Parabolic Equations with a Singular Absorption: Extinction Versus Blow-up

We prove a local existence of weak solutions of semilinear parabolic equations with a strong singular absorption and a source. Moreover, we consider the qualitative behavior of solutions. We show that any solution exists globally and vanishes after a nite time if either the initial data or the source term is small enough. On the other hand, we point out some criteria such that solutions are explosive in a nite time.

Problem Eq. (1) can be considered as a limit of mathematical models describing enzymatic kinetics (see [1]), or the Langmuir-Hinshelwood model of the heterogeneous chemical catalyst (see, e.g. [19] p. 68, [22], [18]). This problem has been studied by the authors in [18], [29], [28], [13], [16], [9], [7], [8], [6], [20], and references therein. These authors have considered the existence and uniqueness, and the qualitative behavior of these solutions. For example, when f = 0, D. Phillips [18] proved the existence of solution for the Cauchy problem associating to Eq. (1). A partial uniqueness of solution of Eq. (1) was proved by J. Davila and M. Montenegro, [9] for a class of solutions with initial data u 0 (x) ≥ Cdist(x, ∂Ω) µ , for µ ∈ (1, 2 1+β ) (see also [5] the uniqueness in a dierent class of solutions). A beautiful result established by M. Winkler, [20], showed that the uniqueness of solution fails in general. One of the interesting behaviors of solutions of Eq. (1) is the extinction that any solution vanishes after a nite time even beginning with a positive initial data, see [18], [28] ( see also [7] for a quasilinear equation of this type). It is known that this phenomenon occurs according to the presence of the nonlinear singular absorption u −β χ {u>0} . One can see the same situation for the nonlinear absorption u β , for β ∈ (0, 1), see [21] and references therein. Equation (1) with source term f (u) satisfying the sublinear condition, i.e: f (u) ≤ C(u + 1), was considered by J. Davila and M. Montenegro, [9]. The authors proved the existence of solution and showed that the measure of the set {(x, t) ∈ Ω × (0, ∞) : u(x, t) = 0} is positive (see also a more general statement in [23]). In other words, the solution may exhibit the quenching behavior. Still in the sublinear case with source term λf (u), M. Montenegro [17] proved that there is a real number λ 0 > 0 such that for any λ ∈ (0, λ 0 ), there is t 0 > 0 such that And it is called the complete quenching phenomenon.
From our knowledge, Eq. (1) with a general source term f (u) has not been studied completely, such as f (u) = u 2 . Thus, we would like to investigate existence, and the qualitative behavior of solutions of Eq. (1) for source term f (u). It is well known that nonlinear parabolic equations with general source f (u) may cause the nite time blow-up, i.e: there is a time T 0 > 0 such that lim t→T0 u(t) ∞ → +∞. As mentioned above, the nonlinear absorption u −β χ {u>0} causes the complete quenching phenomenon. Thus, it is interesting to see when the complete quenching prevails the blow-up, and conversely. We also note that the similar questions were studied by [7], [8], [6] for the quasilinear parabolic equations of this type. To be simple, we consider f (u) = u p , p ≥ 1 through this paper, although our analysis can be applied to a general source f (u), which is a locally Lipschitz function on [0, ∞). Before discussing the behaviors of solutions of Eq. (1), it is necessary to introduce a notion of weak solution, and establish rst a local existence of solutions of Eq. (1).
Remark 1. The result of Theorem 1 implies that u is continuous up to the boundary. Furthermore, u is continuous up to t = 0 if provided Our next purpose is to study the global existence of solutions. Particularly, we are interested in the complete quenching phenomenon that any solution vanishes identically after anite time under some circumstances. To state our global existence result and the complete quenching phenomenon, we consider Eq. (1) with source term λu p , for any λ > 0. In some of our considerations, a crucial role is played by the rst eigenvalue λ 1 of the Dirichlet problem: where Φ is the rst normalized eigenfunction, It is known that λ 1 decreases when the measure of the spatial domain Ω increases. Then, we have a result of global existence of solution.
Theorem 2. Let u 0 ∈ L ∞ (Ω), and f (u) = λu p , for λ > 0. Assume that there are an open bounded domain Ω 0 , and a positive real number k 0 such that Ω ⊂⊂ Ω 0 , and where λ 1,Ω0 and Φ Ω0 are the rst eigenvalue and the rst eigenfunction of problem Eq. (5) corresponding to Ω 0 . Then, any solution v of Eq. (1) exists globally and satises Remark 2. For a given λ > 0; if u 0 is small enough then Eq. (6) holds, and conversely. Thus, we obtain the global existence of solutions if provided either u 0 or λ is small enough.
Next, we give the complete quenching results.
For a given λ > 0, then every solution of Eq. (1) is extinct after a nite time if provided that u 0 ∞ is small enough. Theorem 4. For a given u 0 ∈ L ∞ (Ω), there is a real number λ 0 > 0 such that every solution of Eq. (1) quenches after a nite time if provided λ ∈ (0, λ 0 ).
Finally, we study the global nonexistence of solutions of Eq. (1), the so called nite time blow-up, see [15], [16], [12], [11], [25], [26], [30]. In this paper, we point out some criteria on initial data u 0 in order to guarantee the blow-up of solution in a nite time. Thus, it is convenient to introduce the energy functional Then we have a blowing up result as follows: . Suppose that f (u) = u p , for p > 1, and E(0) ≤ 0. Let u be a solution of Eq. (1). Then, u blows up in a nite time.
The paper is organized as follows: In the next section, we prove some gradient estimates for the approximating solutions. In Sec. 3. we shall prove the local existence results. The global existence of solutions and the complete quenching phenomenon will be considered in Sec. 4. Section 5. is devoted to study the non-global existence of solution. In the nal Section, we give some simulation in order to illustrate the interesting phenomenon: quenching versus blow-up.
Several notations which will be used through this paper are the following: we denote by C a general positive constant, possibly varying from line to line. Furthermore, the constants which depend on parameters will be emphasized by using parentheses. For example, C = C(p, β, τ ) means that C depends on p, β, τ . And Supp(f ) is denoted as the support compact of f .

2.
Gradient estimate for the approximate solutions In this section, we consider the regularized problem (P ε,η ) of Eq. (1) as follows: , and ψ ∈ C ∞ (R) is a nondecreasing function on R such that ψ(s) = 0 for s ≤ 1, and ψ(s) = 1 for s ≥ 2. Note that g ε is a globally Lipschitz function for any ε > 0. We will show that solution u ε,η of equation (P ε,η ) tends to a solution of Eq. (1) as η, ε → 0. In passing to the limit, we need to derive some gradient estimates for solution u ε,η . Then, we have the following result: Lemma 1. Let u 0 ∈ C ∞ c (Ω), u 0 = 0. There exists a classical unique solution u ε,η of (P ε,η ) in Ω × (0, T ).

Local existence
In this section, we consider a local existence of solution of problem Eq. (1). We give the proof of Theorem 1.
This implies that u ε1 is a sub-solution of the equation satised by u ε2 . Therefore, the comparison principle yields so the conclusion follows. Consequently, there is a nonnegative function u such that u ε ↓ u as ε → 0 + .
Obviously, we have from the comparison principle where n is the unit outward normal vector of ∂Ω.
Since ∇u ε .n ≤ 0, we obtain This inequality and the boundedness of u ε above imply that g ε (u ε ) L 1 (Ω×(0,T )) is bounded by a constant not depending on ε.

Actually, we shall prove
On the other hand, we use a result of gradient convergence of Boccardo et al., [3], [2] in order to obtain Now, it suces to demonstrate that u satises Eq. (1) in the sense of distribution.
For any η > 0 xed, we use the test function ψ η (u ε )φ, for any φ ∈ C ∞ c (Ω × (0, T )), to the equation satised by u ε . Then, using integration by parts yields Note that ψ η (.) plays a role in avoiding the singularity of the term u −β χ {u>0} as u is near 0. Thus, there is no problem of going to the limit as ε → 0 in the indicated equation in order to obtain Next, we go to the limit as η → 0 in the last equation.
By Eq. (14), Eq. (15), and the integration of u p φdxds. (16) Next, we show that In fact, since u satises estimate Eq. (3), we have where Supp(φ) means the support compact of φ, and the constant C > 0 is independent of η.
The conclusion u ∈ C([0, T ]; L 1 (Ω)) is well known, so we skip its proof and refer to the compactness result of J. Simon, [33]. Thus, u is a weak solution of Eq. (1).
To complete the proof of Theorem 1, it remains to show that u is the maximal solution of Eq. (1).
In fact, we observe that Thus, which implies that v is a sub-solution of equation (P ε ).

4.
Global existence and complete quenching phenomenon In this part, we study the global existence of solution and the complete quenching phenomenon through proving Theorem 2, and Theorem 3.
Since u is the maximal solution, then it suces to work on u.
Proof of Theorem 2. Let u be the maximal solution of Eq. (1) in Ω × (0, T ). To prove that u exists globally, we show that u is bounded by a constant not depending on t.
Next, we will show that for a given λ > 0, the maximal solution u must vanish identically after a nite time if u 0 ∞ is small enough.
Proof of Theorem 3. Since u 0 ∞ is small enough, we can choose a real number k 0 > 0 small as well, and an open bounded domain Ω 0 containing Ω, such that Eq. (6) holds. Thanks to Theorem 2, the maximal solution u exists globally, and it is bounded by Using the test function u to Eq. (1) gives us Since u ≤ M , we have where c M = λM p+β . Note that M is as small It follows from Eq. (22) that Now, using Garliardo-Nirenberg's inequality [31] yields Then, we obtain with c 2 = c 2 (β, M, N ) > 0.
If we can show that there exists a time t 0 ∈ [0, ∞) such that y(t 0 ) = 0, it follows then from Eq. (25) that y(t) = 0, for any t ≥ t 0 , thereby proves Theorem 3. Indeed, we assume a contradiction that y(t) > 0, for any t > 0. Solving the ordinary dierential inequality Eq. (25) yields with c 4 = c3 N +3 . This leads to a contradiction as t is suciently large. Thus, u must quench after a nite time.
Similarly, for a given initial data u 0 , we also obtain the complete quenching result for the case λ small. Thus, Theorem 4 follows. Remark 3. Inequality Eq. (26) implies that the extinction time of u, denoted by T ≤

5.
Non-global existence of solutions In this section, we study the non-global existence of solutions of Eq. (1). We give the proof of Theorem 5.

Simulation results
In this part, we will illustrate our theoretical results with some numerical experiences. In the sequel, we consider Eq. (1) with p = 2, β = 0.8, I = (0, L), and u 0 (x) = x(1 − x/L). Our numerical scheme mimics the one in the paper of [10]. Similarly, we use the linear nite elements with mass lumping in a uniform mess for the space variable to discretize our Eq. (1). The reader who is interested in detail can nd in [10].
We x L = 2.
With λ = 14, the maximal solution of Eq.