Complicated Asymptotic Behavior of Solutions for Heat Equation in Some Weighted Space

and Applied Analysis 3 Endowed with the obvious norms, ∥ ∥φ ∥ ∥ Y 0 R N ∥ ∥ ∥ ∥φ · ( 1 |·| )−σ/2∥ ∥ ∥ L∞ RN , ∥ ∥φ ∥ ∥ L p σ RN ∥ ∥ ∥ ∥φ · ( 1 |·| )−σ/2∥ ∥ ∥ Lp RN , 2.3 the spaces Y 0 R N and Lpσ R are both Banach spaces. Notice that if σ 0, then Y 0 0 ( R N ) C0 ( R N ) , L0 ( R N ) L∞ ( R N ) , L p 0 ( R N ) L ( R N ) . 2.4 Next we give the definition of the ω-limit set ω σ u0 which is our main study object in this paper . Definition 2.1. Let σ ≥ 0, μ, β > 0, and suppose that u0 ∈ Y 0 R . The ω-limit set ω μ,β σ u0 is given by ω μ,β σ u0 ≡ { f ∈ Y 0 ( R N ) ; ∃tn −→ ∞ s.t. D tn S tn u0 n→∞ −−−−−→ f in Y 0 ( R N )} . 2.5 Here D λ φ x ≡ λφ λ2βx for φ ∈ Lloc R and λ > 0. In the rest of this section, we will consider the properties of the solutions for the problem 1.1 when the initial value u0 ∈ Lpσ R or u0 ∈ Y 0 R . The following theorem can be seen as some extension of the maximum principle for the problem 1.1 . Theorem 2.2. Let 0 ≤ σ < ∞. Suppose that u0 ∈ Lσ ( R N ) 2.6 and that u x, t S t u0 x are the mild solutions of the problem 1.1 . Then u t S t u0 ∈ Lσ ( R N ) for t > 0. 2.7 Moreover, if t > 1, then ‖S t u0‖L∞σ RN ≤ Ct‖u0‖L∞σ RN , 2.8 or if 0 < t ≤ 1, then ‖S t u0‖L∞σ RN ≤ C‖u0‖L∞σ RN . 2.9 4 Abstract and Applied Analysis Remark 2.3. Let σ 0. From Theorem 2.2, we can obtain the well-known result maximum principle that if u0 ∈ L0 R L∞ R , then ‖S t u0‖L∞ RN ≤ C‖u0‖L∞ RN . 2.10 Proof. To prove this theorem, we need the fact that if φ x M ( 1 |x| )σ/2 for some M > 0, 2.11 then there exists a constant C such that S t φ x ≤ C ( 1 t |x| )σ/2 , 2.12 which proof can be found in 17 ; we give the proof here for completeness. Consider the following problem: ∂v ∂t −Δv 0, in R × 0,∞ , v x, 0 v0 x M|x|, in R. 2.13 For λ > 0, from 2.1 , we can get that D−σ/2,1/2 λ S λt v0 x λ −σ/2 S λt v0 ( λ1/2x ) S t [ D−σ/2,1/2 λ v0 ] x S t v0 x . 2.14 By the existence and the regularity theories of the solutions, we can obtain that, for t > 0, 0 < S t v0 ∈ C∞ ( 0,∞ × R ) , 2.15 see 10, 18 . Now taking t 1, λ s and g x S 1 v0 x in the expression 2.14 , we have S s v0 x sσ/2g ( s−1/2x ) . 2.16 The fact that S s v0 x ∈ C 0,∞ × R \ 0, 0 clearly implies that, for |x| 1, sσ/2g ( s−1/2x ) S s v0 x −→ v0 x M|x| M 2.17


Introduction
In this paper, we consider the asymptotic behavior of solutions to the Cauchy problem of the heat equation where N ≥ 1 and the initial value u 0 ∈ Y σ 0 R N .Whether complexity occurs in the asymptotic behavior of solutions for some evolution equations or not mainly depends on the work spaces that one selects 1-9 .In the space L p R N with 1 ≤ p < ∞, the problem 1.1 under consideration is well posed and the asymptotic behavior of the solutions is rather simple, reflecting the simple structure of the heat equation.Considering, for instance, the problem 1.1 with the initial value u 0 ∈ L 1 R N , 2 Abstract and Applied Analysis it is well-known that the solutions u x, t converge as t → ∞ toward a multiple of the fundamental solution, the one which has the same integral, u x, t S t u 0 x G t * u 0 x −→ MG x, t , 1.2 where G x, t 4πt −N/2 exp −|x| 2 /4t and M R N u 0 x dx, see 10, 11 .It was first found in 2002 12 by Vázquez and Zuazua that the bounded function space L ∞ R N provides a setting where complicated asymptotic behavior of solutions may take place for the heat equation.In fact, they proved that, for any bounded sequence {g j , j 1, 2, . ..} in L ∞ R N , there exists an initial value u 0 ∈ L ∞ R N and a sequence t j k → ∞ as k → ∞ such that uniformly on any compact subset of R N as k → ∞.Subsequently, Cazenave et al. showed that, in the bounded continuous function space C 0 R N , the solutions of the heat equation may present more complex asymptotic behavior 13-15 .Meanwhile, considerable attention has also been paid to study the complicated asymptotic behavior of solutions for the porous medium equation and other evolution equations in some bounded function spaces such as C 0 R N and L ∞ R N see, e.g., 3, 7, 9, 12, 16 and the references therein .
In this paper we find that, even in the unbounded function space Y σ 0 R N with 0 < σ < N, the complicated asymptotic behavior of solutions for the heat equation can also occur.For this purpose, we need to establish the L p σ -L ∞ σ smoothing effect and other estimates for the solutions of the problem 1.1 when the initial value The rest of this paper is organized as follows.In the next section, we give some definitions and some estimates of the solutions to the problem 1.1 .Section 3 is devoted to study the complicated asymptotic behavior of the solutions.

Main Estimates
In this section, we investigate some properties of solutions for the problem 1.1 when the initial value u 0 belongs to some weighted spaces.For these purposes, we first introduce the mild solutions u x, t of the problem 1.1 which are defined as Letting σ ≥ 0 and 1 ≤ p ≤ ∞, we define two weighted spaces Y σ 0 R N and L p σ R N as follows:

2.2
Abstract and Applied Analysis 3 Endowed with the obvious norms,

2.4
Next we give the definition of the ω-limit set ω μ,β σ u 0 which is our main study object in this paper .
Definition 2.1.Let σ ≥ 0, μ, β > 0, and suppose that u 0 ∈ Y σ 0 R N .The ω-limit set ω μ,β σ u 0 is given by Here D μ,β λ ϕ x ≡ λ μ ϕ λ 2β x for ϕ ∈ L 1 loc R N and λ > 0. In the rest of this section, we will consider the properties of the solutions for the problem 1.1 when the initial value The following theorem can be seen as some extension of the maximum principle for the problem 1.1 .
and that u x, t S t u 0 x are the mild solutions of the problem 1.1 .Then 2.9 Remark 2.3.Let σ 0. From Theorem 2.2, we can obtain the well-known result maximum principle that if

2.10
Proof.To prove this theorem, we need the fact that if for some M > 0, 2.11 then there exists a constant C such that 12 which proof can be found in 17 ; we give the proof here for completeness.Consider the following problem:

2.13
For λ > 0, from 2.1 , we can get that

2.14
By the existence and the regularity theories of the solutions, we can obtain that, for t > 0, see 10, 18 .Now taking t 1, λ s and g x S 1 v 0 x in the expression 2.14 , we have The fact that S s v 0 x ∈ C 0, ∞ × R N \ 0, 0 clearly implies that, for |x| 1, By 2.16 , we thus have

2.23
Therefore, by comparison principle and 2.22 , we can get that, for all t ≥ 0, there exists constant C > 0 such that

2.24
So we complete the proof of 2.12 .For any t > 0, from 2.1 and 2.12 , we thus obtain that

2.27
From this, we can get 2.9 .So we complete the proof of this theorem.
S t u 0 x are the solutions of the problem 1.1 .Then 29

2.30
Remark 2.5.If σ 0, then Theorem 2.4 captures the result L p -L ∞ smoothing effect for the heat equation.
Proof.For any t > 0, from 2.1 and Theorem 2.2, we thus obtain that

2.31
Here 1/p 1/p 1.From this, we can get that, if t ≥ 1, then

2.33
So we complete the proof of this theorem.
In the following theorem, we consider the property of the solutions u x, t of 1.1 with the initial data Proof.For σ 0, the above theorem is a well-known result.So, in the rest of this proof, we assume that 0 < σ < ∞.From 2.1 , we have

2.35
For any ε > 0, from u 0 ∈ Y σ 0 R N , we obtain that there exists an M > 0 such that if |y| > M, then

2.36
So, from 2.12 , we have

2.39
Notice also that

2.40
So, there exists a constant C such that sup |y|≤M

2.41
This means that So, there exists an M 1 > 0 such that if |x| > M 1 , then I 1 x < ε.

2.44
So we complete the proof of this theorem.

Complicated Asymptotic Behavior
In this section, we investigate the asymptotic behavior of solutions for the problem 1.1 and give the fact that the weighted space Y σ 0 R N with 0 ≤ σ < N can provide a setting where complexity occurs in the asymptotic behavior of solutions.

then there exists an initial value u
Abstract and Applied Analysis 9 Proof.Suppose that a > 2 is a constant.Then let for j > 1.

3.3
Now we define the initial-value u 0 as So, Here we have used the fact that if 0 < λ ≤ 1, then

3.7
Therefore, the sequence of 3.4 is convergent in Y σ 0 R N .This means that As a result of 2.1 , we see that, for 0 < t < T < ∞, 3.9 where

3.11
we thus obtain that Consequently, for any t > 0, we can select N large enough to satisfy that if n > N, then 0 < λ 2−4β n t < 1.

3.14
From 3.3 , we thus obtain that as n → ∞.From 3.7 and the definition of w n x , we have

3.16
Applying 2.9 to S tλ 2−4β n w n , we thus have as n → ∞.Here we have used 3.3 and the fact that 0 < λ 2−2β n t < 1 for n > N. At present, we want to verify the claim that, for 0 ≤ t < ∞,

3.19
Notice that

3.22
The fact

3.30
So, there exists a sequence {ϕ j } j≥1 ⊂ F such that i for any φ ∈ F, there exists a subsequence {ϕ j k } k≥1 of the sequence {ϕ j } j≥1 satisfying

3.34
The initial-value u 0 is given by

3.36
This means that the sequence 3.35 is convergent in Y σ 0 R N .So, Y σ 0 R N .

3.41
So we complete the proof of this theorem.

3.42
Therefore, from Theorem 3.2, we can get that the weighted space Y σ 0 R N provides a setting where complexity occurs in the asymptotic behavior of solutions for the problem 1.1 .
3.37Similar to the proof of Theorem 2.2, we can prove that, for any φ ∈ F and 0 < t < ∞, there exists a sequence λ j k → ∞ as k → ∞ such that