The extinction versus the blow-up: Global and non-global existence - - PowerPoint PPT Presentation

Introduction Main results

The extinction versus the blow-up: Global and non-global existence of solutions of source types

• f degenerate parabolic equations with a singular

absorption

Nguyen Anh Dao (joint with Professor J. I. Diaz)

Hutech University

January 3, 2019

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

Let I = (x1, x2) be an interval in R. We consider nonnegative solutions of the following equation:    ∂tu − (|ux|p−2ux)x + u−βχ{u>0} = f (u, x, t), in I × (0, T), u(x1, t) = u(x2, t) = 0, t ∈ (0, T), u(x, 0) = u0(x), x ∈ I, (1) with p > 2, β ∈ (0, 1), and χ{u>0} = 1, if u > 0, 0, if u ≤ 0. Note that lim

u↓0 u−βχ{u>0} = +∞. And we impose tactically

u−βχ{u>0} = 0 whenever u = 0.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

f : [0, ∞) × I × [0, ∞) − → R is a nonnegative function satisfying the following hypothesis: (H)    f ∈ C1 [0, ∞) × I × [0, ∞)

• , f (0, x, t) = 0, ∀(x, t) ∈ I × (0, ∞), and

f (u, x, t) ≤ h(u), ∀(x, t) ∈ I × (0, ∞), for some h ∈ C1([0, ∞)).

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results 1 Local existence of solutions of equation (1). 2 Global existence of solutions of equation (1). In particular, we

prove the complete quenching phenomenon of solutions under some additional conditions.

3 Blowing up of solutions. Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

To show the local existence result, we prove a pointwise estimate for |ux| (Bernstein’s estimates). One of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t ∈ (0, ∞), the so called global solutions, or by the contrary a finite time blow-up τ0 > 0 arises such that lim

t→τ0 u(t)L∞(I) = +∞.

Moreover, we prove that any global solution must vanish identically after a finite time if provided that either the initial data

• r the source term is small enough.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

In the case N-dimension and p = 2, equation (1) becomes    ∂tu − ∆u + u−βχ{u>0} = f (u, x, t) in Ω × (0, T), u = 0

• n ∂Ω ∈ (0, T),

u(x, 0) = u0(x) in Ω, (2) where Ω is a bounded domain in RN. Problem (2) can be considered as a limit of mathematical models describing enzymatic kinetics, or the Langmuir-Hinshelwood model of the heterogeneous chemical catalyst (see Strieder and Aris (1973)). This case was studied by the authors, see e.g. Phillips (1987), Kawohl (1996), Levine (1990, 1993), Davila (2004), Winkler (2007).

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

• D. Phillips (1987) proved the existence of solution for the Cauchy

problem associating (2) in the case f = 0. The case in that f (u) is sub-linear, was considered by Davila and Montenegro (2004). The authors showed that the measure of the set {(x, t) ∈ Ω × (0, ∞) : u(x, t) = 0} is positive. In other words, the solution may exhibit the quenching (or the extinction) behavior. Moreover, Winkler (2007) showed that equation (2) with f = 0 has no uniqueness solution in general.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

As mentioned above, to show a local existence result, we first prove a priori pointwise estimate for |ux| involving a certain power

• f u as follows:

|ux(x, t)|p ≤ Cu1−β(x, t), for (x, t) ∈ I × (0, T), (3) for some positive constant C > 0. It is known that such an estimate (3) plays an important role in proving the existence of solution for equations of this type. For instance, in the case p = 2 and f = 0, estimate (3) was obtained by Phillips (1987), namely |∇u(x, t)|2 ≤ Cu1−β(x, t), for (x, t) ∈ Ω × (0, T).

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

To illustrate the global existence result, we first consider equation (1) with the simplest model λf (u) = λuq−1. In some of our considerations, a crucial role is played by the first eigenvalue λI of the Dirichlet problem:

• −∂x(|∂xφI|p−2∂xφI) = λIφp−1

I

in I, φI(x1) = φI(x2) = 0, (4) where φI is the first eigenfunction.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

Note that the value of λI is computed as follows: λI = (p − 1)

• πp

x2 − x1 p, with πp = 2 π/p sin(π/p), (5) see Biezuner, Ercole, Martins (2009). Then λI decreases when the measure of the spatial domain I increases, and conversely.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

For our purpose later, let us remind some classical results on the global and non-global existence of solutions of equation (1) without the singular absorption:    ∂tu − (|ux|p−2ux)x = λuq−1 in I × (0, T), u(x1, t) = u(x2, t) = 0 t ∈ (0, T), u(x, 0) = u0(x) in I. (6)

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

Tsutsumi (1973) proved that if q < p, then problem (6) has global nonnegative solutions whenever initial data u0 belongs to some Sobolev space. The case q ≥ p is quite delicate that there are both nonnegative global solutions, and solutions which blow up in a finite time.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results

Indeed, J. N. Zhao (1993) showed that when q ≥ p, equation (6) has a global solution if the measure of I is small enough, and it has no global solution if the measure of I is large enough. The fact that the first eigenvalue λI decreases with increasing domain can be also used as an alternative explanation for Zhao’s

• result. For example, in the critical case q = p, Y. Li and C. Xie

(2003) showed that if λI > λ, equation (6) has then a unique globally bounded solution. While, the unique solution blows up in a finite time if λI < λ. We also note that the solution is globally bounded if provided that λI = λ and initial data u0(x) ≤ κφI(x), for some κ > 0.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Let us introduce first the notion of a weak solution of equation (1). Definition Let u0 ∈ L∞(I). A function u is called a weak solution of equation (1) if u−βχ{u>0} ∈ L1(I × (0, T)), and u ∈ Lp(0, T; W 1,p (I)) ∩ L∞(I × (0, T)) ∩ C([0, T); L1(I)) satisfies equation (1) in the sense of distribution, i.e: T

• I
• −uφt + |ux|p−2uxφx + u−βχ{u>0}φ − f (u, x, t)φ
• dxdt = 0,

∀φ ∈ C∞

c (I × (0, T)).

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Let Γ(t) be the solution of the equation: ∂tΓ = h(Γ), in (0, T), Γ(0) = u0∞, (7) where T is the maximal existence time of Γ(t), and it depends on u0L∞.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Theorem (Local existence) Let u0 ∈ L∞(I). Then, there exists a time T0 > 0 such that equation (1) has a maximal solution u in I × (0, T0). Moreover, there is a positive constant C = C(β, p) such that |ux(x, τ)|p ≤ Cu1−β(x, τ)

• Γ

1+βγ γ (2T0)Θ(Dxf , Γ(2T0))+

Γ1+β(2T0)Θ(Duf , Γ(2T0)) + τ −1Γ1+β(2T0) + 1

• ,

(8) for a.e (x, τ) ∈ I × (0, T0), with Θ(G, r) = max

0≤u≤r,(x,t)∈I×[0,2T0]

{|G(u, x, t)|}.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Continuity of Theorem

Besides, if (u

1 γ

0 )x ∈ L∞(I), then there is a positive constant

C = C(β, p, (u

1 γ

0 )x∞) such that

|ux(x, τ)|p ≤ Cu1−β(x, τ)

• Γ1+β(2T0)Θ(Duf , Γ(2T0))+

Γ

1+βγ γ (2T0)Θ(Dxf , Γ(2T0)) + 1

• ,

(9) for a.e (x, t) ∈ I × (0, T0).

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

• Remark. As a consequence of (9), we obtain

|u(x, t)−u(y, s)| ≤ C

• |x − y| + |t − s|1/3

, ∀(x, t) ∈ I ×(0, T).

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Next, we have a global existence result for the source λf (u, x, t). Theorem (Global existence) Let u0 ∈ L∞(I), and λ > 0. Assume that there are an open bounded interval I0, and a positive real number κ0 such that I ⊂⊂ I0, and        u0(x) ≤ κ0φI0(x), for a.e x ∈ I, and λI0κp−1 φp−1

I0

(x) + κ−β

0 φ−β I0 (x) ≥ λf (κ0φI0(x), x, t),

∀(x, t) ∈ I × (0, ∞). (10)

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Continuity of Theorem (global existence)

(Note that λI0 and φI0 are the first eigenvalue and the first eigenfunction of problem (4) in I0. Thus, inf

x∈I{φI0} > 0).

Then, any solution u of equation (1) exists globally and u(x, t) ≤ κ0φI0(x), in I × (0, ∞). (11)

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Theorem (Extinction result) Let u0 ∈ L∞(I), and h(0) = 0 in (H). Then, every weak solution of equation (1) vanishes identically after a finite time T ⋆ if provided that either u0∞ or λ is small enough. Moreover, we have T ⋆ ≤ Cu0σ

L2(I),

σ = σ(p) ∈ (0, 1).

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

Concerning the non-global existence of solutions of equation (1), we consider the source f (u) for a simplicity. Then, we have the following result. Theorem (Blow-up) Let u0 ∈ W 1,p (I), and T > 0. Assume that f (u, x, t) = f (u), and

F(u) up

is nondecreasing on (0, ∞). Then, the maximal solution u blows up in a finite time T0 ∈ (0, T] if provided pE(0) + 4(3p − 1) T(p − 2)2

• I

u2

0dx ≤ 0,

(12) with F(u) = u f (s)ds.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

As a consequence of the above Theorem, we have Corollary Assume as in the Theorem above. Then, the maximal solution u is explosive in a finite time if provided E(0) < 0. Moreover, the blow-up time T0 satisfies T0 ∈

• 0,

4(3p − 1) −pE(0)(p − 2)2

• I

u2

0dx

• .

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence

Introduction Main results Existence results Extinction of solution Explosive solution

References

1 J. Davila, M. Montenegro, Existence and asymptotic behavior

for a singular parabolic equation, Trans. Amer. Math. Soc. 357 (2004), 1801-1828.

2 B. Kawohl, R. Kersner, On degenerate diffusion with very

strong absorption, Math. Methods Appl. Sci., 15 (1992), 469-477.

3 N. A. Dao, J. I. Diaz, The extinction versus the blow-up:

Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption, JDE, 263 (2017), 6764-6804.

4 N. A. Dao, J. I. Diaz, H. V. Kha, Complete quenching

phenomenon and instantaneous shrinking of support of solutions of degenerate parabolic equations with nonlinear singular absorption, Proc. Royal Soc. Edin., (2019), 1-24.

Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence