Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 463082 | https://doi.org/10.1155/2012/463082

Liangwei Wang, Jingxue Yin, "Complicated Asymptotic Behavior of Solutions for Heat Equation in Some Weighted Space", Abstract and Applied Analysis, vol. 2012, Article ID 463082, 15 pages, 2012. https://doi.org/10.1155/2012/463082

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

Academic Editor: Toka Diagana
Received05 Jun 2012
Accepted29 Jul 2012
Published29 Aug 2012

Abstract

We investigate the asymptotic behavior of solutions for the heat equation in the weighted space π‘ŒπœŽ0(ℝ𝑁)≑{πœ‘βˆˆπΆ(ℝ𝑁)∢lim|π‘₯|β†’βˆž(1+|π‘₯|2)βˆ’πœŽ/2πœ‘(π‘₯)=0}. Exactly, we find that the unbounded function space π‘ŒπœŽ0(ℝ𝑁) 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 πœ•π‘’πœ•π‘‘βˆ’Ξ”π‘’=0,(π‘₯,𝑑)βˆˆβ„π‘Γ—(0,∞),𝑒(π‘₯,0)=𝑒0(π‘₯),π‘₯βˆˆβ„π‘,(1.1) where 𝑁β‰₯1 and the initial value 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁).

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 𝐿𝑝(ℝ𝑁) 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 𝑒0∈𝐿1(ℝ𝑁), it is well-known that the solutions 𝑒(π‘₯,𝑑) converge as π‘‘β†’βˆž toward a multiple of the fundamental solution, the one which has the same integral, 𝑒(π‘₯,𝑑)=𝑆(𝑑)𝑒0(π‘₯)=𝐺(𝑑)βˆ—π‘’0(π‘₯)βŸΆπ‘€πΊ(π‘₯,𝑑),(1.2) where 𝐺(π‘₯,𝑑)=(4πœ‹π‘‘)βˆ’π‘/2exp(βˆ’|π‘₯|2/4𝑑) and βˆ«π‘€=ℝ𝑁𝑒0(π‘₯)dπ‘₯, see [10, 11].

It was first found in 2002 [12] by VΓ‘zquez and Zuazua that the bounded function space 𝐿∞(ℝ𝑁) 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 {𝑔𝑗,𝑗=1,2,…} in 𝐿∞(ℝ𝑁), there exists an initial value 𝑒0∈𝐿∞(ℝ𝑁) and a sequence π‘‘π‘—π‘˜β†’βˆž as π‘˜β†’βˆž such that 𝑒𝑑𝑗1/2π‘˜π‘₯,π‘‘π‘—π‘˜ξ‚ξ€·π‘‘=π‘†π‘—π‘˜ξ€Έπ‘’0ξ€·π‘‘π‘—π‘˜π‘₯ξ€ΈβŸΆπ‘†(1)𝑔𝑗(π‘₯)(1.3) uniformly on any compact subset of ℝ𝑁 as π‘˜β†’βˆž. Subsequently, Cazenave et al. showed that, in the bounded continuous function space 𝐢0(ℝ𝑁), 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 𝐢0(ℝ𝑁) and 𝐿∞(ℝ𝑁) (see, e.g., [3, 7, 9, 12, 16] and the references therein).

In this paper we find that, even in the unbounded function space π‘ŒπœŽ0(ℝ𝑁) 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 πΏπ‘πœŽβ€“πΏβˆžπœŽ smoothing effect and other estimates for the solutions of the problem (1.1) when the initial value 𝑒0βˆˆπΏπ‘πœŽ(ℝ𝑁)≑{πœ‘βˆΆ(1+|β‹…|2)βˆ’πœŽ/2πœ‘(β‹…)βˆˆπΏπ‘(ℝ𝑁)} with 1<π‘β‰€βˆž or 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁).

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 𝑒0 belongs to some weighted spaces. For these purposes, we first introduce the mild solutions 𝑒(π‘₯,𝑑) of the problem (1.1) which are defined as 𝑒(π‘₯,𝑑)=𝑆(𝑑)𝑒0(π‘₯)=(4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘ξƒ©βˆ’||||expπ‘₯βˆ’π‘¦2ξƒͺ𝑒4𝑑0(𝑦)d𝑦.(2.1) Letting 𝜎β‰₯0 and 1β‰€π‘β‰€βˆž, we define two weighted spaces π‘ŒπœŽ0(ℝ𝑁) and πΏπ‘πœŽ(ℝ𝑁) as follows: π‘ŒπœŽ0ξ€·β„π‘ξ€Έβ‰‘ξ‚»ξ€·β„πœ‘(π‘₯)βˆˆπΆπ‘ξ€ΈβˆΆlim|π‘₯|β†’βˆžξ€·πœ‘(π‘₯)1+|π‘₯|2ξ€Έβˆ’πœŽ/2ξ‚Ό,𝐿=0π‘πœŽξ€·β„π‘ξ€Έβ‰‘ξ‚†ξ€·πœ‘βˆΆπœ‘(β‹…)1+|β‹…|2ξ€Έβˆ’πœŽ/2βˆˆπΏπ‘ξ€·β„π‘ξ€Έξ‚‡.(2.2) Endowed with the obvious norms, β€–πœ‘β€–π‘ŒπœŽ0(ℝ𝑁)=β€–β€–ξ€·πœ‘(β‹…)1+|β‹…|2ξ€Έβˆ’πœŽ/2β€–β€–πΏβˆž(ℝ𝑁),β€–πœ‘β€–πΏπ‘πœŽ(ℝ𝑁)=β€–β€–ξ€·πœ‘(β‹…)1+|β‹…|2ξ€Έβˆ’πœŽ/2‖‖𝐿𝑝(ℝ𝑁),(2.3) the spaces π‘ŒπœŽ0(ℝ𝑁) and πΏπ‘πœŽ(ℝ𝑁) are both Banach spaces. Notice that if 𝜎=0, then π‘Œ00ℝ𝑁=𝐢0ℝ𝑁,𝐿∞0ℝ𝑁=πΏβˆžξ€·β„π‘ξ€Έ,𝐿𝑝0ℝ𝑁=𝐿𝑝ℝ𝑁.(2.4) Next we give the definition of the πœ”-limit set πœ”πœŽπœ‡,𝛽(𝑒0) which is our main study object in this paper.

Definition 2.1. Let 𝜎β‰₯0, πœ‡,  𝛽>0, and suppose that 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁). The πœ”-limit set πœ”πœŽπœ‡,𝛽(𝑒0) is given by πœ”πœŽπœ‡,𝛽𝑒0ξ€Έβ‰‘ξ‚»π‘“βˆˆπ‘ŒπœŽ0ℝ𝑁;βˆƒπ‘‘π‘›βŸΆβˆžs.t.π·βˆšπœ‡,𝛽𝑑𝑛𝑆𝑑𝑛𝑒0ξ€»π‘›β†’βˆžβˆ’βˆ’βˆ’βˆ’β†’π‘“inπ‘ŒπœŽ0ℝ𝑁.(2.5)
Here π·πœ†πœ‡,π›½πœ‘(π‘₯)β‰‘πœ†πœ‡πœ‘(πœ†2𝛽π‘₯) for πœ‘βˆˆπΏ1loc(ℝ𝑁) and πœ†>0.
In the rest of this section, we will consider the properties of the solutions for the problem (1.1) when the initial value 𝑒0βˆˆπΏπ‘πœŽ(ℝ𝑁) or 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁).
The following theorem can be seen as some extension of the maximum principle for the problem (1.1).

Theorem 2.2. Let 0β‰€πœŽ<∞. Suppose that 𝑒0βˆˆπΏβˆžπœŽξ€·β„π‘ξ€Έ(2.6) and that 𝑒(π‘₯,𝑑)=𝑆(𝑑)𝑒0(π‘₯) are the mild solutions of the problem (1.1). Then 𝑒(𝑑)=𝑆(𝑑)𝑒0βˆˆπΏβˆžπœŽξ€·β„π‘ξ€Έfor𝑑>0.(2.7) Moreover, if 𝑑>1, then ‖‖𝑆(𝑑)𝑒0β€–β€–πΏβˆžπœŽ(ℝ𝑁)β‰€πΆπ‘‘πœŽβ€–β€–π‘’0β€–β€–πΏβˆžπœŽ(ℝ𝑁),(2.8) or if 0<𝑑≀1, then ‖‖𝑆(𝑑)𝑒0β€–β€–πΏβˆžπœŽ(ℝ𝑁)‖‖𝑒≀𝐢0β€–β€–πΏβˆžπœŽ(ℝ𝑁).(2.9)

Remark 2.3. Let 𝜎=0. From Theorem 2.2, we can obtain the well-known result (maximum principle) that if 𝑒0∈𝐿∞0(ℝ𝑁)=𝐿∞(ℝ𝑁), then ‖‖𝑆(𝑑)𝑒0β€–β€–πΏβˆž(ℝ𝑁)‖‖𝑒≀𝐢0β€–β€–πΏβˆž(ℝ𝑁).(2.10)

Proof. To prove this theorem, we need the fact that if ξ€·πœ‘(π‘₯)=𝑀1+|π‘₯|2ξ€ΈπœŽ/2forsome𝑀>0,(2.11) then there exists a constant 𝐢 such that 𝑆(𝑑)πœ‘(π‘₯)≀𝐢1+𝑑+|π‘₯|2ξ€ΈπœŽ/2,(2.12) which proof can be found in [17]; we give the proof here for completeness. Consider the following problem: πœ•π‘£πœ•π‘‘βˆ’Ξ”π‘£=0,inℝ𝑁×(0,∞),𝑣(π‘₯,0)=𝑣0(π‘₯)=𝑀|π‘₯|𝜎,inℝ𝑁.(2.13) For πœ†>0, from (2.1), we can get that π·πœ†βˆ’πœŽ/2,1/2𝑆(πœ†π‘‘)𝑣0(π‘₯)=πœ†βˆ’πœŽ/2𝑆(πœ†π‘‘)𝑣0πœ†ξ€»ξ€·1/2π‘₯𝐷=𝑆(𝑑)πœ†βˆ’πœŽ/2,1/2𝑣0ξ€»(π‘₯)=𝑆(𝑑)𝑣0(π‘₯).(2.14) By the existence and the regularity theories of the solutions, we can obtain that, for 𝑑>0, 0<𝑆(𝑑)𝑣0βˆˆπΆβˆžξ€·(0,∞)×ℝ𝑁,(2.15) see [10, 18]. Now taking 𝑑=1, πœ†=𝑠 and 𝑔(π‘₯)=𝑆(1)𝑣0(π‘₯) in the expression (2.14), we have 𝑆(𝑠)𝑣0(π‘₯)=π‘ πœŽ/2π‘”ξ€·π‘ βˆ’1/2π‘₯ξ€Έ.(2.16) The fact that 𝑆(𝑠)𝑣0(π‘₯)∈𝐢([0,∞)×ℝ𝑁⧡(0,0)) clearly implies that, for |π‘₯|=1, π‘ πœŽ/2π‘”ξ€·π‘ βˆ’1/2π‘₯ξ€Έ=𝑆(𝑠)𝑣0(π‘₯)βŸΆπ‘£0(π‘₯)=𝑀|π‘₯|𝜎=𝑀(2.17) as 𝑠→0. Let 𝑦=π‘ βˆ’1/2π‘₯.(2.18) So, ||𝑦||⟢∞asπ‘ βŸΆ0.(2.19) Therefore, ||𝑦||βˆ’πœŽπ‘”(𝑦)βˆ’π‘€βŸΆ0(2.20) as |𝑦|β†’βˆž. So, there exists constant 0<𝐢<∞ such that ξ€·0≀𝑔(π‘₯)≀𝐢1+|π‘₯|2ξ€ΈπœŽ/2.(2.21) By (2.16), we thus have 𝑆(𝑠)𝑣0ξ€·(π‘₯)≀𝐢𝑠+|π‘₯|2ξ€ΈπœŽ/2.(2.22) Notice that ξ€·0β‰€πœ‘(π‘₯)≀𝐢1+𝑣0ξ€Έ(π‘₯).(2.23) Therefore, by comparison principle and (2.22), we can get that, for all 𝑑β‰₯0, there exists constant 𝐢>0 such that ξ€·0≀𝑆(𝑑)πœ‘(π‘₯)≀𝐢1+𝑑+|π‘₯|2ξ€ΈπœŽ/2.(2.24) So we complete the proof of (2.12). For any 𝑑>0, from (2.1) and (2.12), we thus obtain that ||𝑒||=||𝑆(π‘₯,𝑑)(𝑑)𝑒0||=||||(π‘₯)(4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)𝑒0||||=||||((𝑦)d𝑦4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)ξ‚€||𝑦||1+2ξ‚πœŽ/2𝑒0(ξ‚€||𝑦||𝑦)1+2ξ‚βˆ’πœŽ/2||||≀‖‖𝑒d𝑦0β€–β€–π‘ŒπœŽβˆž(ℝ𝑁)(4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)ξ‚€||𝑦||1+2ξ‚πœŽ/2ξ€·d𝑦≀𝐢1+𝑑+|π‘₯|2ξ€ΈπœŽ/2‖‖𝑒0β€–β€–π‘ŒπœŽβˆž(ℝ𝑁).(2.25) Therefore, if 𝑑>1, then ||||𝑒(π‘₯,𝑑)≀𝐢1+|π‘₯|2ξ€ΈπœŽ/2π‘‘πœŽβ€–β€–π‘’0β€–β€–π‘ŒπœŽβˆž(ℝ𝑁).(2.26) This clearly illustrates (2.8). If 0<𝑑≀1, then ||||𝑒(π‘₯,𝑑)≀𝐢1+|π‘₯|2ξ€ΈπœŽ/2‖‖𝑒0β€–β€–π‘ŒπœŽβˆž(ℝ𝑁).(2.27) From this, we can get (2.9). So we complete the proof of this theorem.

Theorem 2.4 (πΏπ‘πœŽβ€“πΏβˆžπœŽ smoothing effect). Let 1<𝑝<∞ and 0β‰€πœŽ<∞. Suppose 𝑒0βˆˆπΏπ‘πœŽ(ℝ𝑁) and that 𝑒(π‘₯,𝑑)=𝑆(𝑑)𝑒0(π‘₯) are the solutions of the problem (1.1). Then 𝑒(𝑑)=𝑆(𝑑)𝑒0βˆˆπΏβˆžπœŽξ€·β„π‘ξ€Έfor𝑑>0.(2.28) Moreover, if 𝑑>1, then ‖‖𝑆(𝑑)𝑒0β€–β€–πΏβˆžπœŽ(ℝ𝑁)β‰€πΆπ‘‘πœŽβˆ’π‘/2𝑝‖‖𝑒0β€–β€–πΏπ‘πœŽ(ℝ𝑁),(2.29) or if 0<𝑑≀1, then ‖‖𝑆(𝑑)𝑒0β€–β€–πΏβˆžπœŽ(ℝ𝑁)β‰€πΆπ‘‘βˆ’π‘/2𝑝‖‖𝑒0β€–β€–πΏπ‘πœŽ(ℝ𝑁).(2.30)

Remark 2.5. If 𝜎=0, then Theorem 2.4 captures the result πΏπ‘β€“πΏβˆž smoothing effect for the heat equation.

Proof. For any 𝑑>0, from (2.1) and Theorem 2.2, we thus obtain that ||𝑒||=||𝑆(π‘₯,𝑑)(𝑑)𝑒0||=||||(π‘₯)(4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)𝑒0||||=||||((𝑦)d𝑦4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)ξ‚€||𝑦||1+2ξ‚πœŽ/2𝑒0(ξ‚€||𝑦||𝑦)1+2ξ‚βˆ’πœŽ/2||||≀d𝑦(4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)ξ‚€||𝑦||1+2ξ‚π‘ξ…žπœŽ/2ξ‚Ήd𝑦1/π‘ξ…žΓ—ξ‚Έ(4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)ξ‚€||𝑦||1+2ξ‚βˆ’π‘πœŽ/2𝑒𝑝0ξ‚Ή(𝑦)d𝑦1/𝑝≀𝐢1+𝑑+|π‘₯|2ξ€ΈπœŽ/2π‘‘βˆ’π‘/2𝑝‖‖𝑒0β€–β€–πΏπ‘πœŽ(ℝ𝑁).(2.31) Here 1/𝑝+1/𝑝′=1. From this, we can get that, if 𝑑β‰₯1, then ‖𝑒(𝑑)β€–πΏβˆžπœŽ(ℝ𝑁)β‰€πΆπ‘‘πœŽ/2βˆ’π‘/2𝑝‖‖𝑒0β€–β€–πΏπ‘πœŽ(ℝ𝑁),(2.32) or if 0<𝑑<1, then ‖𝑒(𝑑)β€–πΏβˆžπœŽ(ℝ𝑁)β‰€πΆπ‘‘βˆ’π‘/2𝑝‖‖𝑒0β€–β€–πΏπ‘πœŽ(ℝ𝑁).(2.33)
So we complete the proof of this theorem.

In the following theorem, we consider the property of the solutions 𝑒(π‘₯,𝑑) of (1.1) with the initial data 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁).

Theorem 2.6. Let 0β‰€πœŽ<∞. If 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁), then 𝑆(𝑑)𝑒0βˆˆπ‘ŒπœŽ0ℝ𝑁for𝑑β‰₯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 ξ€·1+|π‘₯|2ξ€Έβˆ’πœŽ/2||||=𝑒(π‘₯,𝑑)1+|π‘₯|2ξ€Έβˆ’πœŽ/2||||(4πœ‹π‘‘)βˆ’π‘/2ξ€œβ„π‘exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)𝑒0||||=ξ€·(𝑦)d𝑦1+|π‘₯|2ξ€Έβˆ’πœŽ/2||||(4πœ‹π‘‘)βˆ’π‘/2ξ€œ|𝑦|≀𝑀exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)𝑒0||||+ξ€·(𝑦)d𝑦1+|π‘₯|2ξ€Έβˆ’πœŽ/2||||(4πœ‹π‘‘)βˆ’π‘/2ξ€œ|𝑦|≀𝑀exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)𝑒0||||(𝑦)d𝑦=I1(π‘₯)+I2(π‘₯).(2.35) For any πœ€>0, from 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁), we obtain that there exists an 𝑀>0 such that if |𝑦|>𝑀, then ||||𝑒0ξ‚€||𝑦||(𝑦)1+2ξ‚βˆ’πœŽ/2||||<πœ€.(2.36) So, from (2.12), we have I2ξ€·(π‘₯)=1+|π‘₯|2ξ€Έβˆ’πœŽ/2||||(4πœ‹π‘‘)βˆ’π‘/2ξ€œ|𝑦|>𝑀exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)𝑒0ξ‚€||𝑦||(𝑦)1+2ξ‚βˆ’πœŽ/2ξ‚€||𝑦||1+2ξ‚πœŽ/2||||≀d𝑦1+|π‘₯|2ξ€Έβˆ’πœŽ/2sup||𝑦||>𝑀||||𝑒0(ξ‚€||𝑦||𝑦)1+2ξ‚βˆ’πœŽ/2||||(4πœ‹π‘‘)βˆ’π‘/2ξ€œ|𝑦|>𝑀exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)ξ‚€||𝑦||1+2ξ‚πœŽ/2ξ€·d𝑦≀𝐢1+|π‘₯|2ξ€Έβˆ’πœŽ/2ξ€·1+𝑑+|π‘₯|2ξ€ΈπœŽ/2πœ€.(2.37) So, if 𝑑>1, then I2ξ€·(π‘₯)≀𝐢1+|π‘₯|2ξ€Έβˆ’πœŽ/2ξ€·1+𝑑+|π‘₯|2ξ€ΈπœŽ/2πœ€β‰€πΆπ‘‘πœŽ/2πœ€,(2.38) or if 0≀𝑑≀1, then I2ξ€·(π‘₯)≀𝐢1+|π‘₯|2ξ€Έβˆ’πœŽ/2ξ€·1+𝑑+|π‘₯|2ξ€ΈπœŽ/2πœ€β‰€πΆπœ€.(2.39) Notice also that 𝑒0βˆˆπ‘ŒπœŽ0ξ€·β„π‘ξ€Έξ€·β„βŠ‚πΆπ‘ξ€Έ.(2.40) So, there exists a constant 𝐢 such that sup||𝑦||≀𝑀||𝑒0||(𝑦)≀𝐢.(2.41) This means that I1ξ€·(π‘₯)=1+|π‘₯|2ξ€Έβˆ’πœŽ/2||||(4πœ‹π‘‘)βˆ’π‘/2ξ€œ|𝑦|≀𝑀exp(βˆ’|π‘₯βˆ’π‘¦|2/4𝑑)𝑒0||||ξ€·(𝑦)d𝑦≀𝐢1+|π‘₯|2ξ€Έβˆ’πœŽ/2.(2.42) So, there exists an 𝑀1>0 such that if |π‘₯|>𝑀1, then I1(π‘₯)<πœ€.(2.43) Combining this with (2.35), (2.38), and (2.39), we can obtain that, for 𝑑β‰₯0, 𝑒(𝑑)=𝑆(𝑑)𝑒0βˆˆπ‘ŒπœŽ0ℝ𝑁.(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 π‘ŒπœŽ0(ℝ𝑁) with 0β‰€πœŽ<𝑁 can provide a setting where complexity occurs in the asymptotic behavior of solutions.

Theorem 3.1. Let πœ‡>0, 𝜎β‰₯0, 𝑝>1, and 𝛽>1/2. If π‘πœ‡+2π›½πœŽ<𝑝,π‘“βˆˆπ‘ŒπœŽ0ξ€·β„π‘ξ€Έξ™πΏπ‘πœŽξ€·β„π‘ξ€Έ,(3.1) then there exists an initial value 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁) and a sequence π‘‘π‘›β†’βˆž as π‘›β†’βˆž such that π·βˆšπœ‡,𝛽𝑑𝑛𝑆𝑑𝑛𝑒0π‘›β†’βˆžβˆ’βˆ’βˆ’βˆ’β†’π‘“inπ‘ŒπœŽ0ℝ𝑁.(3.2) That is, π‘“βˆˆπœ”πœŽπœ‡,𝛽(𝑒0).

Proof. Suppose that π‘Ž>2 is a constant. Then let πœ†1πœ†=π‘Ž,𝑗𝑗=max2𝑝/(π‘βˆ’π‘(πœ‡+2π›½πœŽ))πœ†(2π›½π‘βˆ’π‘(πœ‡+2π›½πœŽ))/(π‘βˆ’π‘(πœ‡+2π›½πœŽ))π‘—βˆ’1,ξ€·2π‘—πœ†π‘—βˆ’1ξ€Έ1/πœ‡ξ‚for𝑗>1.(3.3) Now we define the initial-value 𝑒0 as 𝑒0(π‘₯)=βˆžξ“π‘—=1πœ†π‘—βˆ’πœ‡π‘“ξƒ©π‘₯πœ†π‘—2𝛽ξƒͺ=βˆžξ“π‘—=1π·πœ†πœ‡,π›½π‘—βˆ’1𝑓(π‘₯).(3.4) Let ξ‚€β€–β„“=maxπ‘“β€–π‘ŒπœŽ0(ℝ𝑁),β€–π‘“β€–πΏπ‘πœŽ(ℝ𝑁).(3.5) So, ‖‖𝑒0β€–β€–π‘ŒπœŽ0(ℝ𝑁)β‰€βˆžξ“π‘—=1πœ†π‘—βˆ’πœ‡β€–β€–β€–β€–π‘“ξƒ©1πœ†π‘—2𝛽⋅ξƒͺβ€–β€–β€–β€–π‘ŒπœŽ0(ℝ𝑁)β‰€βˆžξ“π‘—=12βˆ’π‘—β€–π‘“(π‘₯)β€–π‘ŒπœŽ0(ℝ𝑁)≀ℓ.(3.6) Here we have used the fact that if 0<πœ†β‰€1, then ‖𝑓(πœ†β‹…)β€–π‘ŒπœŽ0(ℝ𝑁)=supπ‘₯βˆˆβ„π‘|||ξ€·1+π‘₯2ξ€Έβˆ’πœŽ/2|||𝑓(πœ†π‘₯)=supπ‘₯βˆˆβ„π‘|||||ξ‚€||||𝑓(πœ†π‘₯)1+πœ†π‘₯2ξ‚βˆ’πœŽ/2||||1+πœ†π‘₯21+|π‘₯|2ξƒͺ𝜎/2|||||≀supπ‘₯βˆˆβ„π‘||||ξ‚€||||𝑓(πœ†π‘₯)1+πœ†π‘₯2ξ‚βˆ’πœŽ/2||||=β€–π‘“β€–π‘ŒπœŽ0(ℝ𝑁).(3.7) Therefore, the sequence of (3.4) is convergent in π‘ŒπœŽ0(ℝ𝑁). This means that 𝑒0βˆˆπ‘ŒπœŽ0ℝ𝑁.(3.8) As a result of (2.1), we see that, for 0<𝑑<𝑇<∞,π·πœ†πœ‡,π›½π‘›ξ€Ίπ‘†ξ€·πœ†2𝑛𝑑𝑒0ξ€»ξ‚€(π‘₯)=π‘†π‘‘πœ†π‘›2βˆ’4𝛽𝑒𝑛+𝑣𝑛+𝑀𝑛=π‘†π‘‘πœ†π‘›2βˆ’4𝛽𝑒𝑛+π‘†π‘‘πœ†π‘›2βˆ’4𝛽𝑣𝑛+π‘†π‘‘πœ†π‘›2βˆ’4𝛽𝑀𝑛,(3.9) where 𝑒𝑛(π‘₯)=π‘›βˆ’1𝑗=1π·πœ†πœ‡,π›½π‘›ξ‚Έπ·πœ†πœ‡,π›½π‘—βˆ’1𝑓(π‘₯)=πœ†πœ‡π‘›π‘›βˆ’1𝑗=1πœ†π‘—βˆ’πœ‡π‘“ξƒ©π‘₯πœ†π‘›2π›½πœ†π‘—2𝛽ξƒͺ,𝑣𝑛(π‘₯)=π·πœ†πœ‡,π›½π‘›ξ‚ƒπ·πœ†πœ‡,π›½π‘›βˆ’1𝑀𝑓(π‘₯)=𝑓(π‘₯),𝑛=βˆžξ“π‘—=𝑛+1π·πœ†πœ‡,π›½π‘›ξ‚Έπ·πœ†πœ‡,π›½π‘—βˆ’1𝑓(π‘₯)=πœ†πœ‡π‘›βˆžξ“π‘—=𝑛+1πœ†π‘—βˆ’πœ‡π‘“ξƒ©π‘₯πœ†π‘›2π›½πœ†π‘—2𝛽ξƒͺ.(3.10) Notice that, if πœ†β‰₯1, then ‖𝑓(πœ†β‹…)β€–π‘ŒπœŽπ‘(ℝ𝑁)=ξ‚΅ξ€œβ„π‘|||ξ€·1+|π‘₯|2ξ€Έβˆ’πœŽ/2|||𝑓(πœ†π‘₯)𝑝dπ‘₯1/π‘β‰€βŽ›βŽœβŽœβŽξ€œβ„π‘|||||ξ‚€||||𝑓(πœ†π‘₯)1+πœ†π‘₯2ξ‚βˆ’πœŽ/2||||1+πœ†π‘₯21+|π‘₯|2ξƒͺ𝜎/2|||||π‘βŽžβŽŸβŽŸβŽ dπ‘₯1/π‘β‰€πœ†πœŽξƒ©ξ€œβ„π‘||||ξ‚€||||𝑓(πœ†π‘₯)1+πœ†π‘₯2ξ‚βˆ’πœŽ/2||||𝑝ξƒͺdπ‘₯1/𝑝=πœ†πœŽβˆ’π‘/π‘β€–π‘“β€–π‘ŒπœŽπ‘(ℝ𝑁),(3.11) we thus obtain that β€–β€–π‘’π‘›β€–β€–π‘ŒπœŽπ‘(ℝ𝑁)β‰€π‘›βˆ’1𝑗=1ξ‚΅πœ†π‘›πœ†π‘—ξ‚Άπœ‡βˆ’2𝛽𝑁/𝑝+2π›½πœŽβ€–π‘“β€–π‘ŒπœŽπ‘(ℝ𝑁)ξ‚΅πœ†β‰€π‘›π‘›πœ†π‘›βˆ’1ξ‚Άπœ‡βˆ’2𝛽𝑁/𝑝+2π›½πœŽβ€–π‘“β€–π‘ŒπœŽπ‘(ℝ𝑁).(3.12) Consequently, for any 𝑑>0, we can select 𝑁 large enough to satisfy that if 𝑛>𝑁, then 0<πœ†π‘›2βˆ’4𝛽𝑑<1.(3.13) Here we have used the hypothesis 𝛽>1/2. So, (2.30), (3.11), and (3.12) indicate β€–β€–π‘†ξ‚€πœ†π‘›2βˆ’4𝛽𝑑𝑒𝑛‖‖(π‘₯)π‘ŒπœŽ0(ℝ𝑁)ξ‚€πœ†β‰€πΆπ‘›2βˆ’4π›½π‘‘ξ‚βˆ’π‘/2𝑝‖‖𝑒𝑛‖‖(π‘₯)πΏπ‘πœŽ(ℝ𝑁)β‰€πΆπœ†2𝛽𝑁/π‘βˆ’2π›½πœŽβˆ’πœ‡π‘›βˆ’1π‘‘βˆ’2𝑁/𝑝(𝑛ℓ)πœ†π‘›πœ‡βˆ’π‘/𝑝+2π›½πœŽ.(3.14) From (3.3), we thus obtain that β€–β€–π‘†ξ‚€πœ†π‘›2βˆ’4𝛽𝑑𝑒𝑛‖‖(π‘₯)π‘ŒπœŽ0(ℝ𝑁)β‰€πΆπ‘‘βˆ’2𝑁/π‘β„“π‘›βˆ’1⟢0(3.15) as π‘›β†’βˆž. From (3.7) and the definition of 𝑀𝑛(π‘₯), we have β€–β€–π‘€π‘›β€–β€–π‘ŒπœŽ0(ℝ𝑁)β‰€β€–β€–β€–β€–βˆžξ“π‘—=𝑛+1ξ‚΅πœ†π‘›πœ†π‘—ξ‚Άπœ‡π‘“ξƒ©ξ‚΅πœ†π‘›πœ†π‘—ξ‚Ά2𝛽⋅ξƒͺβ€–β€–β€–β€–π‘ŒπœŽ0(ℝ𝑁)β‰€βˆžξ“π‘—=𝑛+1ξ‚΅πœ†π‘›πœ†π‘—ξ‚Άπœ‡β„“.(3.16)
Applying (2.9) to 𝑆(π‘‘πœ†π‘›2βˆ’4𝛽)𝑀𝑛, we thus have β€–β€–π‘†ξ‚€π‘‘πœ†π‘›2βˆ’4π›½ξ‚π‘€π‘›β€–β€–π‘ŒπœŽ0(ℝ𝑁)β‰€πΆπœ†πœ‡π‘›βˆžξ“π‘—=𝑛+1πœ†π‘—+1βˆ’πœ‡β‰€πΆβ„“βˆžξ“π‘—=𝑛+12βˆ’π‘—βŸΆ0(3.17) as π‘›β†’βˆž. Here we have used (3.3) and the fact that 0<πœ†π‘›2βˆ’2𝛽𝑑<1 for 𝑛>𝑁. At present, we want to verify the claim that, for 0≀𝑑<∞, ξ‚€π‘†ξ‚€πœ†π‘›2βˆ’4π›½π‘‘ξ‚π‘£π‘›ξ‚βˆ’π‘“π‘ŒπœŽ0(ℝ𝑁)π‘›β†’βˆžβˆ’βˆ’βˆ’βˆ’β†’0.(3.18) In fact, if 0<𝑑<∞, then β€–β€–π‘†ξ‚€πœ†π‘›2βˆ’4π›½π‘‘ξ‚π‘£π‘›β€–β€–βˆ’π‘“π‘ŒπœŽ0(ℝ𝑁)≀sup|π‘₯|≀𝑀1+|π‘₯|2ξ€Έβˆ’πœŽ/2|||π‘†ξ‚€πœ†π‘›2βˆ’4𝛽𝑑𝑣𝑛|||(π‘₯)βˆ’π‘“(π‘₯)+sup|π‘₯|>𝑀1+|π‘₯|2ξ€Έβˆ’πœŽ/2|||π‘†ξ‚€πœ†π‘›2βˆ’4𝛽𝑑𝑣𝑛(|||π‘₯)+sup|π‘₯|>𝑀1+|π‘₯|2ξ€Έβˆ’πœŽ/2||||.𝑓(π‘₯)(3.19) Notice that 𝑣𝑛=π‘“βˆˆπ‘ŒπœŽ0ℝ𝑁.(3.20) Therefore, from Theorem 2.6, we have π‘†ξ‚€πœ†π‘›2βˆ’4π›½π‘‘ξ‚π‘£π‘›βˆˆπ‘ŒπœŽ0ℝ𝑁.(3.21) So, for any πœ€>0, from (2.35), (2.39), and (2.43), we get that if 𝑛>𝑁, there exists a constant 𝑀 independing on 𝑛 and 𝑑 such that sup|π‘₯|>𝑀1+|π‘₯|2ξ€Έβˆ’πœŽ/2|||π‘†ξ‚€πœ†π‘›2βˆ’4𝛽𝑑𝑣𝑛|||<πœ€(π‘₯)3,sup|π‘₯|>𝑀1+|π‘₯|2ξ€Έβˆ’πœŽ/2||𝑓||<πœ€(π‘₯)3.(3.22) The fact 𝑣𝑛=π‘“βˆˆπΆ(ℝ𝑁) means that 𝑆(𝑑)𝑣𝑛[∈𝐢0,∞)×ℝ𝑁.(3.23) Therefore, for any πœ€>0, there exists an 𝑁1>0 such that if 𝑛>𝑁1, then sup|π‘₯|≀𝑀1+|π‘₯|2ξ€Έβˆ’πœŽ/2|||π‘†ξ‚€πœ†π‘›2βˆ’4𝛽𝑑𝑣𝑛|||<πœ€(π‘₯)βˆ’π‘“(π‘₯)3.(3.24)
So, from (3.19), (3.22), and (3.24), we thus obtain that β€–β€–π‘†ξ‚€πœ†π‘›2βˆ’4π›½π‘‘ξ‚π‘£π‘›β€–β€–βˆ’π‘“π‘ŒπœŽ0(ℝ𝑁)⟢0(3.25) as π‘›β†’βˆž. From (3.9), (3.15), (3.17), and (3.18), we obtain that for any fixed 0<𝑑<∞, π·πœ†πœ‡,π›½π‘›π‘†ξ€·πœ†2𝑛𝑑𝑒0ξ‚€πœ†=𝑆𝑛2βˆ’2𝛽𝑑𝑒𝑛+𝑣𝑛+π‘€π‘›ξ€Έπ‘›β†’βˆžβˆ’βˆ’βˆ’βˆ’β†’π‘“inπ‘ŒπœŽ0ℝ𝑁.(3.26) Equation (3.2) follows from (3.26) by setting 𝑑=1 and 𝑑𝑛=πœ†2𝑛. The proof of this theorem is complete.

Theorem 3.2. Let πœ‡>0, 𝜎β‰₯0, and 𝛽>1/2. If 0<πœ‡+2π›½πœŽ<𝑁,(3.27) then there exists an initial value 𝑒0βˆˆπ‘ŒπœŽ0(ℝ𝑁) such that πœ”πœŽπœ‡,𝛽𝑒0ξ€Έ=π‘ŒπœŽ0ℝ𝑁.(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 π‘πœ‡+2π›½πœŽ<𝑝.(3.29) Therefore, there exists a countable dense subset 𝐹 of π‘ŒπœŽ0(ℝ𝑁) such that πΉβŠ‚π‘ŒπœŽ0ξ€·β„π‘ξ€Έξ™πΏπ‘πœŽξ€·β„π‘ξ€Έ.(3.30) So, there exists a sequence {πœ‘π‘—}𝑗β‰₯1βŠ‚πΉ such that (i)for any πœ™βˆˆπΉ, there exists a subsequence {πœ‘π‘—π‘˜}π‘˜β‰₯1 of the sequence {πœ‘π‘—}𝑗β‰₯1 satisfying πœ‘π‘—π‘˜(π‘₯)=πœ™βˆ€π‘˜β‰₯1,(3.31)(ii)there exists a constant 𝐢>0 satisfying ξ‚€β€–β€–πœ‘maxπ‘—β€–β€–π‘ŒπœŽ0(ℝ𝑁),β€–β€–πœ‘π‘—β€–β€–πΏπ‘πœŽ(ℝ𝑁)≀𝐢𝑗for𝑗β‰₯1.(3.32)Now we select a constant π‘Ž>2(3.33) and then take πœ†π‘—=ξƒ―ξ‚€π‘—π‘Žif𝑗=1,max4𝑝/(π‘βˆ’π‘(πœ‡+2π›½πœŽ))πœ†(2π›½π‘βˆ’π‘(πœ‡+2π›½πœŽ))/(π‘βˆ’π‘(πœ‡+2π›½πœŽ))π‘—βˆ’1,ξ€·2π‘—πœ†π‘—βˆ’1𝑗1/πœ‡ξ‚if𝑗>1.(3.34) The initial-value 𝑒0 is given by 𝑒0(π‘₯)=βˆžξ“π‘—=1πœ†π‘—βˆ’πœ‡πœ‘π‘—ξƒ©π‘₯πœ†π‘—2𝛽ξƒͺ=βˆžξ“π‘—=1π·πœ†πœ‡,π›½π‘—βˆ’1πœ‘π‘—(π‘₯).(3.35)
From (3.7), (3.32), and (3.34), we have ‖‖𝑒0β€–β€–π‘ŒπœŽ0(ℝ𝑁)β‰€βˆžξ“π‘—=1πœ†π‘—βˆ’πœ‡β€–β€–πœ‘π‘—β€–β€–π‘ŒπœŽ0(ℝ𝑁)β‰€πΆβˆžξ“π‘—=12βˆ’π‘—β‰€πΆ.(3.36)
This means that the sequence (3.35) is convergent in π‘ŒπœŽ0(ℝ𝑁). So, 𝑒0βˆˆπ‘ŒπœŽ0ℝ𝑁.(3.37) Similar to the proof of Theorem 2.2, we can prove that, for any πœ™βˆˆπΉ and 0<𝑑<∞, there exists a sequence πœ†π‘—π‘˜β†’βˆž as π‘˜β†’βˆž such that π·πœ†πœ‡,π›½π‘—π‘˜π‘†ξ‚€πœ†2π‘—π‘˜π‘‘ξ‚π‘’0β†’πœ™inπ‘ŒπœŽ0ℝ𝑁(3.38) as π‘˜β†’βˆž. Let π‘‘π‘›π‘˜=πœ†π‘›1/2π‘˜ and 𝑑=1 in (2.1); we thus have π·βˆšπœ‡,π›½π‘‘π‘—π‘˜π‘†ξ€·π‘‘π‘—π‘˜ξ€Έπ‘’0βŸΆπœ™inπ‘ŒπœŽ0ℝ𝑁.(3.39) Therefore, πΉβŠ‚πœ”πœŽπœ‡,𝛽𝑒0ξ€ΈβŠ‚π‘ŒπœŽ0ℝ𝑁.(3.40) Notice that 𝐹 is dense in π‘ŒπœŽ0(ℝ𝑁) and that πœ”πœŽπœ‡,𝛽(𝑒0) is a closed subset of π‘ŒπœŽ0(ℝ𝑁). We thus obtain from (3.40) that πœ”πœŽπœ‡,𝛽𝑒0ξ€Έ=π‘ŒπœŽ0ℝ𝑁.(3.41) So we complete the proof of this theorem.

Remark 3.4. For any 0β‰€πœŽ<𝑁, there exist constants πœ‡>0 and 𝛽>1/2 such that 0<πœ‡+2π›½πœŽ<𝑁.(3.42)
Therefore, from Theorem 3.2, we can get that the weighted space π‘ŒπœŽ0(ℝ𝑁) 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.

References

  1. A. Gmira and L. Véron, β€œLarge time behaviour of the solutions of a semilinear parabolic equation in ℝN,” Journal of Differential Equations, vol. 53, no. 2, pp. 258–276, 1984. View at: Publisher Site | Google Scholar
  2. T.-Y. Lee and W.-M. Ni, β€œGlobal existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem,” Transactions of the American Mathematical Society, vol. 333, no. 1, pp. 365–378, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. J. L. Vázquez, β€œAsymptotic beahviour for the porous medium equation posed in the whole space,” Journal of Evolution Equations, vol. 3, no. 1, pp. 67–118, 2003. View at: Publisher Site | Google Scholar
  4. J. A. Carrillo and J. L. Vázquez, β€œAsymptotic complexity in filtration equations,” Journal of Evolution Equations, vol. 7, no. 3, pp. 471–495, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. N. Hayashi, E. I. Kaikina, and P. I. Naumkin, β€œSubcritical nonlinear heat equation,” Journal of Differential Equations, vol. 238, no. 2, pp. 366–380, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. T. Cazenave, F. Dickstein, and F. B. Weissler, β€œSign-changing stationary solutions and blowup for the nonlinear heat equation in a ball,” Mathematische Annalen, vol. 344, no. 2, pp. 431–449, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. J. Yin, L. Wang, and R. Huang, β€œComplexity of asymptotic behavior of the porous medium equation in ℝN,” Journal of Evolution Equations, vol. 11, no. 2, pp. 429–455, 2011. View at: Publisher Site | Google Scholar
  8. N. Hayashi and P. I. Naumkin, β€œAsymptotics for nonlinear heat equations,” Nonlinear Analysis, vol. 74, no. 5, pp. 1585–1595, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. J. Yin, L. Wang, and R. Huang, β€œComplexity of asymptotic behavior of solutions for the porous medium equation with absorption,” Acta Mathematica Scientia B, vol. 30, no. 6, pp. 1865–1880, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. D. V. Widder, The Heat Equation, vol. 67, Academic Press, New York, NY, USA, 1975, Pure and Applied Mathematics.
  11. T. Cazenave, F. Dickstein, M. Escobedo, and F. B. Weissler, β€œSelf-similar solutions of a nonlinear heat equation,” Journal of Mathematical Sciences, vol. 8, no. 3, pp. 501–540, 2001. View at: Google Scholar | Zentralblatt MATH
  12. J. L. Vázquez and E. Zuazua, β€œComplexity of large time behaviour of evolution equations with bounded data,” Chinese Annals of Mathematics B, vol. 23, no. 2, pp. 293–310, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. T. Cazenave, F. Dickstein, and F. B. Weissler, β€œUniversal solutions of the heat equation on ℝN,” Discrete and Continuous Dynamical Systems A, vol. 9, no. 5, pp. 1105–1132, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. T. Cazenave, F. Dickstein, and F. B. Weissler, β€œUniversal solutions of a nonlinear heat equation on ℝN,” Annali della Scuola Normale Superiore di Pisa, vol. 2, no. 1, pp. 77–117, 2003. View at: Google Scholar | Zentralblatt MATH
  15. T. Cazenave, F. Dickstein, and F. B. Weissler, β€œNonparabolic asymptotic limits of solutions of the heat equation on ℝN,” Journal of Dynamics and Differential Equations, vol. 19, no. 3, pp. 789–818, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. T. Cazenave, F. Dickstein, and F. B. Weissler, β€œChaotic behavior of solutions of the Navier-Stokes system in ℝN,” Advances in Differential Equations, vol. 10, no. 4, pp. 361–398, 2005. View at: Google Scholar
  17. L. Wang and J. Yin, β€œGrow-up rate of solutions for heat equation with sublinear source,” 2011. View at: Google Scholar
  18. Z. Wu, J. Yin, and C. Wang, Elliptic and Parabolic Equations, World Scientific Publishing, Hackensack, NJ, USA, 2006.

Copyright © 2012 Liangwei Wang and Jingxue Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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