Blow-Up Phenomena for Porous Medium Equation with Nonlinear Flux on the Boundary

Abstract

We investigate the blow-up phenomena for nonnegative solutions of porous medium equation with Neumann boundary conditions. We find that the absorption and the nonlinear flux on the boundary have some competitions in the blow-up phenomena.

1. Introduction

In this paper, we are concerned with the blow-up of solutions of porous medium equations with nonlinear flux on the boundary. Consider 𝑢_𝑡=Δ𝑢^𝑚−𝑓(𝑢),(𝑥,𝑡)∈Ω×0,𝑡^∗,𝜕𝑢^𝑚(𝑢),()𝜕𝜈=𝑔𝑥,𝑡∈𝜕Ω×0,𝑡^∗,𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈Ω, (1)(2)(3) where 𝑚>1, the nonnegative initial value 𝑢₀(𝑥)∈𝐶(Ω)⋂𝐿^∞(Ω), Ω is a bounded region in ℝ^𝑁 (𝑁≥2) with the sufficiently smooth boundary 𝜕Ω, 𝜈 is the unit normal vector on 𝜕Ω, 𝑡^∗ is the blow-up time if blow-up occurs, or else 𝑡^∗=∞.

The blow-up phenomena for the nonnegative solutions of the heat equation with nonlinear sources (𝑚=1 and 𝑓(𝑢)=−𝑢^𝑝 in (1)) in the whole space ℝ^𝑁 was first found by Fujita in 1966, see [1]. He proved the following results: (a) if 1<𝑝<1+(2/𝑁), then (1) has no global positive solutions; (b) if 𝑝>1+(2/𝑁), then there exist global positive solutions.

The critical case 𝑝=1+(2/𝑁) was proved to belong to the blow-up case in 1970’s by several authors [2–4]. In 1980, Galaktionov and others [5] considered the nonnegative solutions of (1) (with 𝑚>1 and 𝑓(𝑢)=−𝑢^𝑝) in whole space ℝ^𝑁. They found some results similar to those for the heat equation (𝑚=1) as follows(a) if 1<𝑝<𝑚+(2/𝑁), then (1) has no global solutions; (b) if 𝑝>𝑚+(2/𝑁), then there exist global positive solutions that decay like 𝑡^{−1/(𝑝−1)}.

In [6, 7], Galaktionov, Mochizuki and Suzuki, had also revealed that the critical case 𝑝=𝑚+(2/𝑁) belongs to the blow-up case, see also [8, 9].

In 2010, Payne et al. [10] considered a semilinear heat equation with nonlinear boundary condition (𝑚=1 in (1)) and established conditions on nonlinearities sufficient to guarantee that 𝑢(𝑥,𝑡) exists for all time 𝑡>0 as well as conditions on data forcing the solution 𝑢(𝑥,𝑡) to blow up at some finite time 𝑡^∗. When 𝑁=1, the blow-up phenomena for the solutions of the porous medium equation with nonlinear flux on the boundary had also been studied by several authors [11, 12]. For other interesting results on the large time behavior on the solutions of the porous medium equation, we refer the reader to papers [13–16].

Inspired by the above papers, we will study the blow-up phenomena for the solutions of the porous medium equation with nonlinear flux on the boundary in higher dimensional space (𝑁≥2). In fact, we find that if the absorption is more powerful than the boundary flux, then the solutions of the problem (1)–(3) exist for all time on a bounded star-shaped region. On the other hand, if the boundary flux is more powerful, then the solutions of the problem (1)–(3) blow-up at a finite time. Moreover, we will give the upper-bound estimates for the blow-up time.

The paper is organized as follows. In Section 2, we concentrate our attention on the conditions of the global existence for the solutions of the problem (1)–(3). Section 3 is devoted to the investigation of the blow-up phenomena for the solutions of the problem (1)–(3).

2. Criterion for Global Existence

In this section, we investigate the global solutions of problem (1)–(3). The main result of this section is the following theorem.

Theorem 1. Let Ω be a bounded star-shaped region and assume that 𝑞>𝑚 satisfy 2𝑞<𝑚+𝑝. (4) If 𝑓 and 𝑔 satisfy the following conditions: 𝑓(𝜉)≥𝑘₁𝜉^𝑝(𝜉),𝜉≥0,0≤𝑔≤𝑘₂𝜉^𝑞,𝜉≥0, (5)(6) where 𝑘₁, 𝑘₂ are nonnegative constants, then the nonnegative solutions 𝑢(𝑥,𝑡) of the problem (1)–(3) do not blow up.

Proof. Let Φ(𝑡)=_Ω𝑢²𝑑𝑥. (7) Differentiating (7) and making use of (1), we obtain that Φ^′(𝑡)=2_Ω𝑢𝑢_𝑡𝑑𝑥=2_Ω𝑢Δ𝑢^𝑚−𝑓(𝑢)𝑑𝑥. (8) From the hypothesis (5), we get Φ^′(𝑡)≤2_Ω𝑢Δ𝑢^𝑚−𝑘₁𝑢^𝑝𝑑𝑥. (9) By (2), (6) and the divergence theorem, we have _Ω𝑢Δ𝑢^𝑚𝑑𝑥=_𝜕Ω𝑢∇𝑢^𝑚⋅𝜈𝑑𝑠−_Ω∇𝑢⋅∇𝑢^𝑚𝑑𝑥=_𝜕Ω𝑢𝜕𝑢^𝑚𝜕𝜈𝑑𝑠−_Ω∇𝑢⋅∇𝑢^𝑚𝑑𝑥≤𝑘_2𝜕Ω𝑢^𝑞+1𝑑𝑠−_Ω∇𝑢⋅∇𝑢^𝑚𝑑𝑥. (10) Here we used the identity div(𝑢∇𝑢^𝑚)=𝑢Δ𝑢^𝑚+∇𝑢⋅∇𝑢^𝑚. By the divergence theorem again, we get _𝜕Ω𝑢^𝑞+1𝑥⋅𝜈𝑑𝑠=_Ωdiv𝑢^𝑞+1𝑥𝑑𝑥. (11) Let 𝜌₀=min_𝜕Ω(𝑥⋅𝜈),𝑑=max_𝜕Ω|𝑥|. (12) Point out that 𝜌₀ is positive because Ω is star-shaped by hypothesis. Notice also that div𝑢^𝑞+1𝑥=div𝑢^{(𝑚+1)/22(𝑞+1)/(𝑚+1)}𝑥=𝑁𝑢^𝑞+1+2𝑞+1𝑢𝑚+1^{(2𝑞−𝑚+1)/2}𝑥⋅∇𝑢^(𝑚+1)/2. (13) We thus have _𝜕Ω𝑢^𝑞+1𝑁𝑑𝑠≤𝜌_0Ω𝑢^𝑞+1𝑑𝑥+2𝑑𝑞+1𝜌₀()×𝑚+1_Ω𝑢^{(2𝑞−𝑚+1)/2}∇𝑢^(𝑚+1)/2𝑑𝑥. (14) On the another hand ∇𝑢⋅∇𝑢^𝑚=∇𝑢⋅𝑚𝑢^𝑚−1𝑢∇𝑢=𝑚^(𝑚−1)/2∇𝑢𝑢^(𝑚−1)/2=∇𝑢4𝑚(𝑚+1)²∇𝑢^(𝑚+1)/22. (15) Therefore, from (10)–(15), we have Φ^′(𝑡)≤2𝑘_2𝜕Ω𝑢^𝑞+1𝑑𝑠−2_Ω∇𝑢⋅∇𝑢^𝑚𝑑𝑥−2𝑘_1Ω𝑢^𝑝+1≤𝑑𝑥2𝑘₂𝑁𝜌_0Ω𝑢^𝑞+1+𝑑𝑥4𝑘₂𝑑𝑞+1𝜌₀(𝑚+1)_Ω𝑢^{(2𝑞−𝑚+1)/2}∇𝑢^(𝑚+1)/2−𝑑𝑥8𝑚(𝑚+1)^2Ω∇𝑢^(𝑚+1)/22𝑑𝑥−2𝑘_1Ω𝑢^𝑝+1𝑑𝑥. (16) We obtain from the Young inequality that _Ω𝑢^{(2𝑞−𝑚+1)/2}∇𝑢^(𝑚+1)/2≤𝜎𝑑𝑥2_Ω𝑢^{2𝑞−𝑚+1}1𝑑𝑥+2𝜎_Ω∇𝑢^(𝑚+1)/22𝑑𝑥, (17) where 𝑘𝜎=²𝑑𝑞+1(𝑚+1)4𝑚𝜌₀. (18) This 𝜎 leads to 4𝑘₂𝑑𝑞+1𝜌₀(𝑚+1)_Ω𝑢^{(2𝑞−𝑚+1)/2}∇𝑢^(𝑚+1)/2≤𝑑𝑥8𝑚(𝑚+1)²𝜎²_Ω𝑢^{2𝑞−𝑚+1}+𝑑𝑥8𝑚(𝑚+1)^2Ω∇𝑢^(𝑚+1)/22𝑑𝑥. (19) Combining this with (16), we get Φ^′(𝑡)≤2𝑘₂𝑁𝜌_0Ω𝑢^𝑞+1𝑑𝑥+8𝑚(𝑚+1)²𝜎²×_Ω𝑢^{2𝑞−𝑚+1}𝑑𝑥−2𝑘_1Ω𝑢^𝑝+1𝑑𝑥. (20) Let 𝛼=𝑞−𝑚.𝑝−𝑞 (21) Therefore, the hypotheses that 𝑞>𝑚 and 2𝑞<𝑚+𝑝 imply that 0<𝛼<1. (22) So, by Hölder's inequality, we have _Ω𝑢^{2𝑞−𝑚+1}𝑑𝑥≤_Ω𝑢^𝑞+1𝑑𝑥^𝛼_Ω𝑢^𝑝+1𝑑𝑥^1−𝛼. (23) For 𝜖>0, we obtain from (23) that _Ω𝑢^{2𝑞−𝑚+1}𝑑𝑥≤𝜖_Ω𝑢^𝑝+1𝑑𝑥^1−𝛼𝜖^{(𝛼−1)/𝛼}_Ω𝑢^𝑞+1𝑑𝑥^𝛼≤(1−𝛼)𝜖_Ω𝑢^𝑝+1𝑑𝑥+𝛼𝜖^{(𝛼−1)/𝛼}_Ω𝑢^𝑞+1𝑑𝑥. (24) Thus, inserting (24) in (20), we obtain Φ^′(𝑡)≤2𝑘₂𝑁𝜌_0Ω𝑢^𝑞+1𝑑𝑥+8𝑚(𝑚+1)²𝜎²×(1−𝛼)𝜖_Ω𝑢^𝑝+1𝑑𝑥+𝛼𝜖^{(𝛼−1)/𝛼}_Ω𝑢^𝑞+1𝑑𝑥−2𝑘_1Ω𝑢^𝑝+1=𝑑𝑥2𝑘_2𝑁𝜌₀+8𝑚𝜎²𝛼(𝑚+1)²𝜖^{(𝛼−1)/𝛼}×_Ω𝑢^𝑞+1𝑑𝑥+8𝑚𝜎²𝜖(1−𝛼)(𝑚+1)²−2𝑘₁×_Ω𝑢^𝑝+1𝑑𝑥=𝑀_1Ω𝑢^𝑞+1𝑑𝑥−𝑀_2Ω𝑢^𝑝+1𝑑𝑥, (25) where 𝑀₁=2𝑘_2𝑁𝜌₀+8𝑚𝜎²𝛼(𝑚+1)²𝜖^{(𝛼−1)/𝛼}𝑀>0,₂=2𝑘₁−8𝑚𝜎²𝜖(1−𝛼)(𝑚+1)², (26) and let 𝜖 be sufficiently small to ensure 𝑀₂>0. By Hölder's inequality again, we have _Ω𝑢^𝑞+1𝑑𝑥≤_Ω𝑢^𝑝+1𝑑𝑥^{(𝑞+1)/(𝑝+1)}|Ω|^{(𝑝−𝑞)/(𝑝+1)}, (27) where we assume throughout the paper that |Ω|=∫_Ω𝑑𝑥 is the measure of Ω. Using (25) and (27), we obtain Φ^′(𝑡)≤𝑀_1Ω𝑢^𝑝+1𝑑𝑥^{(𝑞+1)/(𝑝+1)}×|Ω|^{(𝑝−𝑞)/(𝑝+1)}−𝑀₂𝑀_1Ω𝑢^𝑝+1𝑑𝑥^{(𝑝−𝑞)/(𝑝+1)}. (28) Moreover, using Hölder's inequality once more, we have Φ(𝑡)=_Ω𝑢²𝑑𝑥≤_Ω𝑢^𝑝+1𝑑𝑥^2/(𝑝+1)|Ω|^{(𝑝−1)/(𝑝+1)}, (29) that is, _Ω𝑢^𝑝+1𝑑𝑥^{(𝑝−𝑞)/(𝑝+1)}≥Φ(𝑡)^{(𝑝−𝑞)/2}|Ω|^{(1−𝑝)(𝑝−𝑞)/2(𝑝+1)}. (30) Finally, from (28) and (30), we obtain Φ^′(𝑡)≤𝑀_1Ω𝑢^𝑝+1𝑑𝑥^{(𝑞+1)/(𝑝+1)}×|Ω|^{(𝑝−𝑞)/(𝑝+1)}−𝑀₂𝑀₁Φ(𝑡)^{(𝑝−𝑞)/2}|Ω|^{(1−𝑝)(𝑝−𝑞)/2(𝑝+1)}. (31) We deduced from (31) that Φ(𝑡)≤max{Φ(0),(𝑀₂/𝑀₁)^{2/(𝑞−𝑝)}|Ω|}. On the other hand, Φ(𝑡) is nonnegative function by assumption. So that Φ(𝑡) keeps bounded continuously under the conditions given in Theorem 1, the solutions exsit for all time 𝑡>0. That is, we find that the global solution exists when the absorption is more powerful than the nonlinear boundary flux and this accomplishes the proof of Theorem 1.

3. Criterion for Blow-Up

In this section, we concentrate on the finite time 𝑡^∗ on which blow-up occurs. We construct two auxiliary functions to redefine 𝑓 and 𝑔, then the nonlinear boundary-flux is more powerful than the absorption, and we obtain the following result.

Theorem 2. Suppose 0≤𝛼≤𝛽. (32) Let 𝐹(𝜉)=^𝜉₀𝑓𝜂𝑚()𝑑𝜂−𝑚−12^𝜉₀∇𝜂²𝜂^𝑚−2𝐺(𝜉)=𝑑𝜂,_𝜉0𝑔𝜂Ψ(𝑡)𝑑𝜂,=2_𝜕Ω𝐺(𝑢)𝑑𝑠−_Ω∇𝑢⋅∇𝑢^𝑚𝑑𝑥−2_Ω𝐹(𝑢)𝑑𝑥. (33) If Ψ(0)(𝜉)()𝐹(𝜉)(𝜉)𝐺(𝜉)>0,𝜉𝑓≤21+𝛼,𝜉≥0,𝜉𝑔≥21+𝛽,𝜉≥0, (34) then the solutions 𝑢(𝑥,𝑡) of the problem (1)–(3) blow up at time 𝑡^∗<𝑇 with Φ(0)𝑇=Ψ(0).2𝛽1+𝛽 (35) Here Φ(𝑡) is defined in (7). Moreover, if 𝛽=0, then 𝑇=∞.

Proof. Differentiating (7) and using the hypothesis (33), we have Φ^′(𝑡)=2_Ω𝑢𝑢_𝑡𝑑𝑥=2_Ω𝑢Δ𝑢^𝑚−𝑓(𝑢)𝑑𝑥=2_𝜕Ω𝑢𝑔(𝑢)𝑑𝑠−2_Ω∇𝑢⋅∇𝑢^𝑚𝑑𝑥−2_Ω𝑢𝑓(𝑢)𝑑𝑥≥21+𝛽Ψ(𝑡). (36) Differentiating (33), we thus obtain from (15) that Ψ^′(𝑡)=2_𝜕Ω𝑔(𝑢)𝑢_𝑡𝑑𝑠−4𝑚(𝑚+1)^2Ω∇𝑢^(𝑚+1)/22_𝑡𝑑𝑥−2_Ω𝑓(𝑢)𝑢_𝑡𝑑𝑥+𝑚(𝑚−1)_Ω|∇𝑢|²𝑢^𝑚−2𝑢_𝑡𝑑𝑥. (37) Note the identity that ∇𝑢^(𝑚+1)/22_𝑡=(𝑚+1)²(𝑚−1)4×|∇𝑢|²𝑢^𝑚−2𝑢_𝑡+(𝑚+1)²2𝑢^𝑚−1∇𝑢⋅∇𝑢_𝑡. (38) So, from (37), we get Ψ^′(𝑡)=2_𝜕Ω𝑔(𝑢)𝑢_𝑡𝑑𝑠−2_Ω∇𝑢_𝑡⋅∇𝑢^𝑚𝑑𝑥−2_Ω𝑓(𝑢)𝑢_𝑡𝑑𝑥. (39) Therefore, Ψ^′(𝑡)=2_Ω𝑢²_𝑡𝑑𝑥>0. (40) Here, we have used the identities div𝑢_𝑡∇𝑢^𝑚=𝑢_𝑡Δ𝑢^𝑚+∇𝑢_𝑡⋅∇𝑢^𝑚,_Ω∇𝑢_𝑡⋅∇𝑢^𝑚𝑑𝑥=_𝜕Ω𝑢_𝑡∇𝑢^𝑚⋅𝜈𝑑𝑠−_Ω𝑢_𝑡Δ𝑢^𝑚𝑑𝑥. (41) So, the hypothesis Ψ(0)>0 implies that for all 𝑡∈(0,𝑡^∗), the following inequality holds (𝑡)>0: Ψ(𝑡)>0. (42) By the Schwarz inequality, we have Φ^′(𝑡)²=4_Ω𝑢𝑢_𝑡𝑑𝑥²≤2Φ(𝑡)Ψ^′(𝑡). (43) Together with (36), we have Φ(𝑡)Ψ^′(𝑡)≥12Φ^′(𝑡)²≥1+𝛽Φ^′(𝑡)Ψ(𝑡). (44) That is, ΨΦ^{−(1+𝛽)′}≥0. (45) Integrating this from 0 to 𝑡, we obtain Ψ(𝑡)(Φ(𝑡))^−(1+𝛽)≥Ψ(0)(Φ(0))^−(1+𝛽)=𝑀. (46) Substituting (46) in (36) we obtain the differential inequality Φ^′(𝑡)≥21+𝛽Ψ≥21+𝛽𝑀Φ^1+𝛽. (47) If 𝛽>0, then (Φ(𝑡))^−𝛽≤(Φ(0))^−𝛽−2𝛽1+𝛽𝑀𝑡. (48) This leads to 𝑡^∗1≤𝑇=𝑀(Φ(02𝛽1+𝛽))^−𝛽=Φ(0)Ψ(0).2𝛽1+𝛽 (49) If 𝛼=𝛽=0, then Φ(𝑡)≥Φ(0)𝑒^2𝑀𝑡 (50) holds for 𝑡>0. This implies that 𝑡^∗=∞ and completes the proof of Theorem 2.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of “CQ CSTC” (cstc2012jjA00013), and Scientific and Technological Research Program of Chongqing Municipal Education Commission.