General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

Abstract

We derive general bounds for the large time size of supnorm values ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ) of solutions to one-dimensional advection-diffusion equations 𝑢_𝑡+(𝑏(𝑥,𝑡)𝑢)_𝑥=𝑢_𝑥𝑥,𝑥∈ℝ,𝑡>0 with initial data 𝑢(⋅,0)∈𝐿^𝑝0(ℝ)∩𝐿^∞(ℝ) for some 1≤𝑝₀<∞ and arbitrary bounded advection speeds 𝑏(𝑥,𝑡), introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.

1. Introduction

In this work, we obtain very general large time estimates for supnorm values of solutions 𝑢(⋅,𝑡) to parabolic initial value problems of the form𝑢_𝑡+(𝑏(𝑥,𝑡)𝑢)_𝑥=𝑢_𝑥𝑥,𝑥∈ℝ,𝑡>0,𝑢(⋅,0)=𝑢₀∈𝐿^𝑝0(ℝ)∩𝐿^∞(ℝ),1≤𝑝₀<∞, (1a)(1b)for arbitrary continuously differentiable advection fields 𝑏∈𝐿^∞(ℝ×[0,∞[). Here, by solution to (1a) and (1b) in some time interval [0,𝑇_∗[,0<𝑇_∗≤∞, we mean a function 𝑢∶ℝ×[0,𝑇_∗[→ℝ which is bounded in each strip 𝑆_𝑇=ℝ×[0,𝑇], 0<𝑇<𝑇_∗, solves (1a) in the classical sense for 0<𝑡<𝑇_∗, and satisfies 𝑢(⋅,𝑡)→𝑢₀ in𝐿¹_loc(ℝ) as 𝑡→0. It follows from the a priori estimates given in Section 2 that all solutions of problem (1a), (1b) are actually globally defined (𝑇_∗=∞), with 𝑢(⋅,𝑡)∈𝐶⁰([0,∞[,𝐿^𝑝(ℝ)) for each 𝑝≥𝑝₀ finite. Given 𝑏∈𝐿^∞(ℝ×[0,∞[), what then can be said about the size of supnorm values ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ) for 𝑡≫1?

When 𝜕𝑏/𝜕𝑥≥0 for all 𝑥∈ℝ,𝑡≥0, it is well known that, for each 𝑝₀≤𝑝≤∞, ‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ) is monotonically decreasing in 𝑡, with ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤𝐾𝑝₀𝑢_{0𝐿𝑝0(ℝ)}𝑡^−1/2𝑝0∀𝑡>0𝑏_𝑥≥0 (2) for some constant 0<𝐾(𝑝₀)<2^−1/𝑝0 that depends only on 𝑝₀; see, for example, [1–5]. For general 𝑏(𝑥,𝑡), however, estimating ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ) is much harder. To see why, let us illustrate with the important case 𝑝₀=1, where one has ‖𝑢(⋅,𝑡)‖_𝐿¹_(ℝ)≤𝑢_0𝐿1(ℝ)∀𝑡>0, (3) as recalled in Theorem 1. Writing (1a) as 𝑢_𝑡+𝑏(𝑥,𝑡)𝑢_𝑥=𝑢_𝑥𝑥−𝑏_𝑥(𝑥,𝑡)𝑢, (4) we observe on the right hand side of (4) that |𝑢(𝑥,𝑡)| is pushed to grow at points (𝑥,𝑡) where 𝑏_𝑥(𝑥,𝑡)<0. If this condition persists long enough, large values of |𝑢(𝑥,𝑡)| might be generated, particularly at sites where −𝑏_𝑥(𝑥,𝑡)≫1. Now, because of constraint (3), any persistent growth in solution size will eventually create long thin structures as shown in Figure 1, which, in turn, tend to be effectively dissipated by viscosity. The final overall behavior that ultimately results from such competition is not immediately clear, either on physical or on mathematical grounds.

Figure 1 

Solution profiles showing typical growth in regions with 𝑏𝑥<0, where 𝑏=5cos𝑥. After reaching maximum height, solution starts decaying very slowly due to its spreading and mass conservation (decay rate is not presently known).

As shown in (4), it is not the magnitude of 𝑏(𝑥,𝑡) itself but instead its oscillation that is relevant in determining ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ). Accordingly, we introduce the quantity 𝐵(𝑡) defined by 𝐵(𝑡)=12sup_𝑥∈ℝ𝑏(𝑥,𝑡)−inf_𝑥∈ℝ𝑏(𝑥,𝑡),𝑡≥0, (5) which plays a fundamental role in the analysis. Our main result is now easily stated.

Main Theorem. For each 𝑝≥𝑝₀, one has¹limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤3√3𝑝2𝜋^1/𝑝⋅ℬ^1/𝑝⋅limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ), (6)where ℬ=limsup_𝑡→∞𝐵(𝑡).

In particular, in the important case 𝑝₀=1 considered above, we obtain, using (3), limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤3√3𝑢2𝜋⋅ℬ⋅_0𝐿1(ℝ), (7) so that 𝑢(⋅,𝑡) stays uniformly bounded for all time in this case.² Estimates similar to (6) can also be shown to hold for the 𝑛-dimensional problem 𝑢_𝑡+div(𝐛(𝑥,𝑡)𝑢)=Δ𝑢,𝑢(⋅,0)∈𝐿^𝑝ℝ^𝑛∩𝐿^∞ℝ^𝑛, (8) but to simplify our discussion we consider here the case 𝑛=1 only. Our derivation of (6), which improves some unpublished results by the third author, uses the 1D inequality ‖𝚟‖_𝐿^∞_(ℝ)≤𝐶_∞‖𝚟‖_𝐿^1/3_1(ℝ)𝚟_𝑥𝐿^2/3_2(ℝ),𝚟∈𝐿¹(ℝ)∩𝐻¹(ℝ), (9) where 𝐶_∞=(3/4)^2/3, and can be readily extended to other problems of interest like 1D systems of viscous conservation laws [6, Ch. 9] or the more general equation 𝑢_𝑡+(𝑏(𝑥,𝑡,𝑢)𝑢)_𝑥=𝑎(𝑥,𝑡,𝑢)𝑢_𝑥𝑥,𝑎(𝑥,𝑡,𝑢)≥𝜇(𝑡)>0, (10) with bounded values 𝑏(𝑥,𝑡,𝑢); provided that we assume ∫^∞𝜇(𝑡)𝑑𝑡=∞: using a similar argument, we get the estimate³ [7, Ch. 2] limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤3√3𝑝2𝜋^1/𝑝⋅ℬ_𝜇^1/𝑝⋅limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ), (11) for each 𝑝≥𝑝₀, whereℬ_𝜇=limsup_𝑡→∞𝐵(𝑡)𝜇(𝑡),𝐵(𝑡)=12sup_𝑥∈ℝ𝑏(𝑥,𝑡,𝑢(𝑥,𝑡))−inf_𝑥∈ℝ𝑏(𝑥,𝑡,𝑢(𝑥,𝑡)). (12a)(12b)More involving applications, such as problems with superlinear advection or degenerate diffusion, which require considerable extra work, will be studied in the future.

2. A Priori Estimates

This section contains some preliminary results on the solutions of problem (1a) and (1b) needed later for our derivation of estimate (6), which is completed in Section 3. (Recall that a solution on some given time interval [0,𝑇_∗[, 0<𝑇_∗≤∞, is a function𝑢(⋅,𝑡)∈𝐿^∞_loc([0,𝑇_∗[,𝐿^∞(ℝ)) which is smooth (𝐶² in 𝑥, 𝐶¹ in 𝑡) in ℝ×]0,𝑇_∗[ and solves (1a) there, verifying the initial condition in the sense of𝐿¹_loc(ℝ), i.e., ‖𝑢(⋅,𝑡)−𝑢₀‖_𝐿1(𝕂)→0 as 𝑡→0 for each compact 𝕂⊂ℝ. Local existence theory can be found in, e.g., [8, Ch. 6].) We start with a simple Gronwall-type estimate for ‖𝑢(⋅,𝑡)‖_𝐿^𝑞_(ℝ), 𝑝₀≤𝑞<∞. The corresponding result for the supnorm (𝑞=∞) is more difficult to obtain and will be given at the end of Section 2; see Theorem 4.

Theorem 1. If𝑢(⋅,𝑡)∈𝐿^∞_loc([0,𝑇_∗[,𝐿^∞(ℝ)) solves problem (1a), (1b), then 𝑢(⋅,𝑡)∈𝐶⁰([0,𝑇_∗[,𝐿^𝑞(ℝ)) for each 𝑝₀≤𝑞<∞, and ‖𝑢(⋅,𝑡)‖_𝐿^𝑞_(ℝ)≤‖𝑢(⋅,0)‖_𝐿^𝑞_(ℝ)1⋅exp2𝑞−1^𝑡₀𝐵(𝜏)²𝑑𝜏 (13) for all 0<𝑡<𝑇_∗.

Proof. The proof is standard, so we will only sketch the basic steps. Taking 𝑆∈𝐶¹(ℝ) such that 𝑆^′(v)≥0 for all v, 𝑆(0)=0,𝑆(v)=sgn(v) for|v|≥1, let (given 𝛿>0)𝐿_𝛿(u)=∫^u₀𝑆(v/𝛿)𝑑v, so that𝐿_𝛿(u)→|u| as 𝛿→0, uniformly inu. LetΦ_𝛿(u)=𝐿_𝛿(u)^𝑞. Given 𝑅>0, 0<𝜖≤1, let 𝜁_𝑅(⋅) be the cut-off function 𝜁_𝑅(𝑥)=0 for |𝑥|≥𝑅, 𝜁_𝑅(𝑥)=exp{−𝜖√1+𝑥²}−exp{−𝜖1+𝑅²} for |𝑥|<𝑅. Multiplying (1a) by Φ^′_𝛿(𝑢(𝑥,𝑡))⋅𝜁_𝑅(𝑥) if 𝑞≠2, or 𝑢(𝑥,𝑡)⋅𝜁_𝑅(𝑥) if 𝑞=2, and integrating the result on ℝ×[0,𝑡], we obtain, letting 𝛿→0 and then 𝑅→∞, since 𝑢∈𝐿^∞(ℝ×[0,𝑡]),𝚄_𝜖(𝑡)+𝑉_𝜖(𝑡)≤𝚄_𝜖(0)+^𝑡₀𝐺_𝜖(𝜏)𝚄_𝜖(𝜏)𝚄𝑑𝜏,_𝜖(𝑡)=_ℝ|𝑢(𝑥,𝑡)|^𝑞𝑤_𝜖(𝑥)𝑑𝑥, (14a) where 𝑤_𝜖(𝑥)=exp{−𝜖√1+𝑥²}, 𝐺_𝜖(𝑡)=(1/2)𝑞(𝑞−1)𝐵(𝑡)²+𝜖2𝑞⋅sup_{0≤𝜏≤𝑡}‖𝑢(⋅,𝜏)‖_𝐿^∞_(ℝ) +𝜖, and 𝑉_𝜖(𝑡)=12𝑞×𝑞−1_{𝑡0𝑢≠0}|𝑢(𝑥,𝜏)|^𝑞−2𝑢_𝑥(𝑥,𝜏)²𝑤_𝜖(𝑥)𝑑𝑥𝑑𝜏,if𝑞≠2,_𝑡0ℝ𝑢_𝑥(𝑥,𝜏)²𝑤_𝜖(𝑥)𝑑𝑥𝑑𝜏,if𝑞=2. (14b)By Gronwall’s lemma, (14a) and (14b) giveU_𝜖(𝑡)≤U_𝜖(0)⋅exp{∫^𝑡₀𝐺_𝜖(𝜏)𝑑𝜏}, from which we obtain (13) by simply letting 𝜖→0. This shows, in particular, that𝑢(⋅,𝑡)∈𝐿^∞_loc([0,𝑇_∗[,𝐿^𝑞(ℝ)) if 𝑝₀≤𝑞<∞. Now, to get 𝑢(⋅,𝑡)∈𝐶⁰([0,𝑇_∗[,𝐿^𝑞(ℝ)), it is sufficient to show that, given 𝜀>0 and 0<𝑇<𝑇_∗ arbitrary, we can find 𝑅=𝑅(𝜀,𝑇)≫1 large enough so that we have ‖𝑢(⋅,𝑡)‖_𝐿^𝑞_{(|𝑥|>𝑅)}<𝜀 for any 0≤𝑡≤𝑇. Taking 𝜓∈𝐶²(ℝ) with 0≤𝜓≤1 and 𝜓(𝑥)=0 for all 𝑥≤0, 𝜓(𝑥)=1 for all 𝑥≥1, let Ψ_𝑅,𝑀∈𝐶²(ℝ) be the cut-off function given by Ψ_𝑅,𝑀(𝑥)=0 if |𝑥|≤𝑅−1, Ψ_𝑅,𝑀(𝑥)=𝜓(|𝑥|−𝑅+1) if 𝑅−1<|𝑥|<𝑅, and Ψ_𝑅,𝑀(𝑥)=1 if 𝑅≤|𝑥|≤𝑅+𝑀, Ψ_𝑅,𝑀(𝑥)=𝜓(𝑅+𝑀+1−|𝑥|) if 𝑅+𝑀<|𝑥|<𝑅+𝑀+1, Ψ_𝑅,𝑀(𝑥)=0 if |𝑥|≥𝑅+𝑀+1, where 𝑅>1, 𝑀>0 are given. Multiplying (1a) by Φ^′_𝛿(𝑢(𝑥,𝑡))⋅Ψ_𝑅,𝑀(𝑥) if 𝑞≠2, or 𝑢(𝑥,𝑡)⋅Ψ_𝑅,𝑀(𝑥) if 𝑞=2, and integrating the result on ℝ×[0,𝑡], 0<𝑡≤𝑇, we obtain, as in (14a) and (14b), by letting 𝛿→0, 𝑀→∞, that ‖𝑢(⋅,𝑡)‖_𝐿^𝑞_{(|𝑥|>𝑅)}<𝜀/2+‖𝑢(⋅,0)‖_𝐿^𝑞_{(|𝑥|>𝑅−1)} for all 0≤𝑡≤𝑇, provided that we take 𝑅>1 sufficiently large. This gives the continuity result, and the proof is complete.

An important by-product of the proof above is that we have (letting 𝜖→0 in (14a) and (14b), and using (13)), for each 0<𝑇<𝑇_∗ and 𝑞≥max{𝑝₀,2}, ^𝑇_0ℝ|𝑢(𝑥,𝜏)|^𝑞−2𝑢_𝑥(𝑥,𝜏)²𝑑𝑥𝑑𝜏<∞. (15) Therefore, if we repeat the steps above leading to (14a) and (14b), we obtain (letting 𝛿→0, 𝑅→∞, 𝜖→0, in this order, taking (13) and (15) into account) the identity ‖𝑢(⋅,𝑡)‖^𝑞_{𝐿𝑞(ℝ)}+𝑞𝑞−1^𝑡_0ℝ|𝑢(𝑥,𝜏)|^𝑞−2𝑢_𝑥(𝑥,𝜏)²=‖𝑢(𝑑𝑥𝑑𝜏⋅,0)‖^𝑞_{𝐿𝑞(ℝ)}+𝑞𝑞−1^𝑡_0ℝ𝑏(𝑥,𝜏)−𝛽(𝜏)|𝑢(𝑥,𝜏)|^𝑞−2×𝑢(𝑥,𝜏)𝑢_𝑥(𝑥,𝜏)𝑑𝑥𝑑𝜏 (16) for every 0<𝑡<𝑇_∗ and max{𝑝₀,2}≤𝑞<∞, where 𝛽(𝑡)=12sup_𝑥∈ℝ𝑏(𝑥,𝑡)+inf_𝑥∈ℝ𝑏(𝑥,𝑡),𝑡≥0. (17) The core of the difficulty in the analysis of (1a) and (1b) is apparent here: under the sole assumption that 𝑏 is bounded, it is not much clear how one should go about the last term in (16) in order to get more than (13) above. Actually, it will be convenient to consider (16) in the (equivalent) differential form, that is, 𝑑‖𝑢(𝑑𝑡⋅,𝑡)‖^𝑞_{𝐿𝑞(ℝ)}+𝑞𝑞−1_ℝ|𝑢(𝑥,𝑡)|^𝑞−2𝑢_𝑥(𝑥,𝑡)²𝑑𝑥=𝑞𝑞−1_ℝ𝑏(𝑥,𝑡)−𝛽(𝑡)|𝑢(𝑥,𝑡)|^𝑞−2×𝑢(𝑥,𝑡)𝑢_𝑥(𝑥,𝑡)𝑑𝑥 (18) for all 𝑡∈[0,𝑇_∗[⧵𝐸_𝑞, where 𝐸_𝑞⊂[0,𝑇_∗[ has zero measure. We then readily obtain, using (9) and the one-dimensional Nash inequality [9] ‖𝚟‖_𝐿²_(ℝ)≤𝐶₂‖𝚟‖_𝐿^2/3_1(ℝ)𝚟_𝑥𝐿^1/3_2(ℝ),𝐶₂=3√34𝜋^1/3, (19) where the value given above for 𝐶₂ is optimal [10], the following result.

Theorem 2. Let 𝑞≥2𝑝₀. If 𝑡̂∈[0,𝑇_∗[⧵𝐸_𝑞 is such that (𝑑/𝑑𝑡)‖𝑢(⋅,𝑡)‖^𝑞_{𝐿𝑞(ℝ)|̂𝑡𝑡=}≥0, then𝑢⋅,𝑡̂_{𝐿𝑞(ℝ)}≤𝑞2𝐶³₂^1/𝑞𝐵𝑡̂^1/𝑞𝑢⋅,𝑡̂_{𝐿𝑞/2(ℝ)},𝑢⋅,𝑡̂_𝐿∞(ℝ)≤𝑞2𝐶₂𝐶_∞^2/𝑞𝐵𝑡̂^2/𝑞𝑢⋅,𝑡̂_{𝐿𝑞/2(ℝ)}. (20a)(20b)

Proof. Consider (20a) first. From (5), (17), and (18), we have _ℝ𝑢𝑥,𝑡̂^𝑞−2𝑢_𝑥𝑥,𝑡̂²𝑡̂𝑑𝑥≤𝐵_ℝ𝑢𝑥,𝑡̂^𝑞−1𝑢_𝑥𝑥,𝑡̂𝑑𝑥. (21) This gives _ℝ𝑢𝑥,𝑡̂^𝑞−2𝑢_𝑥𝑥,𝑡̂²𝑑𝑥≤𝐵𝑡̂²𝑢⋅,𝑡̂^𝑞_{𝐿𝑞(ℝ)}, (22) or, in terms of 𝑣̂∈𝐿¹(ℝ)∩𝐿^∞(ℝ) defined by 𝑣(̂𝑥)=|𝑢(𝑥,𝑡)̂|^𝑞/2 if 𝑞>2, 𝑣(̂𝑥)=𝑢(𝑥,𝑡)̂ if 𝑞=2, 𝑣_{𝑥𝐿2(ℝ)}≤𝑞2𝐵𝑡̂‖𝑣‖̂_𝐿²_(ℝ). (23) Using (19), we then get ‖𝑣‖̂²_𝐿2(ℝ)≤(𝑞/2)𝐶³₂𝐵(𝑡)̂‖𝑣‖̂²_𝐿1(ℝ), which is equivalent to (20a). Similarly, (20b) can be obtained, using (9).

Thus, we can use (20a) and (20b) when ‖𝑢(⋅,𝑡)‖_𝐿^𝑞_(ℝ) is not decreasing. If it is decreasing, (18) becomes useless but at least we know in such case that ‖𝑢(⋅,𝑡)‖_𝐿^𝑞_(ℝ) is not increasing, which should be useful too. Different values of 𝑞 have different scenarios, which we will have to piece together in some way. The next result shows us just how. To this end, it is convenient to introduce the quantities 𝔹(𝑡₀;𝑡) and 𝕌_𝑝(𝑡₀;𝑡) defined by 𝔹𝑡₀;𝑡=sup𝐵(𝜏)∶𝑡₀≤𝜏≤𝑡,𝕌_𝑝𝑡₀;𝑡=sup‖𝑢(⋅,𝜏)‖_𝐿^𝑝_(ℝ)∶𝑡₀≤𝜏≤𝑡, (24)(25) given 𝑝≥𝑝₀,0≤𝑡₀≤𝑡<𝑇_∗ arbitrary.

Theorem 3. Let 𝑞≥2𝑝₀. For each 0≤𝑡₀<𝑇_∗, we have 𝕌_𝑞𝑡₀𝑢;𝑡≤max⋅,𝑡_{0𝐿𝑞(ℝ)};𝑞2𝐶³₂^1/𝑞𝔹𝑡₀;𝑡^1/𝑞𝕌_𝑞/2𝑡₀;𝑡 (26) for all 𝑡₀≤𝑡<𝑇_∗.

Proof. Set 𝜆_𝑞(𝑡)=((𝑞/2)𝐶³₂)^1/𝑞𝔹(𝑡₀;𝑡)^1/𝑞𝕌_𝑞/2(𝑡₀;𝑡). There are three cases to consider.
Case I. ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ)>𝜆_𝑞(𝑡) for all 𝑡₀≤𝜏≤𝑡. By (20a), Theorem 2, we must then have (𝑑/𝑑𝜏)‖𝑢(⋅,𝜏)‖^𝑞_{𝐿𝑞(ℝ)}<0 for all 𝜏∈[𝑡₀,𝑡]⧵𝐸_𝑞, so that ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ) is monotonically decreasing in [𝑡₀,𝑡]. In particular, 𝕌_𝑞(𝑡₀;𝑡)=‖𝑢(⋅,𝑡₀)‖_{𝐿𝑞(ℝ)} in this case, and (26) holds.
Case II. ‖𝑢(⋅,𝑡₀)‖_{𝐿𝑞(ℝ)}>𝜆_𝑞(𝑡) and ‖𝑢(⋅,𝑡₁)‖_{𝐿𝑞(ℝ)}≤𝜆_𝑞(𝑡) for some 𝑡₁∈]𝑡₀,𝑡]. In this case, let 𝑡₂∈]𝑡₀,𝑡] be such that we have ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ)>𝜆_𝑞(𝑡) for all 𝑡₀≤𝜏<𝑡₂, while ‖𝑢(⋅,𝑡₂)‖_{𝐿𝑞(ℝ)}=𝜆_𝑞(𝑡). We claim that ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ)≤𝜆_𝑞(𝑡) for every 𝑡₂≤𝜏≤𝑡: in fact, if this were not true, we could then find 𝑡₃,𝑡₄ with 𝑡₂≤𝑡₃<𝑡₄≤𝑡 such that ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ)>𝜆_𝑞(𝑡) for all 𝑡₃<𝜏≤𝑡₄, ‖𝑢(⋅,𝑡₃)‖_{𝐿𝑞(ℝ)}=𝜆_𝑞(𝑡). By (20a), Theorem 2, this would require (𝑑/𝑑𝜏)‖𝑢(⋅,𝜏)‖^𝑞_{𝐿𝑞(ℝ)}<0 for all 𝜏∈]𝑡₃,𝑡₄]⧵𝐸_𝑞, so that ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ) could not increase anywhere on [𝑡₃,𝑡₄]. This contradicts ‖𝑢(⋅,𝑡₃)‖_{𝐿𝑞(ℝ)}<‖𝑢(⋅,𝑡₄)‖_{𝐿𝑞(ℝ)}, and so we have ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ)≤𝜆_𝑞(𝑡) for every 𝑡₂≤𝜏≤𝑡, as claimed. On the other hand, by (20a), ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ) has to be monotonically decreasing on [𝑡₀,𝑡₂], just as in Case I. Therefore, we have 𝕌_𝑞(𝑡₀;𝑡)=‖𝑢(⋅,𝑡₀)‖_{𝐿𝑞(ℝ)} in this case again, which shows (26).
Case III. Consider ‖𝑢(⋅,𝑡₀)‖_{𝐿𝑞(ℝ)}≤𝜆_𝑞(𝑡). This gives ‖𝑢(⋅,𝜏)‖_𝐿^𝑞_(ℝ)≤𝜆_𝑞(𝑡) for every 𝑡₀≤𝜏≤𝑡, by repeating the argument used on the interval [𝑡₂,𝑡] in Case II. It follows that we must have 𝕌_𝑞(𝑡₀;𝑡)≤𝜆_𝑞(𝑡) in this case, and the proof of Theorem 3 is complete.

An important application of Theorem 3 is the following result.

Theorem 4. Let 𝑝₀≤𝑝<∞, 0≤𝑡₀<𝑇_∗. Then ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤2𝑝^1/𝑝⋅max𝑢⋅,𝑡_{0𝐿∞(ℝ)};𝔹𝑡₀;𝑡^1/𝑝𝕌_𝑝𝑡₀;𝑡 (27) for any 𝑡₀≤𝑡<𝑇_∗, where 𝔹(𝑡₀;𝑡) and 𝕌_𝑝(𝑡₀;𝑡) are given in (24) and (25) above.

Proof. Let 𝑘∈ℤ, 𝑘≥2. Applying (26) successively with 𝑞=2𝑝,4𝑝,…,2^𝑘𝑝, we obtain‖𝑢(⋅,𝑡)‖_𝐿2^𝑘_𝑝(ℝ)≤max𝐮⋅,𝑡_0𝐿2^𝑘_𝑝(ℝ);𝐾(𝑘,ℓ)^1/𝑝⋅𝔹𝑡₀;𝑡^{(1/𝑝)(2−ℓ−2−𝑘)}⋅𝑢⋅,𝑡_0𝐿2^ℓ_𝑝(ℝ)𝐾(),1≤ℓ≤𝑘−1;𝑘,0^1/𝑝⋅𝔹𝑡₀;𝑡^{(1/𝑝)(1−2−𝑘)}⋅𝕌_𝑝𝑡₀;𝑡, (28a) where 𝐾(𝑘,ℓ)=^𝑘_𝑗=ℓ+12^𝑗−1𝑝𝐶³₂^2−𝑗,0≤ℓ≤𝑘−1. (28b)Now, for 1≤ℓ≤𝑘−1, 𝔹𝑡₀;𝑡^{(1/𝑝)(2−ℓ−2−𝑘)}⋅𝑢⋅,𝑡_0𝐿2^ℓ_𝑝(ℝ)≤𝔹𝑡₀;𝑡^{(1/𝑝)(2−ℓ−2−𝑘)}⋅𝑢(⋅,𝑡₀)^{(2−ℓ−2−𝑘)/(1−2−𝑘)}_{𝐿𝑝(ℝ)}⋅𝑢⋅,𝑡₀^{(1−2−ℓ)/(1−2−𝑘)}_{𝐿2𝑘𝑝(ℝ)}≤max𝑢⋅,𝑡_0𝐿2^𝑘_𝑝(ℝ);𝔹𝑡₀;𝑡^{(1/𝑝)(1−2−𝑘)}⋅𝑢⋅,𝑡_{0𝐿𝑝(ℝ)} (29) by Young’s inequality (see, e.g., [11, page 622]); in particular, we get, from (28a) and (28b), ‖𝑢(⋅,𝑡)‖_𝐿2^𝑘_𝑝(ℝ)≤2𝑝^1/𝑝⋅max𝑢⋅,𝑡_0𝐿2^𝑘_𝑝(ℝ);𝔹𝑡₀;𝑡^{(1/𝑝)(1−2−𝑘)}⋅𝕌_𝑝𝑡₀;𝑡, (30) since 𝐾(𝑘,ℓ)≤2𝑝 for all 0≤ℓ≤𝑘−1. Letting 𝑘→∞, (27) is obtained.

It follows from Theorems 1 and 4 that 𝑢(⋅,𝑡) is globally defined (𝑇_∗=∞). Now, from (27), we immediately obtain, letting 𝑡→∞, limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤2𝑝^1/𝑝⋅max𝑢⋅,𝑡_{0𝐿∞(ℝ)};𝔹𝑡₀^1/𝑝𝕌_𝑝𝑡₀ (31) for any 𝑡₀≥0, where 𝔹(𝑡₀) and 𝕌_𝑝(𝑡₀) are given by 𝔹𝑡₀=sup𝐵(𝑡)∶𝑡≥𝑡₀,𝕌_𝑝𝑡₀=sup‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ)∶𝑡≥𝑡₀. (32)(33) Taking (𝑡₀^(𝑛))_𝑛 such that 𝑡₀^(𝑛)→∞ and ‖𝑢(⋅,𝑡₀^(𝑛))‖_𝐿∞(ℝ)→liminf_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ), and applying (31) with 𝑡₀=𝑡₀^(𝑛) for each 𝑛, we then obtain, letting 𝑛→∞, limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤2𝑝^1/𝑝⋅maxliminf_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ);ℬ^1/𝑝⋅𝒰_𝑝, (34) where ℬ and 𝒰_𝑝 are given by ℬ=limsup_𝑡→∞𝐵(𝑡),𝒰_𝑝=limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ). (35)

3. Large Time Estimates

In this section, we use the results obtained above to derive two basic large time estimates (given in Theorems 5 and 6) for solutions 𝑢(⋅,𝑡) of problem (1a), (1b), which represent important intermediate steps that will ultimately lead to the main result stated in Theorem 7.

Theorem 5. Let 𝑞≥2𝑝₀, and ℬ≥0 be as defined in (35). Then limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑞_(ℝ),≤𝑞2𝐶³₂^1/𝑞⋅ℬ^1/𝑞⋅limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑞/2_(ℝ), (36) where 𝐶₂=(3√3/(4𝜋))^1/3 is the constant in the Nash inequality (19).

Proof. We set 𝑝=𝑞/2 and assume that 𝒰_𝑝 is finite. As in the proof of Theorem 2, we take 𝑣∈𝐿^∞(ℝ×[0,∞[) given by 𝑣(𝑥,𝑡)=|𝑢(𝑥,𝑡)|^𝑝 if 𝑝>1, 𝑣(𝑥,𝑡)=𝑢(𝑥,𝑡) if 𝑝=1. It follows that ‖𝑣(⋅,𝑡)‖²_𝐿2(ℝ)=‖𝑢(⋅,𝑡)‖_𝐿^2𝑝_2𝑝(ℝ),𝑣_𝑥(⋅,𝑡)²_𝐿2(ℝ)=𝑝²_ℝ|𝑢(𝑥,𝑡)|^2𝑝−2𝑢_𝑥(𝑥,𝑡)²𝑑𝑥. (37) Therefore, from (18), we have, for some null set 𝐸_2𝑝⊂[0,∞[, 𝑑‖𝑣(𝑑𝑡⋅,𝑡)‖²_𝐿2(ℝ)1+41−𝑣2𝑝^𝑥(⋅,𝑡)²_𝐿2(ℝ)1≤4𝑝1−𝐵(𝑡𝑣(2𝑝)‖⋅,𝑡)‖_𝐿²_(ℝ)𝑣_𝑥(⋅,𝑡)_𝐿2(ℝ) (38) for all 𝑡∈[0,∞[⧵𝐸_2𝑝, and so, by (19), 𝑑‖𝑣(𝑑𝑡⋅,𝑡)‖²_𝐿2(ℝ)1+41−𝑣2𝑝^𝑥(⋅,𝑡)²_𝐿2(ℝ)≤4𝑝𝐶₂11−𝐵(𝑡𝑣(2𝑝)‖⋅,𝑡)‖_𝐿^2/3_1(ℝ)𝑣_𝑥(⋅,𝑡)_𝐿^4/3_2(ℝ). (39) This gives, by Young’s inequality ([11, page 622]), for all 𝑡∈[0,∞[⧵𝐸_2𝑝, 𝑑‖𝑣(𝑑𝑡⋅,𝑡)‖²_𝐿2(ℝ)+4311−𝑣2𝑝^𝑥(⋅,𝑡)²_𝐿2(ℝ)≤4311−2𝑝𝑝𝐶₂³𝐵(𝑡)³‖𝑣(⋅,𝑡)‖²_𝐿1(ℝ). (40) Setting 𝜆_𝑝=limsup_𝑡→∞𝑔(𝑡),𝑔(𝑡)=𝑝𝐶³₂^1/2𝐵(𝑡)^1/2‖𝑣(⋅,𝑡)‖_𝐿¹_(ℝ), (41) we claim that limsup_𝑡→∞‖𝑣(⋅,𝑡)‖_𝐿²_(ℝ)≤𝜆_𝑝. (42) In fact, let us argue by contradiction. If (42) is false, we can pick 0<𝜂≪1 and a sequence (𝑡_𝑗)_𝑗≥0, 𝑡_𝑗→∞, such that ‖𝑣(⋅,𝑡_𝑗)‖_𝐿2(ℝ)>𝜆_𝑝+𝜂 (for all 𝑗≥0) and 𝑔(𝑡)≤𝜆_𝑝+𝜂/2 for all 𝑡≥𝑡₀. From (20a), Theorem 2, it will then follow that ‖𝑣(⋅,𝑡)‖_𝐿²_(ℝ)>𝜆_𝑝+𝜂,∀𝑡≥𝑡₀. (43) In fact, suppose that (43) were false, so that we had ‖𝑣(⋅,𝑡)̃‖_𝐿²_(ℝ)≤𝜆_𝑝+𝜂 for some 𝑡̃>𝑡₀. Taking 𝑗≫1 with 𝑡_𝑗>𝑡̃, we could then find 𝑡̂∈[𝑡,̃𝑡_𝑗[ such that ‖𝑣(⋅,𝑡)‖_𝐿²_(ℝ)>𝜆_𝑝+𝜂 for all 𝑡∈]𝑡,̂𝑡_𝑗], while ‖𝑣(⋅,𝑡)̂‖_𝐿²_(ℝ)=𝜆_𝑝+𝜂, and so there would exist 𝑡_∗∈[𝑡,̂𝑡_𝑗]⧵𝐸_2𝑝 with (𝑑/𝑑𝜏)‖𝑣(⋅,𝑡)‖²_𝐿2(ℝ) positive at 𝑡=𝑡_∗. By (20a), we would have ‖𝑣(⋅,𝑡_∗)‖_𝐿2(ℝ)≤𝜆_𝑝, but this would contradict the fact that ‖𝑣(⋅,𝑡)‖_𝐿²_(ℝ)≥𝜆_𝑝+𝜂 everywhere on [𝑡,̂𝑡_𝑗]. Thus, we conclude that (43) cannot be false, as claimed. We then obtain, from (19), (40), and (43), ‖𝑣(⋅,𝑡)‖⁶_𝐿2(ℝ)≤𝐶⁶₂‖𝑣(⋅,𝑡)‖⁴_𝐿1(ℝ)𝑣_𝑥(⋅,𝑡)²_𝐿2(ℝ)≤𝑔(𝑡)⁶+2𝑝‖𝑣(2𝑝−1⋅,𝑡)‖⁴_𝐿1(ℝ)−𝑑‖𝑣(𝑑𝑡⋅,𝑡)‖²_𝐿2(ℝ) (44) for all 𝑡∈[𝑡₀,∞[⧵𝐸_2𝑝. Recalling that ‖𝑣(⋅,𝑡)‖_𝐿²_(ℝ)>𝜆_𝑝+𝜂, 𝑔(𝑡)≤𝜆_𝑝+𝜂/2, for  all 𝑡≥𝑡₀, this gives −𝑑‖𝑣(𝑑𝑡⋅,𝑡)‖²_𝐿2(ℝ)≥𝐾𝜂,∀𝑡∈𝑡₀,∞⧵𝐸_2𝑝 (45) for some constant 𝐾(𝜂)>0 independent of 𝑡, which cannot be, since this implies 𝑣⋅,𝑡₀²_𝐿2(ℝ)≥𝐾𝜂⋅𝑡−𝑡₀∀𝑡>𝑡₀. (46) This contradiction shows (42), which is equivalent to (36), and the proof is complete.

Applying (36) successively with 𝑞=2𝑝,4𝑝,…,2^𝑘𝑝, we get limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿2^𝑘_𝑝(ℝ)≤^𝑘_𝑗=12^𝑗−1𝑝𝐶³₂^{2−𝑗1/𝑝}⋅ℬ^{(1/𝑝)(1−2−𝑘)}⋅𝒰_𝑝 (47) for 𝑘≥1 arbitrary, where 𝒰_𝑝=limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ). Letting 𝑘→∞, this suggestslimsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤𝐾𝑝⋅ℬ^1/𝑝⋅limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ), (48a) where 𝐾𝑝=^∞_𝑗=12^𝑗−1𝑝𝐶³₂^{2−𝑗1/𝑝}=3√3𝑝2𝜋^1/𝑝 (48b)(cf. (6) above), as long as the limit processes 𝑘→∞, 𝑡→∞ can be interchanged. That this is indeed the case is a consequence of (34) and the following result.

Theorem 6. Let 𝑝≥𝑝₀. Then liminf_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤𝑝𝐶₂𝐶_∞^1/𝑝⋅ℬ^1/𝑝⋅limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ), (49) where 𝐶₂, 𝐶_∞ are the constants given in (19) and (9).

Proof. Again, assuming 𝒰_𝑝 finite (otherwise, (49) is obvious; cf. endnote⁴), we introduce, as in the previous proof, 𝑣∈𝐿^∞(ℝ×[0,∞[) given by 𝑣(𝑥,𝑡)=|𝑢(𝑥,𝑡)|^𝑝 if 𝑝>1, and 𝑣(𝑥,𝑡)=𝑢(𝑥,𝑡) if 𝑝=1. Thus, (40) is valid, and setting 𝜆_𝑝∈ℝ, 𝑔∈𝐿^∞([0,∞[) by 𝜆_𝑝=limsup_𝑡→∞𝑔(𝑡),𝑔(𝑡)=𝑝𝐶₂𝐵(𝑡)‖𝐯(⋅,𝑡)‖_𝐿¹_(ℝ), (50) we have that (49) is obtained if we show that liminf_𝑡→∞‖𝑣(⋅,𝑡)‖_𝐿^∞_(ℝ)≤𝐶_∞⋅𝜆_𝑝. (51) We argue by contradiction and assume that (51) is false. Taking then 0<𝜂≪1,𝑡₀≫1 so that ‖𝑣(⋅,𝑡)‖_𝐿^∞_(ℝ)≥𝐶_∞⋅(𝜆_𝑝+𝜂) and 𝑔(𝑡)≤𝜆_𝑝+𝜂/2 hold for all 𝑡≥𝑡₀, we get, by (9) and (40), ‖𝑣(⋅,𝑡)‖³_𝐿^∞_(ℝ)≤𝐶³_∞‖𝑣(⋅,𝑡)‖_𝐿¹_(ℝ)𝑣_𝑥(⋅,𝑡)²_𝐿2(ℝ)≤𝐶³_∞𝑔(𝑡)³+𝐶³_∞2𝑝‖𝑣(2𝑝−1⋅,𝑡)‖_𝐿¹_(ℝ)×−𝑑‖𝑣(𝑑𝑡⋅,𝑡)‖²_𝐿2(ℝ) (52) for all 𝑡∈[𝑡₀,∞[⧵𝐸_2𝑝. Since ‖𝑣(⋅,𝑡)‖_𝐿^∞_(ℝ)≥𝐶_∞⋅(𝜆_𝑝+𝜂), 𝑔(𝑡)≤𝜆_𝑝+𝜂/2, this gives −𝑑‖𝑣(𝑑𝑡⋅,𝑡)‖²_𝐿2(ℝ)≥𝐾𝜂,∀𝑡∈𝑡₀,∞⧵𝐸_2𝑝 (53) for some constant 𝐾(𝜂)>0 independent of 𝑡. As before, this implies that ‖𝑣(⋅,𝑡₀)‖²_𝐿2(ℝ)≥𝐾(𝜂)⋅(𝑡−𝑡₀) for all 𝑡≥𝑡₀, which is impossible because ‖𝑣(⋅,𝑡₀)‖_𝐿2(ℝ) is finite. This contradiction establishes (51) above, completing the proof of Theorem 6.

We are finally in good position to derive (6), (48a), and (48b). Combining (34) and (49) above, we obtain limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤(2𝑝²)^1/𝑝⋅ℬ^1/𝑝⋅𝒰_𝑝 (54) for each 𝑝≥𝑝₀, so that we have, in particular, limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤2^2𝑘+1𝑝^21/2𝑘𝑝⋅ℬ^1/2𝑘𝑝⋅𝒰₂^𝑘_𝑝 (55) for each 𝑘≥0. By (47), we then get limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)≤2^2𝑘+1𝑝^22−𝑘⋅^𝑘_𝑗=12^𝑗−1𝑝𝐶³₂^{2−𝑗1/𝑝}⋅ℬ^1/𝑝⋅𝒰_𝑝 (56) for all 𝑘. Letting 𝑘→∞, Theorem 7 is obtained, and our argument is complete.

Theorem 7. Let 𝑝≥𝑝₀. Assuming 𝑏∈𝐿^∞(ℝ×[0,∞[), then (6), (48a), and (48b) hold.

It is worth noticing that the corresponding estimate for the 𝑛-dimensional problem (8), namely, limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ^𝑛₎≤𝐾𝑛,𝑝⋅ℬ^𝑛/𝑝⋅limsup_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(ℝ^𝑛₎, (57) where ℬ≥0 is similarly defined, can be also derived in arbitrary dimension 𝑛>1.

4. Concluding Remarks

We close our discussion of the problem (1a), (1b), given 𝑏∈𝐿^∞(ℝ×[0,∞[), 1≤𝑝₀<∞, indicating a few questions which were not answered by our analysis:(a)characterize all 𝑏∈𝐿^∞(ℝ×[0,∞[) for which it is true that ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)→0 (as 𝑡→∞) for every solution 𝑢(⋅,𝑡) of problem (1a) and (1b);(b)same question as (a) above, but requiring only that limsup‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)<∞ (as 𝑡→∞) for every solution 𝑢(⋅,𝑡) of problem (1a) and (1b), in case 𝑝₀>1;⁵(c)given 𝑝₀>1, characterize all 𝑏∈𝐿^∞(ℝ×[0,∞[) such that ‖𝑢(⋅,𝑡)‖_𝐿^𝑝_0(ℝ)→0 (as 𝑡→∞) for every solution 𝑢(⋅,𝑡) of problem (1a) and (1b);(d)same question as (c) above, but requiring only that limsup‖𝑢(⋅,𝑡)‖_𝐿^𝑝_0(ℝ)<∞ (as 𝑡→∞) for every solution 𝑢(⋅,𝑡) of problem (1a) and (1b);(e)for 𝑝₀=1, characterize all 𝑏∈𝐿^∞(ℝ×[0,∞[) such that ‖𝑢(⋅,𝑡)‖_𝐿¹_(ℝ)→|𝑚| (as 𝑡→∞) for every solution 𝑢(⋅,𝑡), where 𝑚=∫_ℝ𝑢₀(𝑥)𝑑𝑥 is the solution mass;(f)for 𝑝₀=1, and 𝑏∈𝐿^∞(ℝ×[0,∞[) not satisfying property (e), what are the values of lim_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿¹_(ℝ) in case of initial states that change sign?

These questions can be similarly posed for solutions 𝑢(⋅,𝑡) of autonomous problems 𝑢_𝑡+(𝑏(𝑥)𝑢)_𝑥=𝑢_𝑥𝑥,𝑢(⋅,0)∈𝐿^𝑝0(ℝ)∩𝐿^∞(ℝ), (58) where 𝑏∈𝐿^∞(ℝ) does not depend on the time variable. For (58), question (e) has been answered in [12] (see also [13]). Another interesting question is the following: (g)when (58) admits no stationary solutions other than the trivial solution 𝑢=0, is it true that lim_𝑡→∞‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)=0 for every solution 𝑢(⋅,𝑡)?

Moreover, for solutions 𝑢(⋅,𝑡) of (1a) and (1b) or (58) with ‖𝑢(⋅,𝑡)‖_𝐿^∞_(ℝ)→0 as 𝑡→∞, there is the question of determining the proper decay rate.⁶ As suggested by Figure 1, solution decay may sometimes happen at remarkably slow rates.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) for their financial support.

Endnotes

1. In (6), (11), and other similar expressions in the text, it is assumed that 0⋅∞=∞.
2. The constants (3√3𝑝/(2𝜋))^1/𝑝 in (6) and (7) are not optimal; minimal values are not known.
3. In (6), (11), and other similar expressions in the text, it is assumed that 0⋅∞=∞.
4. In (6), (11), and other similar expressions in the text, it is assumed that 0⋅∞=∞.
5. For 𝑝₀=1, any 𝑏∈𝐿^∞(ℝ×[0,∞[) satisfies property (b); compare (7) in Section 1.
6. In case we have 𝑏_𝑥≥0 for all 𝑥, 𝑡, the answer is given in (2) above.