Introduction Main results The extinction versus the blow-up: Global and non-global existence of solutions of source types of 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 = ( x 1 , x 2 ) be an interval in R . We consider nonnegative solutions of the following equation: ∂ t u − ( | u x | p − 2 u x ) x + u − β χ { u > 0 } = f ( u , x , t ) , in I × (0 , T ) , u ( x 1 , t ) = u ( x 2 , t ) = 0 , t ∈ (0 , T ) , u ( x , 0) = u 0 ( x ) , x ∈ I , (1) � 1 , if u > 0 , with p > 2, β ∈ (0 , 1), and χ { u > 0 } = 0 , if u ≤ 0 . u ↓ 0 u − β χ { u > 0 } = + ∞ . And we impose tactically Note that lim u − β χ { 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: f ∈ C 1 � � [0 , ∞ ) × I × [0 , ∞ ) , f (0 , x , t ) = 0 , ∀ ( x , t ) ∈ I × (0 , ∞ ) , and ( H ) f ( u , x , t ) ≤ h ( u ) , ∀ ( x , t ) ∈ I × (0 , ∞ ) , for some h ∈ C 1 ([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 | u x | (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 τ 0 > 0 arises such that lim t → τ 0 � 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 or 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 ∂ t u − ∆ u + u − β χ { u > 0 } = f ( u , x , t ) in Ω × (0 , T ) , u = 0 on ∂ Ω ∈ (0 , T ) , (2) u ( x , 0) = u 0 ( x ) in Ω , where Ω is a bounded domain in R N . 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 | u x | involving a certain power of u as follows: | u x ( x , t ) | p ≤ Cu 1 − 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 ≤ Cu 1 − 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 λ f ( u ) = λ u q − 1 . In some of our considerations, a crucial role is played by the first eigenvalue λ I of the Dirichlet problem: − ∂ x ( | ∂ x φ I | p − 2 ∂ x φ I ) = λ I φ p − 1 � in I , I (4) φ I ( x 1 ) = φ I ( x 2 ) = 0 , where φ I 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 λ I is computed as follows: π p π/ p � p , with π p = 2 � λ I = ( p − 1) sin( π/ p ) , (5) x 2 − x 1 see Biezuner, Ercole, Martins (2009). Then λ I 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: ∂ t u − ( | u x | p − 2 u x ) x = λ u q − 1 in I × (0 , T ) , u ( x 1 , t ) = u ( x 2 , t ) = 0 t ∈ (0 , T ) , (6) u ( x , 0) = u 0 ( x ) in I . 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 u 0 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 λ I 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 λ I > λ , equation (6) has then a unique globally bounded solution. While, the unique solution blows up in a finite time if λ I < λ . We also note that the solution is globally bounded if provided that λ I = λ and initial data u 0 ( x ) ≤ κφ I ( 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
Existence results Introduction Extinction of solution Main results Explosive solution Let us introduce first the notion of a weak solution of equation (1). Definition Let u 0 ∈ L ∞ ( I ). A function u is called a weak solution of equation (1) if u − β χ { u > 0 } ∈ L 1 ( I × (0 , T )), and u ∈ L p (0 , T ; W 1 , p ( I )) ∩ L ∞ ( I × (0 , T )) ∩ C ([0 , T ); L 1 ( I )) satisfies 0 equation (1) in the sense of distribution, i.e: � T � � � − u φ t + | u x | p − 2 u x φ x + u − β χ { u > 0 } φ − f ( u , x , t ) φ dxdt = 0 , 0 I ∀ 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
Existence results Introduction Extinction of solution Main results Explosive solution Let Γ( t ) be the solution of the equation: � ∂ t Γ = h (Γ) , in (0 , T ) , (7) Γ(0) = � u 0 � ∞ , where T is the maximal existence time of Γ( t ), and it depends on � u 0 � L ∞ . Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence
Existence results Introduction Extinction of solution Main results Explosive solution Theorem (Local existence) Let u 0 ∈ L ∞ ( I ) . Then, there exists a time T 0 > 0 such that equation (1) has a maximal solution u in I × (0 , T 0 ) . Moreover, there is a positive constant C = C ( β, p ) such that � 1+ βγ | u x ( x , τ ) | p ≤ Cu 1 − β ( x , τ ) γ (2 T 0 )Θ( D x f , Γ(2 T 0 ))+ Γ (8) � Γ 1+ β (2 T 0 )Θ( D u f , Γ(2 T 0 )) + τ − 1 Γ 1+ β (2 T 0 ) + 1 , for a.e ( x , τ ) ∈ I × (0 , T 0 ) , with Θ( G , r ) = max {| G ( u , x , t ) |} . 0 ≤ u ≤ r , ( x , t ) ∈ I × [0 , 2 T 0 ] Nguyen Anh Dao (joint with Professor J. I. Diaz) The extinction versus the blow-up: Global and non-global existence
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