) is such that 0 ≤ p(x) ≤ dhKd(x) 2σ 1−σ in Ω, (3.15) for some K > 0 small enough. Then, there exists a constant C∗σ > 0 such that 0 ≤ ĥ(x) ≤ C∗σd(x) 2 1−σ in Ω. (3.16) In particular, ĥ is a flat solution. Proof. As in the proof of Theorem 3.2, we apply the method of local supersolutions such as presented in [7]. Let x0 ∈ ∂Ω and define Ωx0,R = Ω ∩ BR(x0) for some R > 0 to be determined later. Observe that since d(x) ≤ |x− x0|, we have −∆ĥ+ θr dhc(1 + r) ĥσ ≤ φ(x) ≤ K|x− x0| 2σ 1−σ in Ωx0,R. Let h(x : x0) = C|x − x0| 2 1−σ . As a consequence of [7, Theorem 1.15], if we set ξ = θr dhc(1+r) , then −∆h+ ξh σ = [ ξCσ − 2(2σ +N(1− σ)) (1− σ)2 C ] |x− x0| 2σ 1−σ , (in our model N = 2 but the result applies to any arbitrary N ≥ 1). The function Ψ(C) = ξCσ − 2(2σ +N(1− σ)) (1− σ)2 C takes nonnegative values for C ∈ [0, CN,ξ,σ] with CN,ξ,σ = [ ξ(1− σ)2 2(2σ +N(1− σ)) ] 1 1−σ , (notice that Ψ(CN,ξ,σ) = 0). Moreover Ψ(C) attains its maximum at some C∗N,ξ,σ. Then, a good choice of the constant K mentioned in (3.15) is K = Ψ(C∗N,ξ,σ) dh . In that case we know that −∆ĥ+ ξĥσ ≤ −∆h+ ξh σ in Ωx0,R. Clearly ĥ ≤ h on ∂Ωx0,R ∩ ∂Ω and we have ĥ ≤ h on ∂Ωx0,R \ ∂Ω if, for instance, ‖ĥ‖L∞(Ω) ≤ C∗N,ξ,αR 2 1−σ . (3.17) Finally, we assume R “large enough” so that R ≥ [‖ĥ‖L∞(Ω) C∗N,ξ ](1−σ)/2 and then (3.17) holds. In conclusion, by the maximum principle, 0 ≤ ĥ(x) ≤ C∗N,ξ,σ|x− x0| 2 1−σ in Ωx0,R, and since x0 ∈ ∂Ω is arbitrary, this implies (3.16). � 18 J. I. DÍAZ, D. HILHORST, P. KYRIAZOPOULOS EJDE-2021/08 Remark 3.7. An easy modification of the proof of Theorem 3.2, by using the special constant C∗N,ξ,σ of the above proof, allows to show the infinite waiting time property when the initial datum h0(x) grows at most as d(x, ∂S(h0)) 2 m−α near the boundary of its support S(h0). Indeed, it suffices to use the same arguments of [3, Theorem 3.1]. Other qualitative behavior can be proved by using the methods of [2, 3]. Acknowledgments. J. I. Dı́az was partially supported by the project ref. MTM 2017-85449-P of the DGISPI (Spain) and the Research Group NOMAT (Ref. 910480) of the UCM. References [1] M. Alfaro, H. Izuhara, M. Mimura; On a nonlocal system for vegetation in drylands, Journal of Mathematical Biology 77 (2018), no. 6-7, 1761–1793. [2] L. Alvarez, J. I. Dı́az; On the initial growth of interfaces in reaction-diffusion equations with strong absorption, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 123 (1993), no. 5, 803–817. [3] L. Alvarez, J. I. 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EJDE-2021/08 A PARABOLIC SYSTEM WITH STRONG ABSORPTION 19 [20] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva; Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, R.I., 1967. [21] E. Meron; Nonlinear physics of ecosystems, CRC Press, 2015. [22] R. Temam; Navier–stokes equations and nonlinear functional analysis, Society for Industrial and Applied Mathematics, 1995. [23] J. L. Vázquez; The porous medium equation, Oxford University Press, 2007. [24] I. I. Vrabie; Compactness methods for nonlinear evolutions, CRC Press, 1995. Jesús Ildefonso D́ıaz Instituto de Matemática Interdisciplinar, Univ. Complutense de Madrid, Plaza de las Ciencias 3 28040 Madrid, Spain Email address: jidiaz@ucm.es Danielle Hilhorst CNRS and Laboratoire de Mathématiques, University Paris-Saclay, Orsay Cedx 91405, France Email address: danielle.hilhorst@universite-paris-saclay.fr Paris Kyriazopoulos Department of Mathematics, University of the Aegean, 832 00 Karlovassi, Samos, Greece Email address: pkyriazopoulos@aegean.gr 1. Introduction 2. Existence of solutions 2.1. Regularized system for (PD) 2.2. Neumann boundary conditions (PN) 3. Qualitative properties of the surface water component 3.1. Extinction in finite time 3.2. Estimates on the support of h(t,) Acknowledgments References