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

Abstract

We investigate the asymptotic behavior of solutions for the heat equation in the weighted space 𝑌^𝜎0(ℝ^𝑁)≡{𝜑∈𝐶(ℝ^𝑁)∶lim_|𝑥|→∞(1+|𝑥|²)^−𝜎/2𝜑(𝑥)=0}. Exactly, we find that the unbounded function space 𝑌^𝜎0(ℝ^𝑁) with 0<𝜎<𝑁 can provide a setting where complexity occurs in the asymptotic behavior of solutions for the heat equation.

1. Introduction

In this paper, we consider the asymptotic behavior of solutions to the Cauchy problem of the heat equation 𝜕𝑢𝜕𝑡−Δ𝑢=0,(𝑥,𝑡)∈ℝ^𝑁𝑢(𝑥,0)=𝑢₀×(0,∞),(𝑥),𝑥∈ℝ^𝑁,(1.1) where 𝑁≥1 and the initial value 𝑢₀∈𝑌^𝜎0(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 𝑢₀∈𝐿¹(ℝ^𝑁), 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)𝑢₀(𝑥)=𝐺(𝑡)∗𝑢₀(𝑥)⟶𝑀𝐺(𝑥,𝑡),(1.2) where 𝐺(𝑥,𝑡)=(4𝜋𝑡)^−𝑁/2exp(−|𝑥|²/4𝑡) and 𝑀=∫_ℝ𝑁𝑢₀(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 𝑢₀∈𝐿^∞(ℝ^𝑁) and a sequence 𝑡_𝑗𝑘→∞ as 𝑘→∞ such that 𝑢𝑡^𝑗1/2_𝑘𝑥,𝑡_𝑗𝑘=𝑆𝑡^𝑗_𝑘𝑢₀𝑡_𝑗𝑘𝑥⟶𝑆(1)𝑔_𝑗(𝑥)(1.3) uniformly on any compact subset of ℝ^𝑁 as 𝑘→∞. Subsequently, Cazenave et al. showed that, in the bounded continuous function space 𝐶₀(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 𝐶₀(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^𝜎0(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^𝑝𝜎–𝐿^∞𝜎 smoothing effect and other estimates for the solutions of the problem (1.1) when the initial value 𝑢₀∈𝐿^𝑝𝜎(ℝ^𝑁)≡{𝜑∶(1+|⋅|²)^−𝜎/2𝜑(⋅)∈𝐿^𝑝(ℝ^𝑁)} with 1<𝑝≤∞ or 𝑢₀∈𝑌^𝜎0(ℝ^𝑁).

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.

2. Main Estimates

In this section, we investigate some properties of solutions for the problem (1.1) when the initial value 𝑢₀ 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 𝑢(𝑥,𝑡)=𝑆(𝑡)𝑢₀(𝑥)=(4𝜋𝑡)^−𝑁/2_ℝ𝑁exp−||𝑥−𝑦||²4𝑡𝑢⁰(𝑦)d𝑦.(2.1) Letting 𝜎≥0 and 1≤𝑝≤∞, we define two weighted spaces 𝑌^𝜎0(ℝ^𝑁) and 𝐿^𝑝𝜎(ℝ^𝑁) as follows: 𝑌^𝜎0ℝ^𝑁𝐿≡𝜑(𝑥)∈𝐶ℝ^𝑁∶lim_|𝑥|→∞𝜑(𝑥)1+|𝑥|²^−𝜎/2=0_𝑝𝜎,ℝ^𝑁≡𝜑∶𝜑(⋅)1+|⋅|²^−𝜎/2∈𝐿^𝑝ℝ^𝑁.(2.2) Endowed with the obvious norms, ‖‖𝜑𝜑‖‖_𝑌^𝜎0_(ℝ^𝑁_{)𝐿𝑝𝜎(ℝ𝑁)}=‖‖𝜑(⋅)1+|⋅|²^−𝜎/2‖‖_{𝐿∞(ℝ𝑁)},=‖‖𝜑(⋅)1+|⋅|²^−𝜎/2‖‖_{𝐿𝑝(ℝ𝑁)},(2.3) the spaces 𝑌^𝜎0(ℝ^𝑁) and 𝐿^𝑝𝜎(ℝ^𝑁) are both Banach spaces. Notice that if 𝜎=0, then 𝑌⁰⁰ℝ^𝑁=𝐶₀𝐿_𝑝0ℝ^𝑁,𝐿^∞0ℝ^𝑁=𝐿^∞ℝ^𝑁,ℝ^𝑁=𝐿^𝑝ℝ^𝑁.(2.4) Next we give the definition of the 𝜔-limit set 𝜔^{𝜎𝜇,𝛽}(𝑢₀) which is our main study object in this paper.

Definition 2.1. Let 𝜎≥0, 𝜇,  𝛽>0, and suppose that 𝑢₀∈𝑌^𝜎0(ℝ^𝑁). The 𝜔-limit set 𝜔^{𝜎𝜇,𝛽}(𝑢₀) is given by 𝜔^{𝜎𝜇,𝛽}𝑢₀≡𝑓∈𝑌^𝜎0ℝ^𝑁;∃𝑡_𝑛⟶∞s.t.𝐷^√𝜇,𝛽_𝑡𝑛𝑆𝑡_𝑛𝑢₀−−−−^𝑛→∞→𝑓in𝑌^𝜎0ℝ^𝑁.(2.5)
Here 𝐷^{𝜆𝜇,𝛽}𝜑(𝑥)≡𝜆^𝜇𝜑(𝜆^2𝛽𝑥) for 𝜑∈𝐿^1loc(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 𝑢₀∈𝐿^𝑝𝜎(ℝ^𝑁) or 𝑢₀∈𝑌^𝜎0(ℝ^𝑁).
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∈𝐿^∞𝜎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∈𝐿^∞𝜎ℝ^𝑁for𝑡>0.(2.7) Moreover, if 𝑡>1, then ‖‖𝑆(𝑡)𝑢₀‖‖_{𝐿∞𝜎(ℝ𝑁)}≤𝐶𝑡^𝜎‖‖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)

Remark 2.3. Let 𝜎=0. From Theorem 2.2, we can obtain the well-known result (maximum principle) that if 𝑢₀∈𝐿^∞0(ℝ^𝑁)=𝐿^∞(ℝ^𝑁), 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|²^𝜎/2forsome𝑀>0,(2.11) then there exists a constant 𝐶 such that 𝑆(𝑡)𝜑(𝑥)≤𝐶1+𝑡+|𝑥|²^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,inℝ^𝑁𝑣(𝑥,0)=𝑣₀×(0,∞),(𝑥)=𝑀|𝑥|^𝜎,inℝ^𝑁.(2.13) For 𝜆>0, from (2.1), we can get that 𝐷^{𝜆−𝜎/2,1/2}𝑆(𝜆𝑡)𝑣₀(𝑥)=𝜆^−𝜎/2𝑆(𝜆𝑡)𝑣₀𝜆^1/2𝑥=𝑆(𝑡)𝐷^{𝜆−𝜎/2,1/2}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^𝑁,(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)=𝑠^𝜎/2𝑔𝑠^−1/2𝑥.(2.16) The fact that 𝑆(𝑠)𝑣₀(𝑥)∈𝐶([0,∞)×ℝ^𝑁⧵(0,0)) clearly implies that, for |𝑥|=1, 𝑠^𝜎/2𝑔𝑠^−1/2𝑥=𝑆(𝑠)𝑣₀(𝑥)⟶𝑣₀(𝑥)=𝑀|𝑥|^𝜎=𝑀(2.17) as 𝑠→0. Let 𝑦=𝑠^−1/2𝑥.(2.18) So, ||𝑦||⟶∞as𝑠⟶0.(2.19) Therefore, ||𝑦||^−𝜎𝑔(𝑦)−𝑀⟶0(2.20) as |𝑦|→∞. So, there exists constant 0<𝐶<∞ such that 0≤𝑔(𝑥)≤𝐶1+|𝑥|²^𝜎/2.(2.21) By (2.16), we thus have 𝑆(𝑠)𝑣₀(𝑥)≤𝐶𝑠+|𝑥|²^𝜎/2.(2.22) Notice that 0≤𝜑(𝑥)≤𝐶1+𝑣₀(𝑥).(2.23) Therefore, by comparison principle and (2.22), we can get that, for all 𝑡≥0, there exists constant 𝐶>0 such that 0≤𝑆(𝑡)𝜑(𝑥)≤𝐶1+𝑡+|𝑥|²^𝜎/2.(2.24) So we complete the proof of (2.12). For any 𝑡>0, from (2.1) and (2.12), we thus obtain that ||𝑢(𝑥,𝑡)||=||𝑆(𝑡)𝑢₀(𝑥)||=||||(4𝜋𝑡)^−𝑁/2_ℝ𝑁=||||(4𝜋𝑡)_−𝑁/2exp^{(−|𝑥−𝑦|2/4𝑡)}𝑢₀(𝑦)d𝑦||||_ℝ𝑁≤‖‖𝑢exp^{(−|𝑥−𝑦|2/4𝑡)}1+||𝑦||²^𝜎/2𝑢₀(𝑦)1+||𝑦||²^−𝜎/2d𝑦₀||||‖‖_{𝑌𝜎∞(ℝ𝑁)}(4𝜋𝑡)^−𝑁/2_ℝ𝑁≤𝐶1+𝑡+|𝑥|₂exp^{(−|𝑥−𝑦|2/4𝑡)}1+||𝑦||²^𝜎/2d𝑦^𝜎/2‖‖𝑢₀‖‖_{𝑌𝜎∞(ℝ𝑁)}.(2.25) Therefore, if 𝑡>1, then ||𝑢(𝑥,𝑡)||≤𝐶1+|𝑥|²^𝜎/2𝑡^𝜎‖‖𝑢₀‖‖_{𝑌𝜎∞(ℝ𝑁)}.(2.26) This clearly illustrates (2.8). If 0<𝑡≤1, then ||𝑢(𝑥,𝑡)||≤𝐶1+|𝑥|²^𝜎/2‖‖𝑢₀‖‖_{𝑌𝜎∞(ℝ𝑁)}.(2.27) From this, we can get (2.9). So we complete the proof of this theorem.

Theorem 2.4 (𝐿^𝑝𝜎–𝐿^∞𝜎 smoothing effect). Let 1<𝑝<∞ and 0≤𝜎<∞. Suppose 𝑢₀∈𝐿^𝑝𝜎(ℝ^𝑁) and that 𝑢(𝑥,𝑡)=𝑆(𝑡)𝑢₀(𝑥) are the solutions of the problem (1.1). Then 𝑢(𝑡)=𝑆(𝑡)𝑢₀∈𝐿^∞𝜎ℝ^𝑁for𝑡>0.(2.28) Moreover, if 𝑡>1, then ‖‖𝑆(𝑡)𝑢₀‖‖_{𝐿∞𝜎(ℝ𝑁)}≤𝐶𝑡^{𝜎−𝑁/2𝑝}‖‖𝑢₀‖‖_𝐿^𝑝𝜎_(ℝ𝑁),(2.29) or if 0<𝑡≤1, then ‖‖𝑆(𝑡)𝑢₀‖‖_{𝐿∞𝜎(ℝ𝑁)}≤𝐶𝑡^{−𝑁/2𝑝}‖‖𝑢₀‖‖_𝐿^𝑝𝜎_(ℝ𝑁).(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 ||𝑢(𝑥,𝑡)||=||𝑆(𝑡)𝑢₀(𝑥)||=||||(4𝜋𝑡)^−𝑁/2_ℝ𝑁=||||(4𝜋𝑡)_−𝑁/2exp^{(−|𝑥−𝑦|2/4𝑡)}𝑢₀(𝑦)d𝑦||||_ℝ𝑁≤(4𝜋𝑡)_−𝑁/2exp^{(−|𝑥−𝑦|2/4𝑡)}1+||𝑦||²^𝜎/2𝑢₀(𝑦)1+||𝑦||²^−𝜎/2d𝑦||||_ℝ𝑁exp^{(−|𝑥−𝑦|2/4𝑡)}1+||𝑦||²^{𝑝𝜎/2}d𝑦^1/𝑝×(4𝜋𝑡)^−𝑁/2_ℝ𝑁exp^{(−|𝑥−𝑦|2/4𝑡)}1+||𝑦||²^{−𝑝𝜎/2}𝑢^𝑝0(𝑦)d𝑦^1/𝑝≤𝐶1+𝑡+|𝑥|²^𝜎/2𝑡^{−𝑁/2𝑝}‖‖𝑢₀‖‖_𝐿^𝑝𝜎_(ℝ𝑁).(2.31) Here 1/𝑝+1/𝑝′=1. From this, we can get that, if 𝑡≥1, then ‖𝑢(𝑡)‖_𝐿^∞𝜎_(ℝ^𝑁₎≤𝐶𝑡^{𝜎/2−𝑁/2𝑝}‖‖𝑢₀‖‖_𝐿^𝑝𝜎_(ℝ𝑁),(2.32) or if 0<𝑡<1, then ‖𝑢(𝑡)‖_𝐿^∞𝜎_(ℝ^𝑁₎≤𝐶𝑡^{−𝑁/2𝑝}‖‖𝑢₀‖‖_𝐿^𝑝𝜎_(ℝ𝑁).(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 𝑢₀∈𝑌^𝜎0(ℝ^𝑁).

Theorem 2.6. Let 0≤𝜎<∞. If 𝑢₀∈𝑌^𝜎0(ℝ^𝑁), then 𝑆(𝑡)𝑢₀∈𝑌^𝜎0ℝ^𝑁for𝑡≥0.(2.34)

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 1+=|𝑥|²1+|𝑥^−𝜎/2|||𝑢₂(𝑥,𝑡)||^−𝜎/2||||(4𝜋𝑡)_−𝑁/2_ℝ𝑁=1+|𝑥|₂exp^{(−|𝑥−𝑦|2/4𝑡)}𝑢₀(𝑦)d𝑦||||^−𝜎/2||||(4𝜋𝑡)_−𝑁/2_{|𝑦|≤𝑀}+1+|𝑥|₂exp^{(−|𝑥−𝑦|2/4𝑡)}𝑢₀(𝑦)d𝑦||||^−𝜎/2||||(4𝜋𝑡)_−𝑁/2_{|𝑦|≤𝑀}=I₁exp^{(−|𝑥−𝑦|2/4𝑡)}𝑢₀(𝑦)d𝑦||||(𝑥)+I₂(𝑥).(2.35) For any 𝜀>0, from 𝑢₀∈𝑌^𝜎0(ℝ^𝑁), we obtain that there exists an 𝑀>0 such that if |𝑦|>𝑀, then ||||𝑢₀(𝑦)1+||𝑦||²^−𝜎/2||||<𝜀.(2.36) So, from (2.12), we have I₂(𝑥)=1+|𝑥|²^−𝜎/2||||(4𝜋𝑡)_−𝑁/2_|𝑦|>𝑀≤1+|𝑥|₂exp^{(−|𝑥−𝑦|2/4𝑡)}𝑢₀(𝑦)1+||𝑦||²^−𝜎/21+||𝑦||²^𝜎/2d𝑦||||^−𝜎/2sup_{||𝑦||>𝑀}||||𝑢₀(𝑦)1+||𝑦||²^−𝜎/2||||(4𝜋𝑡)_−𝑁/2_|𝑦|>𝑀≤𝐶1+|𝑥|₂exp^{(−|𝑥−𝑦|2/4𝑡)}1+||𝑦||²^𝜎/2d𝑦^−𝜎/21+𝑡+|𝑥|²^𝜎/2𝜀.(2.37) So, if 𝑡>1, then I₂(𝑥)≤𝐶1+|𝑥|²^−𝜎/21+𝑡+|𝑥|²^𝜎/2𝜀≤𝐶𝑡^𝜎/2𝜀,(2.38) or if 0≤𝑡≤1, then I₂(𝑥)≤𝐶1+|𝑥|²^−𝜎/21+𝑡+|𝑥|²^𝜎/2𝜀≤𝐶𝜀.(2.39) Notice also that 𝑢₀∈𝑌^𝜎0ℝ^𝑁⊂𝐶ℝ^𝑁.(2.40) So, there exists a constant 𝐶 such that sup_{||𝑦||≤𝑀}||𝑢₀(𝑦)||≤𝐶.(2.41) This means that I₁(𝑥)=1+|𝑥|²^−𝜎/2||||(4𝜋𝑡)_−𝑁/2_{|𝑦|≤𝑀}exp^{(−|𝑥−𝑦|2/4𝑡)}𝑢₀(𝑦)d𝑦||||≤𝐶1+|𝑥|²^−𝜎/2.(2.42) So, there exists an 𝑀₁>0 such that if |𝑥|>𝑀₁, then I₁(𝑥)<𝜀.(2.43) Combining this with (2.35), (2.38), and (2.39), we can obtain that, for 𝑡≥0, 𝑢(𝑡)=𝑆(𝑡)𝑢₀∈𝑌^𝜎0ℝ^𝑁.(2.44) So we complete the proof of this theorem.

3. 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^𝜎0(ℝ^𝑁) with 0≤𝜎<𝑁 can provide a setting where complexity occurs in the asymptotic behavior of solutions.

Theorem 3.1. Let 𝜇>0, 𝜎≥0, 𝑝>1, and 𝛽>1/2. If 𝜇+2𝛽𝜎<𝑁𝑓∈𝑌_𝜎0𝑝,ℝ^𝑁𝐿^𝑝𝜎ℝ^𝑁,(3.1) then there exists an initial value 𝑢₀∈𝑌^𝜎0(ℝ^𝑁) and a sequence 𝑡_𝑛→∞ as 𝑛→∞ such that 𝐷^√𝜇,𝛽_𝑡𝑛𝑆𝑡_𝑛𝑢−−−−_0𝑛→∞→𝑓in𝑌^𝜎0ℝ^𝑁.(3.2) That is, 𝑓∈𝜔^{𝜎𝜇,𝛽}(𝑢₀).

Proof. Suppose that 𝑎>2 is a constant. Then let 𝜆𝜆₁=𝑎,_𝑗=max𝑗^{2𝑝/(𝑁−𝑝(𝜇+2𝛽𝜎))}𝜆^{(2𝛽𝑁−𝑝(𝜇+2𝛽𝜎))/(𝑁−𝑝(𝜇+2𝛽𝜎))𝑗−1},2^𝑗𝜆_𝑗−1^1/𝜇for𝑗>1.(3.3) Now we define the initial-value 𝑢₀ as 𝑢₀(𝑥)=^∞_𝑗=1𝜆_𝑗−𝜇𝑓𝑥𝜆^𝑗2𝛽=^∞_𝑗=1𝐷^{𝜆𝜇,𝛽}_𝑗−1𝑓(𝑥).(3.4) Let ℓ=max‖𝑓‖_𝑌^𝜎0_(ℝ^𝑁₎,‖𝑓‖_𝐿^𝑝𝜎_(ℝ^𝑁₎.(3.5) So, ‖‖𝑢₀‖‖_{𝑌𝜎0(ℝ𝑁)}≤^∞_𝑗=1𝜆_𝑗−𝜇‖‖‖‖𝑓1𝜆^𝑗2𝛽⋅‖‖‖‖_{𝑌𝜎0(ℝ𝑁)}≤^∞_𝑗=12^−𝑗‖𝑓(𝑥)‖_𝑌^𝜎0_(ℝ^𝑁₎≤ℓ.(3.6) Here we have used the fact that if 0<𝜆≤1, then ‖𝑓(𝜆⋅)‖_𝑌^𝜎0_(ℝ^𝑁₎=sup_{𝑥∈ℝ𝑁}|||1+𝑥₂=sup_{𝑥∈ℝ𝑁}^−𝜎/2𝑓(𝜆𝑥)||||||||𝑓(𝜆𝑥)1+||𝜆𝑥||₂^−𝜎/211++||𝜆|𝑥𝑥|||²²^𝜎/2≤sup_{𝑥∈ℝ𝑁}|||||||||𝑓(𝜆𝑥)1+||𝜆𝑥||₂^−𝜎/2=‖𝑓‖_{𝑌𝜎0(ℝ𝑁)}||||.(3.7) Therefore, the sequence of (3.4) is convergent in 𝑌^𝜎0(ℝ^𝑁). This means that 𝑢₀∈𝑌^𝜎0ℝ^𝑁.(3.8) As a result of (2.1), we see that, for 0<𝑡<𝑇<∞,𝐷^{𝜆𝜇,𝛽}_𝑛𝑆𝜆^2𝑛𝑡𝑢₀(𝑥)=𝑆𝑡𝜆^{𝑛2−4𝛽}𝑢^𝑛+𝑣_𝑛+𝑤_𝑛=𝑆𝑡𝜆^{𝑛2−4𝛽}𝑢^𝑛+𝑆𝑡𝜆^{𝑛2−4𝛽}𝑣^𝑛+𝑆𝑡𝜆^{𝑛2−4𝛽}𝑤^𝑛,(3.9) where 𝑢_𝑛(𝑥)=^𝑛−1_𝑗=1𝐷^{𝜆𝜇,𝛽}_𝑛𝐷^{𝜆𝜇,𝛽𝑗−1}𝑓(𝑥)=𝜆^{𝜇𝑛𝑛−1}_𝑗=1𝜆_𝑗−𝜇𝑓𝑥𝜆^𝑛2𝛽𝑣_𝑛𝜆^𝑗2𝛽,𝑤(𝑥)=𝐷^{𝜆𝜇,𝛽}_𝑛𝐷^{𝜆𝜇,𝛽𝑛−1}𝑓(𝑥)=𝑓(𝑥),_𝑛=^∞_𝑗=_𝑛+1𝐷^{𝜆𝜇,𝛽}_𝑛𝐷^{𝜆𝜇,𝛽𝑗−1}𝑓(𝑥)=𝜆^𝜇𝑛∞_𝑗=_𝑛+1𝜆_𝑗−𝜇𝑓𝑥𝜆^𝑛2𝛽𝜆^𝑗2𝛽.(3.10) Notice that, if 𝜆≥1, then ‖𝑓(𝜆⋅)‖_𝑌^𝜎𝑝_(ℝ^𝑁₎=_ℝ𝑁|||1+|𝑥|₂^−𝜎/2𝑓(𝜆𝑥)|||^𝑝d𝑥^1/𝑝≤⎛⎜⎜⎝_ℝ𝑁|||||𝑓(𝜆𝑥)1+||𝜆𝑥||₂^−𝜎/211++||𝜆|𝑥𝑥|||²²^𝜎/2|||||^𝑝d𝑥⎞⎟⎟⎠^1/𝑝≤𝜆^𝜎_ℝ^𝑁||||𝑓(𝜆𝑥)1+||𝜆𝑥||₂^−𝜎/2||||^𝑝d𝑥^1/𝑝=𝜆^{𝜎−𝑁/𝑝}‖𝑓‖_𝑌^𝜎𝑝_(ℝ^𝑁₎,(3.11) we thus obtain that ‖‖𝑢_𝑛‖‖_{𝑌𝜎𝑝(ℝ𝑁)}≤^𝑛−1_𝑗=1𝜆𝜆^𝑛_𝑗^{𝜇−2𝛽𝑁/𝑝+2𝛽𝜎}‖𝑓‖_𝑌^𝜎𝑝_(ℝ^𝑁₎≤𝑛𝜆^𝑛𝜆_𝑛−1^{𝜇−2𝛽𝑁/𝑝+2𝛽𝜎}‖𝑓‖_𝑌^𝜎𝑝_(ℝ^𝑁₎.(3.12) Consequently, for any 𝑡>0, we can select 𝑁 large enough to satisfy that if 𝑛>𝑁, then 0<𝜆^{𝑛2−4𝛽}𝑡<1.(3.13) Here we have used the hypothesis 𝛽>1/2. So, (2.30), (3.11), and (3.12) indicate ‖‖𝑆𝜆^{𝑛2−4𝛽}𝑡𝑢_𝑛(𝑥)‖‖_{𝑌𝜎0(ℝ𝑁)}≤𝐶𝜆^{𝑛2−4𝛽}𝑡^{−𝑁/2𝑝}‖‖𝑢_𝑛(𝑥)‖‖_𝐿^𝑝𝜎_(ℝ𝑁)≤𝐶𝜆^{2𝛽𝑁/𝑝−2𝛽𝜎−𝜇𝑛−1}𝑡^{−2𝑁/𝑝}(𝑛ℓ)𝜆^{𝑛𝜇−𝑁/𝑝+2𝛽𝜎}.(3.14) From (3.3), we thus obtain that ‖‖𝑆𝜆^{𝑛2−4𝛽}𝑡𝑢_𝑛(𝑥)‖‖_{𝑌𝜎0(ℝ𝑁)}≤𝐶𝑡^{−2𝑁/𝑝}ℓ𝑛⁻¹⟶0(3.15) as 𝑛→∞. From (3.7) and the definition of 𝑤_𝑛(𝑥), we have ‖‖𝑤_𝑛‖‖_{𝑌𝜎0(ℝ𝑁)}≤‖‖‖‖^∞_𝑗=_𝑛+1𝜆𝜆^𝑛_𝑗^𝜇𝑓𝜆^𝑛𝜆_𝑗^2𝛽⋅‖‖‖‖_{𝑌𝜎0(ℝ𝑁)}≤^∞_𝑗=_𝑛+1𝜆𝜆^𝑛_𝑗^𝜇ℓ.(3.16)
Applying (2.9) to 𝑆(𝑡𝜆^{𝑛2−4𝛽})𝑤_𝑛, we thus have ‖‖𝑆𝑡𝜆^{𝑛2−4𝛽}𝑤^𝑛‖‖_{𝑌𝜎0(ℝ𝑁)}≤𝐶𝜆^𝜇𝑛∞_𝑗=_𝑛+1𝜆_{𝑗+1−𝜇}≤𝐶ℓ^∞_𝑗=_𝑛+12^−𝑗⟶0(3.17) as 𝑛→∞. Here we have used (3.3) and the fact that 0<𝜆^{𝑛2−2𝛽}𝑡<1 for 𝑛>𝑁. At present, we want to verify the claim that, for 0≤𝑡<∞, 𝑆𝜆^{𝑛2−4𝛽}𝑡𝑣_𝑛−𝑓−−−−_{𝑌𝜎0(ℝ𝑁)𝑛→∞}→0.(3.18) In fact, if 0<𝑡<∞, then ‖‖𝑆𝜆^{𝑛2−4𝛽}𝑡𝑣_𝑛−𝑓‖‖_{𝑌𝜎0(ℝ𝑁)}≤sup_{|𝑥|≤𝑀}1+|𝑥|²^−𝜎/2|||𝑆𝜆_{𝑛2−4𝛽}+sup_|𝑥|>𝑀𝑡𝑣_𝑛(𝑥)−𝑓(𝑥)|||1+|𝑥|²^−𝜎/2|||𝑆𝜆_{𝑛2−4𝛽}𝑡𝑣_𝑛(𝑥)|||+sup_|𝑥|>𝑀1+|𝑥|²^−𝜎/2||𝑓(𝑥)||.(3.19) Notice that 𝑣_𝑛=𝑓∈𝑌^𝜎0ℝ^𝑁.(3.20) Therefore, from Theorem 2.6, we have 𝑆𝜆^{𝑛2−4𝛽}𝑡𝑣_𝑛∈𝑌^𝜎0ℝ^𝑁.(3.21) So, for any 𝜀>0, from (2.35), (2.39), and (2.43), we get that if 𝑛>𝑁, there exists a constant 𝑀 independing on 𝑛 and 𝑡 such that sup_|𝑥|>𝑀1+|𝑥|²^−𝜎/2|||𝑆𝜆_{𝑛2−4𝛽}𝑡𝑣_𝑛(𝑥)|||<𝜀sup_|𝑥|>𝑀3,1+|𝑥|²^−𝜎/2||𝑓(𝑥)||<3𝜀.(3.22) The fact 𝑣_𝑛=𝑓∈𝐶(ℝ^𝑁) means that 𝑆(𝑡)𝑣_𝑛∈𝐶[0,∞)×ℝ^𝑁.(3.23) Therefore, for any 𝜀>0, there exists an 𝑁₁>0 such that if 𝑛>𝑁₁, then sup_{|𝑥|≤𝑀}1+|𝑥|²^−𝜎/2|||𝑆𝜆_{𝑛2−4𝛽}𝑡𝑣_𝑛(𝑥)−𝑓(𝑥)|||<𝜀3.(3.24)
So, from (3.19), (3.22), and (3.24), we thus obtain that ‖‖𝑆𝜆^{𝑛2−4𝛽}𝑡𝑣_𝑛−𝑓‖‖_{𝑌𝜎0(ℝ𝑁)}⟶0(3.25) as 𝑛→∞. From (3.9), (3.15), (3.17), and (3.18), we obtain that for any fixed 0<𝑡<∞, 𝐷^{𝜆𝜇,𝛽}_𝑛𝑆𝜆^2𝑛𝑡𝑢₀=𝑆𝜆^{𝑛2−2𝛽}𝑡𝑢_𝑛+𝑣_𝑛+𝑤_𝑛−−−−^𝑛→∞→𝑓in𝑌^𝜎0ℝ^𝑁.(3.26) Equation (3.2) follows from (3.26) by setting 𝑡=1 and 𝑡_𝑛=𝜆^2𝑛. The proof of this theorem is complete.

Theorem 3.2. Let 𝜇>0, 𝜎≥0, and 𝛽>1/2. If 0<𝜇+2𝛽𝜎<𝑁,(3.27) then there exists an initial value 𝑢₀∈𝑌^𝜎0(ℝ^𝑁) such that 𝜔^{𝜎𝜇,𝛽}𝑢₀=𝑌^𝜎0ℝ^𝑁.(3.28)

Remark 3.3. If 𝜎=0, the above results had been given by Cazenave et al. (see [15, Corollary 6.3]). So our results capture their results. Here we have used some of their ideas.

Proof. By (3.27), there exists a constant 𝑝>1 such that 𝜇+2𝛽𝜎<𝑁𝑝.(3.29) Therefore, there exists a countable dense subset 𝐹 of 𝑌^𝜎0(ℝ^𝑁) such that 𝐹⊂𝑌^𝜎0ℝ^𝑁𝐿^𝑝𝜎ℝ^𝑁.(3.30) So, there exists a sequence {𝜑_𝑗}_𝑗≥1⊂𝐹 such that (i)for any 𝜙∈𝐹, there exists a subsequence {𝜑_𝑗𝑘}_𝑘≥1 of the sequence {𝜑_𝑗}_𝑗≥1 satisfying 𝜑_𝑗𝑘(𝑥)=𝜙∀𝑘≥1,(3.31)(ii)there exists a constant 𝐶>0 satisfying max‖‖𝜑_𝑗‖‖_{𝑌𝜎0(ℝ𝑁)},‖‖𝜑_𝑗‖‖_{𝐿𝑝𝜎(ℝ𝑁)}≤𝐶𝑗for𝑗≥1.(3.32)Now we select a constant 𝑎>2(3.33) and then take 𝜆_𝑗=𝑎max𝑗_{4𝑝/(𝑁−𝑝(𝜇+2𝛽𝜎))}if𝑗=1,𝜆^{(2𝛽𝑁−𝑝(𝜇+2𝛽𝜎))/(𝑁−𝑝(𝜇+2𝛽𝜎))𝑗−1},2^𝑗𝜆_𝑗−1𝑗^1/𝜇if𝑗>1.(3.34) The initial-value 𝑢₀ is given by 𝑢₀(𝑥)=^∞_𝑗=1𝜆_𝑗−𝜇𝜑_𝑗𝜆𝑥^𝑗2𝛽=^∞_𝑗=1𝐷^{𝜆𝜇,𝛽}_𝑗−1𝜑_𝑗(𝑥).(3.35)
From (3.7), (3.32), and (3.34), we have ‖‖𝑢₀‖‖_{𝑌𝜎0(ℝ𝑁)}≤^∞_𝑗=1𝜆_𝑗−𝜇‖‖𝜑_𝑗‖‖_{𝑌𝜎0(ℝ𝑁)}≤𝐶^∞_𝑗=12^−𝑗≤𝐶.(3.36)
This means that the sequence (3.35) is convergent in 𝑌^𝜎0(ℝ^𝑁). So, 𝑢₀∈𝑌^𝜎0ℝ^𝑁.(3.37) Similar to the proof of Theorem 2.2, we can prove that, for any 𝜙∈𝐹 and 0<𝑡<∞, there exists a sequence 𝜆_𝑗𝑘→∞ as 𝑘→∞ such that 𝐷^{𝜆𝜇,𝛽}_𝑗𝑘𝑆𝜆^2𝑗_𝑘𝑡𝑢₀→𝜙in𝑌^𝜎0ℝ^𝑁(3.38) as 𝑘→∞. Let 𝑡_𝑛𝑘=𝜆^𝑛1/2_𝑘 and 𝑡=1 in (2.1); we thus have 𝐷^√𝜇,𝛽_𝑡𝑗𝑘𝑆𝑡_𝑗𝑘𝑢₀⟶𝜙in𝑌^𝜎0ℝ^𝑁.(3.39) Therefore, 𝐹⊂𝜔^{𝜎𝜇,𝛽}𝑢₀⊂𝑌^𝜎0ℝ^𝑁.(3.40) Notice that 𝐹 is dense in 𝑌^𝜎0(ℝ^𝑁) and that 𝜔^{𝜎𝜇,𝛽}(𝑢₀) is a closed subset of 𝑌^𝜎0(ℝ^𝑁). We thus obtain from (3.40) that 𝜔^{𝜎𝜇,𝛽}𝑢₀=𝑌^𝜎0ℝ^𝑁.(3.41) So we complete the proof of this theorem.

Remark 3.4. For any 0≤𝜎<𝑁, there exist constants 𝜇>0 and 𝛽>1/2 such that 0<𝜇+2𝛽𝜎<𝑁.(3.42)
Therefore, from Theorem 3.2, we can get that the weighted space 𝑌^𝜎0(ℝ^𝑁) provides a setting where complexity occurs in the asymptotic behavior of solutions for the problem (1.1).

Acknowledgment

This work is supported by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the Scientific and Technological Projects of Chongqing Municipal Commission of Education.