Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 2, April 2023
Page(s) 99 - 105
DOI https://doi.org/10.1051/wujns/2023282099
Published online 23 May 2023

© Wuhan University 2023

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Let m1, p>1 and q(1,2) be three constants, ΩRN(N>p) be a bounded domain with smooth boundary Ω, and u0(x) be a bounded non-trivial function with |u0|m-1u0W01,p(Ω). We focus our attention here on dealing with the global existence, extinction and non-extinction phenomenon of the following doubly nonlinear parabolic equation

{ u t - d i v ( | ( | u | m - 1 u ) | p - 2 ( | u | m - 1 u ) ) = f ( u ) ,    ( x , t ) Ω × ( 0 , + ) u ( x , t ) = 0 ,                                          ( x , t ) Ω × ( 0 , + ) u ( x , 0 ) = u 0 ( x ) ,                                   x Ω (1)

with

f ( u ) = { | u | q - 2 u l o g | u | ,    i f   u 0 0 ,                             i f   u = 0 (2)

Nonlinear evolutionary problems with logarithmic nonlinearity like model (1) came from inflation cosmology, super symmetric field theories, quantum mechanics and nuclear physics[1-6].

In the past few decades, many mathematical researchers have devoted themselves to investigating the doubly nonlinear parabolic equations, and obtained many meaningful results, such as local and global well-posedness, regularity, blow-up in a finite time and extinction singularity [7-12]. Especially, the authors [13-15] studied the problem

{ u t - d i v ( | u m | p - 2 u m ) = λ u q ,    ( x , t ) Ω × ( 0 , + ) u ( x , t ) = 0 ,                                   ( x , t ) Ω × ( 0 , + ) u ( x , 0 ) = u 0 ( x ) ,                            x Ω (3)

where m>0, p>1, λ>0, q>0 and u0(x) is a non-negative bounded non-trivial function with u0m(x)W01,p(Ω). For the case 0<m(p-1)<1, the authors [14,15] proved that the critical blow-up and extinction exponents are qbc=1 and qec=m(p-1), respectively. Compared with the case 0<m(p-1)<1, the solution for the case m(p-1)1 exhibits completely different properties [13]. On the one hand, for any q>0, the non-trivial solution to problem (3) will never become extinct at a finite time. On the other hand, the critical blow-up exponent becomes qbc*=m(p-1).

Recently, Le and Le [16,17] considered problem (1) with m(p-1)>1 and obtained the existence and non-existence results of the global weak solutions. Precisely speaking, they concluded that if m(p-1)>q-1, then for any |u0|m-1u0W01,p(Ω), problem (1) admits a global solution; if m(p-1)q-1, then there exists a weak solution to problem (1) which is global provided that u0(x) belongs to some specific stable sets, and the weak solution blows up in a finite time provided that u0(x) belongs to some specific unstable sets.

According to our knowledge, there is no extinction result of the solution to problem (1) with 0<m(p-1)<1. Inspired by the above works, we naturally take the following two questions into consideration. Does the solution to problem (1) with 0<m(p-1)<1 exist globally under some certain conditions? If so, is there a critical extinction exponent of the global solutions? In fact, in this paper, we work with the following equivalent formulation of problem (1), obtained by changing variable v=|u|m-1u,

{ ( | v | 1 - m m v ) t - d i v ( | v | p - 2 v ) = f ( v ) ,    ( x , t ) Ω × ( 0 , + ) v ( x , t ) = 0 ,                                            ( x , t ) Ω × ( 0 , + ) v ( x , 0 ) = v 0 = | u 0 | m - 1 u 0 ,                    x Ω (4)

with

f ( v ) = { 1 m | v | q - 1 - m m v l o g | v | ,    i f   v 0 0 ,                                     i f   v = 0 (5)

It is clear that the first equation in problem (4) has degeneracy or singularity at the points where v(x,t)=0 or |v(x,t)|=0, and hence problem (4) might not have classical solution in general. We introduce the definition of the weak solution to problem (4) as follows.

Definition 1   Let T>0. A measurable function v(x,t) defined in Ω×[0,T) is called a weak solution to problem (4) if vL(0,T;W01,p(Ω)), (|v|1-mmv)tL2(0,T;L2(Ω)), v(x,0)=|u0|m-1u0W01,p(Ω) and

Ω [ ( | v | 1 - m m v ) t φ + | v | p - 2 v φ ]   d x = 1 m Ω | v | q - 1 - m m v l o g | v | φ d x (6)

holds for a.e. t(0,T) and any φW01,p(Ω).

Similar to the proof of Theorem 3.3 [17], by Faedo-Galerkin method, we can prove the local existence result of the weak solution to problem (4). Now, we state the main results of this paper as follows.

Theorem 1   Assume that 0<m(p-1)<1. Then the weak solution u(x,t) of problem (1) exists globally.

Theorem 2   Assume that 0<m(p-1)<q-1<1. If

m a x { ( Ω ( | u 0 | m - 1 u 0 ) + m p a + m + 1 m d x ) θ , ( Ω ( | u 0 | m - 1 u 0 ) - m p a + m + 1 m d x ) θ } κ 6 2 κ 7 (7)

with θ=q-1-m(p-1)+mβmpa+m+1, β(0,2-qm) and a>max{-1p,-1mp,(N-p)(m+1)-Npmmp2}.Then the weak solution u(x,t) to problem (1) will vanish in finite time, where κ6 and κ7 are two positive constants, given by (23) and (24), respectively.

Theorem 3   Assume that 0<q-1m(p-1)<1. If

{ Ω | u 0 | m + 1 d x > 0   a n d   E ( | u 0 | m - 1 u 0 ) 0 ,                                             m ( p - 1 ) = q - 1 Ω | u 0 | m + 1 d x > 0   a n d   E ( | u 0 | m - 1 u 0 ) < - m | Ω | [ m ( p - 1 ) - ( q - 1 ) ] e p ( q + m - 1 ) 2 ,    m ( p - 1 ) > q - 1 (8)

Then the weak solution to problem (1) cannot vanish in finite time, where

E ( | u 0 | m - 1 u 0 ) = 1 p Ω | ( | u 0 | m - 1 u 0 ) | p d x - m q + m - 1 Ω | u 0 | q + m - 1 l o g | u 0 | d x [ + m q + m - 12 Ω u 0 q + m - 1 d x ] (9)

Remark 1   From Theorems 2 and 3, we know that the critical extinction exponent of the global solutions to problem (1) is qec=m(p-1)+1.

1 Proofs of the Main Results

Proof of Theorem 1   Multiplying both sides of the first equality in (4) by v(x,t) and integrating over Ω, one gets

1 m + 1 d d t Ω | v | m + 1 m d x + Ω | v | p d x = 1 m Ω | v | m + q - 1 m l o g | v | d x (10)

Remembering that q(1,2) and m1, we can select β(0,2-qm) such that q+mβ(1,2). For this chosen β, we know log|v|1eβ|v|β. Then, from (10), it holds that

1 m + 1 d d t Ω | v | m + 1 m d x + Ω | v | p d x 1 e m β Ω | v | m + q - 1 m + β d x (11)

By using Hölder's inequality, (11) leads to

d d t Ω | v | m + 1 m d x m + 1 e m β Ω | v | m + q - 1 m + β d x κ 1 ( Ω | v | m + 1 m d x ) m β + m + q - 1 m + 1 (12)

where κ1=m+1emβ|Ω|2-q-mβm+1. By a simple calculation, we get

Ω | v | m + 1 m d x [ ( Ω | v 0 | m + 1 m d x ) 2 - q - m β m + 1 + ( 2 - q - m β ) κ 1 m + 1 t ] m + 1 2 - q - m β (13)

It follows from (13) and Hölder's inequality that

Ω | v | m β + m + q - 1 m d x | Ω | 2 - q - m β m + 1 ( Ω | v | m + 1 m d x ) m β + m + q - 1 m + 1 | Ω | 2 - q - m β m + 1 [ ( Ω | v 0 | m + 1 m d x ) 2 - q - m β m + 1 + ( 2 - q - m β ) κ 1 m + 1 t ] m β + m + q - 1 2 - q - m β (14)

On the other hand, multiplying both sides of the first equality in (4) by vt=(|u|m-1u)t and integrating over Ω, then with the help of Hölder's inequality and Cauchy's inequality with ε, we get

4 m 2 ( m + 1 ) 2 Ω [ ( | v | m + 1 2 m ) t ] 2 d x + m p d d t Ω | v | p d x = Ω | v | m + q - 1 m - 2 v l o g | v | v t d x = Ω 1 = { x Ω : | v | 1 } | v | m + q - 1 m - 2 v l o g | v | v t d x + Ω 2 = { x Ω : | v | < 1 } | v | m + q - 1 m - 2 v l o g | v | v t d x 2 m e β ( m + 1 ) Ω 1 | v | 2 m β + 2 q + m - 3 2 m | ( v m + 1 2 m ) t | d x + 4 m 2 e ( m + 1 ) ( m + 2 q - 3 ) Ω 2 | ( v m + 1 2 m ) t | d x m 2 ε 1 e β ( m + 1 ) Ω | v | 2 m β + 2 q + m - 3 2 m d x + 2 m ε 1 e β ( m + 1 ) Ω | ( v m + 1 2 m ) t | 2 d x + m 2 | Ω | e ε 2 ( m + 1 ) ( m + 2 q - 3 ) + 4 m 2 ε 2 e ( m + 1 ) ( m + 2 q - 3 ) Ω | ( v m + 1 2 m ) t | 2 d x (15)

If ε1 and ε2 are sufficiently small such that ε1β+2mε2m+2q-32mem+1, then from (15), it holds that

d d t Ω | v | p d x κ 3 + κ 2 ( Ω | v | m β + m + q - 1 m d x ) 2 m β + 2 q + m - 3 m β + m + q - 1 (16)

where κ2=p2eβε1(m+1)|Ω|2-q-mβmβ+m+q-1 and κ3=mp|Ω|eε2(m+1)(m+2q-3). Combining (14) and (16) tells us that

d d t Ω | v | p d x κ 3 + κ 4 [ ( Ω | v 0 | m + 1 m d x ) 2 - q - m β m + 1 + ( 2 - q - m β ) κ 1 m + 1 t ] 2 m β + 2 q + m - 3 2 - q - m β (17)

where κ4=κ2|Ω|(2-q-mβ)(2mβ+2q+m-3)(m+1)(mβ+m+q-1). Integrating (17), we arrive at

Ω | v | p d x Ω | v 0 | p d x - ( m + 1 ) κ 4 κ 1 ( m β + q + m - 1 ) ( Ω | v 0 | m + 1 m d x ) m β + q + m - 1 m + 1 + κ 3 t + ( m + 1 ) κ 4 κ 1 ( m β + q + m - 1 ) ( ( Ω | v 0 | m + 1 m d x ) 2 - q - m β m + 1 + ( 2 - q - m β ) κ 1 m + 1 t ) m β + q + m - 1 2 - q - m β (18)

which implies that Ω|v|pdx=Ω|(|u|m-1u)|pdx is bounded for all t[0,+). The proof of Theorem 1 is complete.

Proof of Theorem 2   Multiplying both sides of the first equality in (4) by φ=|v|pav+=|u|mpa(|u|m-1u)+, we find that

1 m p a + m + 1 d d t Ω v + m p a + m + 1 m d x + p a + 1 ( a + 1 ) p Ω | v + a + 1 | p d x = 1 m Ω | v | q - 1 - m m v | v | p a v + l o g | v | d x 1 m e β Ω v + m ( p a + β + 1 ) + q - 1 m d x (19)

By virtue of Hölder's inequality and Sobolev embedding inequality, we have

Ω v + m p a + m + 1 m d x | Ω | 1 - ( N - p ) ( m p a + m + 1 ) N p m ( a + 1 ) ( Ω ( v + a + 1 ) m p a + m + 1 m ( a + 1 ) · N p m ( a + 1 ) ( N - p ) ( m p a + m + 1 ) d x ) ( N - p ) ( m p a + m + 1 ) N p m ( a + 1 ) | Ω | 1 - ( N - p ) ( m p a + m + 1 ) N p m ( a + 1 ) κ 5 ( N - p ) ( m p a + m + 1 ) N p m ( a + 1 ) ( Ω | v + a + 1 | p d x ) m p a + m + 1 p m ( a + 1 ) (20)

which implies that

Ω | v + a + 1 | p d x | Ω | ( N - p ) ( m + 1 ) - m p ( N + p a ) N ( m p a + m + 1 ) κ 5 p - N N ( Ω v + m p a + m + 1 m d x ) p m ( a + 1 ) m p a + m + 1 (21)

where κ5=κ5(p,N) is the optimal Sobolev embedding constant. Substituting (21) into (19) and using Hölder's inequality, we get

d d t Ω v + m p a + m + 1 m d x + κ 6 ( Ω v + m p a + m + 1 m d x ) p m ( a + 1 ) m p a + m + 1 κ 7 ( Ω v + m p a + m + 1 m d x ) m ( p a + β + 1 ) + q - 1 m p a + m + 1 (22)

where

κ 6 = ( m p a + m + 1 ) ( p a + 1 ) ( a + 1 ) p κ 5 p - N N | Ω | ( N - p ) ( m + 1 ) - p m ( N + p a ) N ( m p a + m + 1 ) (23)

and

κ 7 = m p a + m + 1 m e β | Ω | 1 - m ( p a + β + 1 ) + q - 1 m p a + m + 1 (24)

Recalling that 0<m(p-1)<q-1 and q+mβ<2, we check that

0 < p m ( a + 1 ) m p a + m + 1 < m ( p a + β + 1 ) + q - 1 m p a + m + 1 < 1 (25)

On the other hand, our assumption (7) tells us that

0 < κ 7 < 1 2 κ 6 ( Ω v 0 + m p a + m + 1 m d x ) m ( p - 1 ) - ( q - 1 ) - m β m p a + m + 1 (26)

Combining (22), (25), (26) and Lemma 1 [18], one can claim that there exist two positive constants ξ and η such that

0 y ( t ) : = Ω v + m p a + m + 1 m d x ξ e - η t ,    t 0 (27)

Choosing

T 0 > m a x { 0 , 1 η l n [ ξ ( 2 κ 7 k 6 ) m p a + m + 1 q - 1 - m ( p - 1 ) + m β ] } (28)

Then it follows from (22) and (27) that

d d t y ( t ) + κ 6 2 ( y ( t ) ) p m ( a + 1 ) m p a + m + 1 0 ,    t T 0 (29)

Integrating above inequality, one has

0 y 1 - p m ( a + 1 ) m p a + m + 1 ( t ) y 1 - p m ( a + 1 ) m p a + m + 1 ( T 0 ) - κ 6 ( m + 1 - p m ) 2 ( m p a + m + 1 ) ( t - T 0 ) (30)

which suggests that there exists a

T 0 ' [ T 0 , T 0 + 2 ( m p a + m + 1 ) κ 6 ( m + 1 - p m ) y m + 1 - p m m p a + m + 1 ( T 0 ) ] (31)

such that

l i m t T 0 ' - y ( t ) = l i m t T 0 ' - Ω v + m p a + m + 1 m ( t ) d x = 0 (32)

Moreover, it can be concluded that

Ω v + ( t ) d x | Ω | m p a + 1 m p a + m + 1 ( Ω v + m p a + m + 1 m d x ) m m p a + m + 1 0   a s   t T 0 ' - (33)

On the other hand, by a similar way, it can be shown that Ωv-(t)dx will vanish in finite time. Thus Ω|v(t)|dx=Ωv+(t)dx+Ωv-(t)dx vanishes in finite time. Recalling that v=|u|m-1u, one can claim that the solution u(x,t) to problem (1) possesses the extinction property. The proof of Theorem 2 is completed.

Proof of Theorem 3   Denoting

E ( v ( t ) ) = 1 p Ω | v | p d x - 1 q + m - 1 Ω | v | q + m - 1 m l o g | v | d x + m ( q + m - 1 ) 2 Ω | v | q + m - 1 m d x (34)

with v=|u|m-1u, then a direct calculation shows that

d E ( v ( t ) ) d t = - 4 m ( m + 1 ) 2 Ω [ ( v m + 1 2 m ) t ] 2 d x (35)

which implies

E ( v ( t ) ) = E ( v 0 ) - 4 m ( m + 1 ) 2 0 t Ω [ ( v m + 1 2 m ) t ] 2 d x d τ (36)

Set

M ( t ) = 1 m + 1 Ω | v | m + 1 m d x (37)

A direct calculation tells us that

M ' ( t ) = - Ω | v | p d x + 1 m Ω | v | q + m - 1 m l o g | v | d x = - p E ( v ( t ) ) + q - 1 - m ( p - 1 ) m ( q + m - 1 ) Ω | v | q + m - 1 m l o g | v | d x   + m p ( q + m - 1 ) 2 Ω | v | q + m - 1 m d x - p E ( v ( t ) ) + q - 1 - m ( p - 1 ) m ( q + m - 1 ) Ω | v | q + m - 1 m l o g | v | d x (38)

By virtue of (36), (38) and Hölder's inequality, one derives

M ' ( t ) - p E ( v 0 ) - m ( p - 1 ) - ( q - 1 ) q + m - 1 Ω 1 = { x Ω : | v | 1 } Ω 2 = { x Ω : | v | < 1 } | v | q + m - 1 m | l o g | v | | d x - p E ( v 0 ) - m ( p - 1 ) - ( q - 1 ) q + m - 1 ( 1 e β Ω 1 | v | m β + m + q - 1 m d x + m | Ω | e ( m + q - 1 ) ) - p E ( v 0 ) - m ( p - 1 ) - ( q - 1 ) e β ( q + m - 1 ) | Ω | 2 - q - m β m + 1 ( Ω | v | m + 1 m d x ) m β + m + q - 1 m + 1 - [ m ( p - 1 ) - ( q - 1 ) ] m | Ω | e ( q + m - 1 ) 2 (39)

If m(p-1)=q-1. Then from (39), one can see that, for any t0,

M ( t ) M ( 0 ) - p E ( v 0 ) t (40)

Keeping in mind that

M ( 0 ) = 1 m + 1 Ω | v 0 | m + 1 m d x = 1 m + 1 Ω | u 0 | m + 1 d x > 0 (41)

and

E ( v 0 ) = E ( | u 0 | m - 1 u 0 ) 0 (42)

then (40) gives us that M(t)>0, which means that the solution u(x,t) to problem (1) cannot vanish in finite time.

If m(p-1)q-1. Remembering that

M ( 0 ) = 1 m + 1 Ω | v 0 | m + 1 m d x = 1 m + 1 Ω | u 0 | m + 1 d x > 0 (43)

and

E ( v 0 ) = E ( | u 0 | m - 1 u 0 ) < - m | Ω | [ m ( p - 1 ) - ( q - 1 ) ] e p ( q + m - 1 ) 2 (44)

combining (39) and Lemma 1.2 [19] gives us that, for any t0,

M ( t ) m i n { M ( 0 ) , ( κ 8 κ 9 ) m + 1 m β + q + m - 1 } > 0 (45)

which means that the solution u(x,t) to problem (1) cannot vanish in finite time, where

κ 8 = - ( p E ( v 0 ) + [ m ( p - 1 ) - ( q - 1 ) ] m | Ω | e ( q + m - 1 ) 2 ) > 0 (46)

κ 9 = m ( p - 1 ) - ( q - 1 ) e β ( q + m - 1 ) | Ω | 2 - q - m β m + 1 > 0 (47)

The proof of Theorem 3 is completed.

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