Skip to main content

Complicated asymptotic behavior of solutions for a porous medium equation with nonlinear sources

Abstract

In this paper, we investigate the complicated asymptotic behavior of the solutions to the Cauchy problem of a porous medium equation with nonlinear sources when the initial value belongs to a weighted L space.

AMS Subject Classification:35K55, 35B40.

1 Introduction

In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem of the porous medium equation with nonlinear sources

u t Δ u m = u p ,in  R N ×(0,),
(1.1)
u(x,0)= u 0 (x),in  R N ,
(1.2)

where m,p>1 and u 0 L ( ρ σ ){φ;φ ρ σ L ( R N )} with ρ σ (x)= ( 1 + | x | 2 ) σ 2 .

It is well known that any positive solutions of problem (1.1)-(1.2) blow up in finite time if 1<p p c m+ 2 N [13], while positive global solutions do exist if p> p c [47]. In 2000, Mukai, Mochizuki and Huang in [6] found that if p>m+ 2 N and 2 p m <σ<N and 0φ C b ( R N ) satisfies lim sup | x | | x | σ φ(x)<, then there exists a constant η(φ)>0 such that for 0<η<η(φ), the solutions u(x,t) of problem (1.1)-(1.2) with the initial value u 0 =ηφ are global and the following estimate holds:

u ( t ) L ( R N ) C t σ σ ( m 1 ) + 2 .
(1.3)

Moreover, if lim | x | | x | σ φ(x)= M 0 >0, then

t σ σ ( m 1 ) + 2 u ( t 1 σ ( m 1 ) + 2 x , t ) t S(1) w 0 (x)

uniformly on R N , where w 0 (x)=M | x | σ . Here S(t) is a semigroup generated by the Cauchy problem of the porous medium equation

w t Δ w m =0,in  R N ×(0,),
(1.4)
w(x,0)= w 0 (x),in  R N
(1.5)

and w 0 (x)=η M 0 | x | σ .

On the other hand, regarding problem (1.4)-(1.5), in 2002, Vázquez and Zuazua [8] found that for any bounded sequence { φ n } n = 1 in L ( R N ), there exists an initial value u 0 L ( R N ) and a sequence t n k as k such that lim k S( t n k ) u 0 ( t n k 1 2 x)=S(1) φ n (x) uniformly on any compact subsets of R N . In our previous papers [9], for any bounded sequence { φ n } n = 1 in C 0 + ( R N ){φ C 0 ( R N );ϕ(x)0}, we have shown that there exists an initial value u 0 C 0 ( R N ) and a sequence t n k as k such that

lim k t n k μ 2 S( t n k ) u 0 ( t n k β x ) =S(1) φ n (x)

uniformly on R N , where 0<μ< N N ( m 1 ) + 2 and β= 2 μ ( m 1 ) 4 . For more details on the study of complicated asymptotic behavior of solutions for the heat equation and other evolution equations, we refer the readers to [1014].

In this paper, we are quite interested in the above mentioned same topic for the equation with strongly nonlinear sources, namely equation (1.1) with p>m+ 2 N . We will show that for any M>0, there is a constant η(M) and an initial value u 0 C η ( M ) σ , + {φC( R N );φ B η ( M ) σ , + } with 2 p m <σ<N such that for any φ C η ( M ) σ , + , there exists a sequence t n as n satisfying

lim n t n σ σ ( m 1 ) + 2 S( t n ) u 0 ( t n 1 σ ( m 1 ) + 2 x ) =S(1)φ(x)

uniformly on R N . Here B η ( M ) σ , + {ϕ=ηφ;0φ L ( ρ σ ), ( 1 + | | 2 ) σ 2 φ ( ) L ( R N ) M and 0ηη(M)}. For this purpose, we first show that if the initial value u 0 B η ( M ) σ , + , then the solutions u(x,t) are global and satisfy

u(x,t)C ( 1 + t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 .
(1.6)

One can easily see that (1.6) captures (1.3). From this, we can follow the framework by Kamin and Peletier [15] to prove that

lim t t 1 σ ( m 1 ) + 2 u ( t σ σ ( m 1 ) + 2 , t ) S ( t ) u 0 ( t 1 σ ( m 1 ) + 2 ) L ( R N ) =0.
(1.7)

So, we can get our results by following the framework in [9] and using (1.6)-(1.7).

The rest of this paper is organized as follows. The next section is devoted to giving a sufficient condition for the global existence of solutions for problem (1.1)-(1.2) and the upper bounded estimates on these solutions. In the last section, we investigate the complicated asymptotic behavior of solutions.

2 Preliminaries and estimates

In this section we state the definition of a weak solution of problem (1.1)-(1.2) and give the upper bounded estimates on the global solutions. We begin with the definition of the weak solution of problem (1.1)-(1.2).

Definition 2.1 [16, 17]

By a weak solution of problem (1.1)-(1.2) in R N ×[0,T), we mean a function u(x,t) in R N ×[0,T) such that

  1. 1.

    u(x,t)0 in R N ×[0,T) and u(x,t)C( R N ×(0,τ]) for each 0<τ<T.

  2. 2.

    For 0<τ<T and any nonnegative φ(x,t) C 2 , 1 ( R N ×[0,T)) which vanishes for large |x|, the following equation holds:

    R N u ( x , τ ) φ ( x , τ ) d x R N u 0 ( x ) φ ( x , 0 ) d x = 0 τ R N u m ( x , t ) Δ φ ( x , t ) d x d t + 0 τ R N u ( x , t ) φ t ( x , t ) d x d t + 0 τ R N u p ( x , t ) φ ( x , t ) d x d t .
    (2.1)

A supersolution [or subsolution] is similarly defined with equality of (2.1) replaced by ≥ [or ≤]. The weak solutions for problem (1.4)-(1.5) can be defined in a similar way as above. It is well known that problem (1.1)-(1.2) has a unique, nonnegative and bounded weak solution, at least locally in time [16, 17]. Now we state the comparison principle for problem (1.1)-(1.2).

Lemma 2.1 [16, 17]

Suppose that for 0<τ<T, u ¯ (x,t), u ̲ (x,t)C( R N ×[0,T)) L ( R N ×[0,τ]) are supersolution and subsolution of the problem (1.1)-(1.2), respectively. If

u ¯ (x,0) u ̲ (x,0)for x R N ,

then, for all (x,t) R N ×(0,T),

u ¯ (x,t) u ̲ (x,t).

To study the asymptotic behavior of solutions for problem (1.1)-(1.2), we adopt the space X 0 and L ( ρ σ ) as that in [1618]. For any σ>0 and ρ σ (x)= ( 1 + | x | 2 ) σ 2 , the L ( ρ σ ) is defined as

L ( ρ σ ) { φ ; φ ρ σ L ( R N ) }

with the obvious norm φ L ( ρ σ ) = φ ρ σ L ( R N ) and the X 0 is given by

X 0 { φ L loc 1 ( R N ) ; φ 1 <  and  ( φ ) = 0 }

with the norm 1 . Here

φ r = sup R r R N ( m 1 ) + 2 m 1 { | x | R } | φ ( x ) | dxand(φ)= lim r φ r .

Hence they are both Banach spaces. The existence and uniqueness of a weak solution of problem (1.4)-(1.5) with the initial-value u 0 X 0 is shown in [16, 17], and this solution satisfies the following proposition.

Proposition 2.1 [17]

Problem (1.4)-(1.5) generates a continuous bounded semigroup in X 0 given by

S(t): w 0 w(x,t).

In other words, S(t) w 0 C([0,); X 0 ). Moreover, if u 0 L 1 ( R N ), then the semigroup S(t) is a contraction.

We now introduce the definitions of scalings and the commutative relations between the semigroup operators and the dilation operators. For any μ,β>0 and v(x) X 0 , the dilation D λ μ , β is defined as follows:

D λ μ , β w(x) λ μ w ( λ β x ) .

From the definitions of the dilation operator and the semigroup operator, we can get that for μ,β>0 and w 0 X 0 ,

D λ μ , β [ S ( λ 2 t ) w 0 ] (x)=S ( λ 2 μ ( m 1 ) 2 β t ) [ D λ μ , β w 0 ] (x);
(2.2)

see details in [19, 20].

In the rest of this section, we give a sufficient condition for the existence of global solutions of problem (1.1)-(1.2) and establish the upper bounded estimates of these solutions.

Theorem 2.1 Let 2 p m <σ<N and M>0. There exists a constant η(M) such that for any 0ηη(M), ϕ(x)0 and ϕ L ( ρ σ ) M, the solutions u(x,t) of problem (1.1)-(1.2) with the initial value u 0 (x)=ηϕ(x) are global. Moreover, the following estimate holds:

0u(x,t)C(M,η) ( 1 + t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 ,
(2.3)

where C(M,η) is a constant dependent only on M and η.

Remark 2.1 Notice that if 0φ C b ( R N ) and lim sup | x | φ(x) | x | σ <, then φ L ( ρ σ ). So, our results capture Theorem 3 in [6]. Here we use some ideas of them.

Proof To prove this theorem, we need the fact that if v 0 =M | x | σ , then

S(t) v 0 (x)C(M) ( t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 ,
(2.4)

which has been given in Lemma 2.6 of [20]. We give the proof here for completeness. In fact,

v 0 r = sup R r R N ( m 1 ) + 2 m 1 B R A | x | σ dxC r σ 2 m 1 0as r.

This means that v 0 X 0 . Therefore, from Proposition 2.1, we obtain that S(t) v 0 (x) is well defined. Taking μ= 2 σ σ ( m 1 ) + 2 and β= 2 σ ( m 1 ) + 2 in (2.2), we have

λ 2 σ σ ( m 1 ) + 2 [ S ( λ 2 s ) v 0 ] ( λ 2 σ ( m 1 ) + 2 x ) =S(s) [ λ 2 σ σ ( m 1 ) + 2 v 0 ( λ 2 σ ( m 1 ) + 2 ) ] (x)=S(s) v 0 (x).
(2.5)

Now taking s=1, λ= t 1 2 and g(x)=S(1) v 0 (x) in (2.5), we obtain that

S(t) v 0 (x)= t σ σ ( m 1 ) + 2 g ( t 1 σ ( m 1 ) + 2 x ) .
(2.6)

The fact that ϕ C ( R N {0}) clearly means that

S(t) v 0 C ( [ 0 , ) × R N { ( 0 , 0 ) } ) C α 2 , α ( ( 0 , ) × R N ) for some α>0;
(2.7)

see [21]. This implies that for |x|=1, the following limit holds:

t σ σ ( m 1 ) + 2 g ( t 1 σ ( m 1 ) + 2 x ) =S(t) v 0 (x)ϕ(x)=M | x | σ =Mas t0.

Let

y= t 1 σ ( m 1 ) + 2 x.

So,

|y|as t0.

Therefore,

| y | σ g(y)M0

as |y|. This means that there exists an M 1 >1 such that if |y| M 1 , then

g(y)2M | y | σ .
(2.8)

From (2.7), for |y| M 1 , there exists a constant C such that

g(y)C.
(2.9)

Combining (2.8) and (2.9), we have

g(x)C(M) ( 1 + | x | 2 ) σ 2 for x R N .

By (2.6), we thus obtain that

S(t) v 0 (x)C(M) ( t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 .

So, we complete the proof of (2.4). Now taking

φ(x)=M ( 1 + | x | 2 ) σ 2 ,

we get that

0<φ(x) v 0 (x)=M | x | σ for x0.

Therefore, by the comparison principle and (2.4), for all t0, we have

S(t)φ(x)S(t) v 0 (x)C(M) ( t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 .
(2.10)

Since S(t)φ(x)C([0,)× R N ) (see [17, 21]), there exists a t 1 >0 such that for all |x|1 and 0t t 1 ,

S(t)φ(x)Cφ(x)C(M) ( 1 + | x | 2 ) σ 2 .

Combining this with (2.6) and using the comparison principle, we can get

S(t)ϕ(x)S(t)φ(x)C(M) ( 1 + t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 .

In other words,

S(t)ϕ(x)C(M) ( ( 1 + t ) 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 .
(2.11)

If η=0, (2.3) clearly holds. In the rest of proof, we can assume that η>0. The hypothesis 2 p m <σ<N indicates

σ(pm)2>0.

Let

η ( M ) m p = 2 C p 1 ( M ) ( p m ) 0 ( 1 + t ) σ ( p 1 ) σ ( m 1 ) + 2 d t = 2 C ( M ) p 1 ( σ ( m 1 ) + 2 ) ( p m ) σ ( p m ) 2 > 0 ,

where C(M) is the constant given by (2.11). For 0<ηη(M), taking

α(t)= [ ( η m p C ( M ) p 1 ( p m ) 0 t ( 1 + s ) σ ( p 1 ) σ ( m 1 ) + 2 d x ) ] 1 p m ,

and

w(x,t)=S(t)ϕ(x),

we obtain from (2.11) that α(t) is an increasing function satisfying

{ a ( 0 ) = η , a ( t ) 2 1 p m η ( M ) for all  t 0 , a ( t ) = C ( M ) p 1 a ( t ) p m + 1 ( 1 + t ) σ ( p 1 ) σ ( m 1 ) + 2 a ( t ) p m + 1 w ( t ) L ( R N ) p 1 .
(2.12)

Now letting b(t) to satisfy

{ b ( t ) = a ( b ( t ) ) m 1 , b ( 0 ) = 0
(2.13)

and then taking

w ¯ (x,t)=a ( b ( t ) ) w(x,t),

one can see that w ¯ (x,t) is a supersolution of the following problem:

u t Δ u m = u p , ( x , t ) R N × ( 0 , ) ; u ( x , 0 ) = u 0 = η ϕ ( x ) , x R N .

Therefore,

u ( x , t ) a ( b ( t ) ) w ( x , t ) 2 1 p m η ( M ) w ( x , b ( t ) ) C ( η , M ) ( 1 + b ( t ) 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 .
(2.14)

(2.12) and (2.13) clearly mean that

η m 1 tb(t) 2 m 1 p m η ( M ) m 1 t.

From this and (2.14), we can get (2.3). So, we complete the proof of this theorem. □

3 Complicated asymptotic behavior

For any M>0, let η(M) be as given by Theorem 2.1. We introduce

B η ( M ) σ , + { φ ( x ) = η ϕ ( x ) : ϕ ( x ) 0 , ϕ L ( ρ σ ) M  and  0 η η ( M ) }

and

C η ( M ) σ , + { φ C ( R N ) ; φ B η ( M ) σ , + } .

In the rest of this section, we show that the complexity may occur in the asymptotic behavior of solutions of problem (1.1)-(1.2) with u 0 C η ( M ) σ , + . Our main result is the following theorem.

Theorem 3.1 Let p>m+ 2 N and 2 p m <σ<N. Then there is a function u 0 C η ( M ) σ , + such that for any φ C η ( M ) σ , + , there exists a sequence t n as n such that

lim n t n σ σ ( m 1 ) + 2 u ( t n 1 σ ( m 1 ) + 2 x , t n ) =S(1)φ(x)

uniformly on R N . Here u(x,t) is the solution of problem (1.1)-(1.2).

To get this theorem, we need to prove the following lemma first.

Lemma 3.1 Suppose p>m+ 2 N and M>0. Let u be a solution of problem (1.1)-(1.2). If 0 u 0 B η ( M ) σ , + with 2 p m <σ<N, then

lim t t σ 2 + σ ( m 1 ) u ( t 1 2 + σ ( m 1 ) , t ) [ S ( t ) u 0 ] ( t 1 2 + σ ( m 1 ) ) L ( R N ) =0.

Proof

We first define the functions

u λ (x,t)= D λ μ , β u(x,λt)= λ μ u ( λ β x , λ 2 t )

and

w λ (x,t)= D λ μ , β w(x,λt)= λ μ w ( λ β x , λ 2 t ) ,

where μ= 2 σ σ ( m 1 ) + 2 and β= 2 σ ( m 1 ) + 2 . Using the comparison principle, we know that for (x,t) R N ×(0,),

w(x,t)u(x,t)

and for all λ1,

w λ (x,t) u λ (x,t).

The results of Theorem 2.1 imply that

u λ ( x , t ) C λ μ [ ( 1 + λ 2 t ) β + λ 2 β | x | 2 ] σ 2 C ( ( λ 2 + t ) 2 β + | x | 2 ) σ 2 C ( λ 2 + t ) μ ( 1 + ( λ 2 + t ) 2 β | x | 2 ) σ 2 .
(3.1)

Here we have used the fact μ=βσ. So,

0 τ B 1 u λ (x,t)dxdtC λ 2 τ + λ 2 s N β μ ds 0 s β r N σ 1 drCτ.

Now we estimate the integral

0 τ B 1 u λ ( x , t ) q dxdt

with q>1 in several steps. For any τ>0, we take λ large enough to satisfy λ 2 τ and assume, without loss of generality, that ( τ + λ 2 ) β >1 in the rest of this proof. Then using the same method as above, we have

0 τ B 1 u λ ( x , t ) q dxdt{ C τ 2 γ β + C τ if  γ > 0  and  N σ q , C τ + C τ ln 1 τ if  γ > 0  and  N = σ q , C τ + C ln ( 1 + λ 2 τ ) if  γ = 0 , C τ + C λ 2 β γ if  γ < 0 ,
(3.2)

where γ=N+σ(m1)σq+2. Similarly, we can get the integral estimates for w λ (x,t), which have been given in [22]. By using the same methods as in [15], we can get that for T>0

u λ (T) w λ (T)0as λ
(3.3)

uniformly on any compact subset of R N . For any T,λ,ϵ>0, we can obtain from (3.1) that there exists a constant R>0 satisfying

u λ ( T ) L ( R N B R ) φ λ ( T ) L ( R N B R ) < ϵ 3
(3.4)

and

w λ ( T ) L ( R N B R ) u λ ( , T ) L ( R N B R ) < ϵ 3 ,
(3.5)

where φ λ (x,t)=C λ μ [ ( 1 + λ 2 t ) 2 σ ( m 1 ) + 1 + | λ β x | 2 ] σ 2 =C [ ( λ 2 + t ) 2 σ ( m 1 ) + 2 + | x | 2 ] σ 2 and B R {x R N ;|x|R}. Taking R as given by (3.4), from (3.3), there exists λ 1 such that for all λ λ 1 ,

u λ ( T ) w λ ( T ) L ( B R ) < ϵ 3 .
(3.6)

Therefore, from (3.4)-(3.6), we have

lim λ u λ ( T ) w λ ( T ) L ( R N ) =0.
(3.7)

Now letting T=1 and λ= t 1 2 in (3.7), we get that

lim t t σ 2 + σ ( m 1 ) [ u ( t 1 2 + σ ( m 1 ) , t ) w ( t 1 2 + σ ( m 1 ) , t ) ] L ( R N ) =0.

So, we complete the proof of this lemma. □

Now we can prove our main result.

Proof of Theorem 3.1

Let

μ= 2 σ σ ( m 1 ) + 2

and

β= 2 σ ( m 1 ) + 2 .

From the definition of C η ( M ) σ , + , we obtain that there exists a countable set F such that

F C η ( M ) σ , + L 1 ( R N )

and for any ϵ>0 and φ C η ( M ) σ , + , there exists a function ϕ ϵ F satisfying

φ ϵ φ L ( R N ) <ϵ.
(3.8)

Therefore, there exists a sequence { φ j } j 1 F such that

  1. I.

    For any ϕF, there exists a subsequence { φ j k } k 1 of the sequence { φ j } j 1 satisfying

    φ j k (x)=ϕfor all k1,
  2. II.

    There exists a constant C>0 satisfying

    max ( φ j L ( R N ) , φ j L 1 ( R N ) ) Cjfor j1.

Now we can follow the methods given in [9] to construct an initial value as follows. Let

u 0 (x)= j = 1 λ j μ χ j ( x / λ j β ) φ j ( x / λ j β ) = j = 1 D λ j 1 μ , β [ χ j ( x ) φ j ( x ) ] .
(3.9)

Here

λ j ={ 2 for  j = 1 , max ( j 4 N ( m 1 ) + 8 2 N μ [ N ( m 1 ) + 2 ] λ j 1 4 β N 2 μ 2 N μ [ N ( m 1 ) + 2 ] , ( 2 j λ j 1 ) 1 μ , λ ¯ j ) for  j > 1 ,
(3.10)

χ j (x) is the cut-off function defined on {x R N ; 2 j <|x|< 2 j } relatively to {x R N ; 2 j + 1 <|x|< 2 j 1 }, and λ ¯ j is selected large enough to satisfy

D λ j μ , β [ S ( λ j 2 t ) u 0 ( x ) ] = D λ j μ , β [ S ( λ j 2 t ) n = 1 j 1 λ n μ χ n ( x / λ n β ) φ n ( x / λ n β ) ] + D λ j μ , β [ S ( λ j 2 t ) λ j μ χ j ( x / λ j β ) φ j ( x / λ j β ) ] + D λ j μ , β [ S ( λ j 2 t ) n = j + 1 λ n μ χ n ( x / λ n β ) φ n ( x / λ n β ) ] .

Notice first that if φ C η ( M ) σ , + , then

φ L ( R N ) η(M), φ L ( ρ σ ) η(M)

and

φ C 0 ( R N ) .

By (3.9) and (3.10), we have

u 0 L ( R N ) u 0 L ( ρ σ ) sup j 1 λ j μ χ j ( x / λ j β ) φ j ( x / λ j β ) L ( ρ σ ) η(M).

So, we have

u 0 C η ( M ) σ , + C 0 ( R N ) .

Using the same method as that in [9], we can get that for any φF, there exists a sequence t n as n such that

t n σ σ ( m 1 ) + 2 [ S ( t n ) u 0 ] ( t n 1 σ ( m 1 ) + 2 x ) n S(1)φ(x)
(3.11)

uniformly on R N . For any ϕ C η ( M ) σ , + , from (1.2), we know that there exists a sequence { φ k }F such that

φ k ϕas k.

Therefore,

S(1) φ k S(1)ϕas k
(3.12)

uniformly on any compact subset of R N . This uses the fact that the map S(1) is regularizing since the images of bounded sets are relatively compact subsets of C α for some α>0 in compact sets of R N [21]. And notice that φ k ,ϕ C η ( M ) σ , + B η ( M ) σ , + . We thus obtain from Theorem 2.1 that for any ε>0, there exists R>0 such that if |x|>R, then

S(1)ϕ(x)< ε 3
(3.13)

and

S(1) φ k (x)< ε 3 for all k1.
(3.14)

Combining (3.12), (3.13) with (3.14), we thus have that

S(1) φ k S(1)ϕas k
(3.15)

uniformly on R N . By Lemma 3.1, (3.11) and (3.15), we can get that for any ϕ C η ( M ) σ , + , there exists a sequence t n as t such that

lim n t n σ σ ( m 1 ) + 2 u ( t n 1 σ ( m 1 ) + 2 x , t n ) =S(1)ϕ(x)

uniformly on R N . So, we complete the proof of Theorem 3.1. □

References

  1. Galaktionov VA, Kurdjumov SP, Mihaĭov AP, Samarskiĭ AA:On unbounded solutions of the Cauchy problem for the parabolic equation u t =( u σ u)+ u β . Sov. Math. Dokl. 1980, 252(6):1362-1364. (Russian)

    Google Scholar 

  2. Galaktionov VA: Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. R. Soc. Edinb. A 1994, 124(3):517-525. (English summary) 10.1017/S0308210500028766

    Article  MATH  MathSciNet  Google Scholar 

  3. Kawanago T:Existence and behaviour of solutions for u t =Δ( u m )+ u . Adv. Math. Sci. Appl. 1997, 7(1):367-400. (English summary)

    MATH  MathSciNet  Google Scholar 

  4. Mochizuki K, Suzuki R: Critical exponent and critical blow-up for quasilinear parabolic equations. Isr. J. Math. 1997, 98: 141-156. (English summary) 10.1007/BF02937331

    Article  MATH  MathSciNet  Google Scholar 

  5. Suzuki R: Asymptotic behavior of solutions of quasilinear parabolic equations with slowly decaying initial data. Adv. Math. Sci. Appl. 1999, 9(1):291-317. (English summary)

    MATH  MathSciNet  Google Scholar 

  6. Mukai K, Mochizuki K, Huang Q: Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values. Nonlinear Anal. 2000, 39(1):33-45. 10.1016/S0362-546X(98)00161-8

    Article  MATH  MathSciNet  Google Scholar 

  7. Suzuki R: Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity. J. Differ. Equ. 2003, 190(1):150-181. (English summary) 10.1016/S0022-0396(02)00086-4

    Article  MATH  Google Scholar 

  8. Vázquez JL, Zuazua E: Complexity of large time behaviour of evolution equations with bounded data. Chin. Ann. Math., Ser. B 2002, 23(2):293-310. 10.1142/S0252959902000274

    Article  MATH  Google Scholar 

  9. Yin J, Liangwei W, Huang R: Complexity of asymptotic behavior of solutions for the porous medium equation with absorption. Acta Math. Sci. 2010, 30(6):1865-1880.

    Article  MATH  MathSciNet  Google Scholar 

  10. Cazenave T, Dickstein F, Weissler FB:Universal solutions of the heat equation on R N . Discrete Contin. Dyn. Syst. 2003, 9(5):1105-1132. (English summary)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cazenave T, Dickstein F, Weissler FB:Universal solutions of a nonlinear heat equation on R N . Ann. Sc. Norm. Super. Pisa, Cl. Sci. 2003, 2(1):77-117. (English summary)

    MATH  MathSciNet  Google Scholar 

  12. Cazenave T, Dickstein F, Weissler FB:Chaotic behavior of solutions of the Navier-Stokes system in R N . Adv. Differ. Equ. 2005, 10(4):361-398.

    MathSciNet  Google Scholar 

  13. Cazenave T, Dickstein F, Weissler FB:Nonparabolic asymptotic limits of solutions of the heat equation on R N . J. Dyn. Differ. Equ. 2007, 19(3):789-818. 10.1007/s10884-007-9076-z

    Article  MATH  MathSciNet  Google Scholar 

  14. Carrillo JA, Vázquez JL: Asymptotic complexity in filtration equations. J. Evol. Equ. 2007, 7(3):471-495. (English summary) 10.1007/s00028-006-0298-z

    Article  MATH  MathSciNet  Google Scholar 

  15. Kamin S, Peletier LA: Large time behaviour of solutions of the heat equation with absorption. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1985, 12(3):393-408.

    MATH  MathSciNet  Google Scholar 

  16. Bénilan P, Crandall MG, Pierre M:Solutions of the porous medium equation in R N under optimal conditions on initial values. Indiana Univ. Math. J. 1984, 33(1):51-87. 10.1512/iumj.1984.33.33003

    Article  MATH  MathSciNet  Google Scholar 

  17. Vázquez JL Oxford Mathematical Monographs. In The Porous Medium Equation. Mathematical Theory. Clarendon, Oxford; 2007.

    Google Scholar 

  18. DiBenedetto E, Herrero MA: On the Cauchy problem and initial traces for a degenerate parabolic equation. Trans. Am. Math. Soc. 1989, 314(1):187-224.

    Article  MATH  MathSciNet  Google Scholar 

  19. Yin J, Wang L, Huang R:Complexity of asymptotic behavior of the porous medium equation in R N . J. Evol. Equ. 2011, 11(2):429-455. 10.1007/s00028-010-0097-4

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang L, Yin J, Jin C: ω -Limit sets for porous medium equation with initial data in some weighted spaces. Discrete Contin. Dyn. Syst., Ser. B 2013, 18(1):223-236.

    Article  MATH  MathSciNet  Google Scholar 

  21. DiBenedetto E: Degenerate Parabolic Equations. Springer, New York; 1993.

    Book  MATH  Google Scholar 

  22. Kamin S, Peletier LA: Large time behaviour of solutions of the porous media equation with absorption. Isr. J. Math. 1986, 55(2):129-146. 10.1007/BF02801989

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by NSFC, the Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of ‘CQ CSTC’ (cstc2012jjA00013), the Scientific and Technological Projects of Chongqing Municipal Commission of Education (KJ121105).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Liangwei Wang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper is the result of joint work of both authors who contributed equally to the final version of the paper. Both authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Wang, L., Yin, J. Complicated asymptotic behavior of solutions for a porous medium equation with nonlinear sources. Bound Value Probl 2013, 35 (2013). https://doi.org/10.1186/1687-2770-2013-35

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2013-35

Keywords