Haixia Li

Wenjie Gao

Yuzhu Han

Han, Yuzhu

Gao, Wenjie

Li, Haixia

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 353 (2015) 825–830Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial differential equationsBlow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux ✩Explosion de solutions de l’équation de la chaleur semi-linéaire avec un terme viscoélastique et un flux de limite non linéaireYuzhu Han a, Wenjie Gao a, Haixia Li ba School of Mathematics, Jilin University, Changchun 130012, PR Chinab School of Mathematics, Changchun Normal University, Changchun 130032, PR Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 19 June 2014Accepted after revision 9 July 2015Available online 4 August 2015Presented by the Editorial BoardIn this article, we study a semilinear heat equationut − u +t∫0g(t − s)u(x, s) ds = 0with a viscoelastic term and a nonlinear flux on the boundary. By defining a modified energy functional and using a concavity argument, a blow-up result for solutions with negative initial energy is proved.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cet article, on étudie une équation de la chaleur semi-linéaireut − u +t∫0g(t − s)u(x, s)ds = 0avec un terme viscoélastique et un flux non linéaire sur la limite. En définissant une modifiée fonctionnelle d’énergie et en utilisant un argument de concavité, un résultat d’explosion des solutions avec énergie initiale négative est prouvé.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.✩ The project is supported by NSFC (11271154, 11401252), by Science and Technology Development Project of Jilin Province (20150201058NY) and by the 985 program of Jilin University. The first author is also supported by Fundamental Research Funds of Jilin University (450060501179).E-mail addresses: yzhan@jlu.edu.cn (Y. Han), wjgao@jlu.edu.cn (W. Gao), lihaixia0611@126.com (H. Li).http://dx.doi.org/10.1016/j.crma.2015.07.0031631-073X/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.826 Y. Han et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 825–8301. IntroductionIn this paper, we investigate the blow-up properties of solutions to the following semilinear initial boundary value problem⎧⎪⎪⎨⎪⎪⎩ut − u +∫ t0 g(t − s)u(x, s)ds = 0, x ∈ , t > 0,∂u∂ν −∫ t0 g(t − s) ∂u(x,s)∂ν ds = |u|p−2u, x ∈ ∂, t > 0,u(x,0) = u0(x), x ∈ ,(1.1)where g: R+ → R+ is a bounded C1 function, p > 2,  ⊂ RN (N ≥ 2) is a bounded domain with smooth boundary ∂ and ν is the unite outward normal on ∂.Problem (1.1) arises from a variety of mathematical models of applied sciences such as viscoelastic fluids and electro-rheological fluids ([1,15]). In particular, when we study heat conduction in materials with memory, the classical Fourier law of heat flux is usually replaced by the following formq = −d∇u −t∫−∞∇[k(x, t)u(x, s)]ds, (1.2)where u is the temperature, d > 0 is the diffusion coefficient and the integral term represents the memory effect of the material. In the past few decades, the study of equations with memory terms has drawn a considerable attention and many results have been obtained on the existence, uniqueness and regularities of weak or classical solutions. We refer the interested readers to [18,19]. From a mathematical point of view, it is hoped that the integral term is dominated by the leading term in the equation and therefore the theory of parabolic equation can be applied to Problem (1.1).Finite-time blow-up is one of the most important properties of many evolutionary equations. When one needs to deter-mine whether the solutions of a given problem blow up in finite time or not, there are many methods to use. For example, the first eigenvalue method introduced by Kaplan in 1963, the concavity method introduced by Levine in the 1970s, and the comparison method based on the comparison principle. In the absence of the memory term (g(t) ≡ 0), properties including blow-up time, blow-up rate, blow-up set and blow-up profiles of solutions to semilinear equations like (1.1) have been in-vestigated by many authors, see [7–10,17] for equations with sources in the interior and [5,6,11,16] for those with sources on the boundary.However, when g 	≡ 0, the blow-up results are much fewer. Messaoudi [12] studied⎧⎪⎨⎪⎩ut − u +∫ t0 g(t − s)u(x, s)ds = |u|p−2u, x ∈ , t > 0,u(x, t) = 0, x ∈ ∂, t > 0,u(x,0) = u0(x), x ∈ ,(1.3)and showed that the solutions to (1.3) blow up in finite time when, among other conditions, the initial energy E(0) ≤ 0, whereE(t) = 12(g  ∇u)(t) + 12(1 −t∫0g(s)ds)‖∇u(t)‖22 −1p‖u(t)‖pp, (1.4)and(g  v)(t) =t∫0g(t − s)‖v(x, t) − v(x, s)‖22 ds. (1.5)The idea of the proof is to establish a connection between ddt ‖u‖2 and ‖u‖r2 for some r > 1. Later, Messaoudi [13] and Fang et al. [4] also proved that the solutions to (1.3) may blow up in finite time when the initial energy is positive, but suitably small.Motivated by the works mentioned above, we shall study the blow-up properties of solutions to (1.1). However, the situations are more complicated. On one hand, the connection between ddt ‖u‖2 and ‖u‖r2, as was established in [12], cannot be obtained here since the source is on the boundary; on the other hand, the appearance of the relaxation function gmakes the powerful comparison principle invalid. To overcome the above two difficulties, we shall define a modified energy functional and use a concavity argument to establish a blow-up criterion. The main result of this paper and its proof will be given in Section 2.Y. Han et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 825–830 8272. Main resultIn order to state and prove the main result, we first define the modified energy functionalE(t) = 12(g  ∇u)(t) + 12(1 −t∫0g(s)ds)‖∇u(t)‖22, −1p‖u(t)‖pp,∂, (2.1)where (g  u)(t) is given in (1.5). For simplicity, we shall omit  and ∂ in the subscript of the norms in the rest of this paper. Throughout this paper, the relaxation function g and the parameter p are supposed to satisfyg(s) ≥ 0, g′(s) ≤ 0, 1 −∞∫0g(s)ds = l > 0, (2.2)and2 < p ≤ 2N − 3N − 2 , N > 3; p > 2, N = 2. (2.3)Before stating our main result, we first give a definition for a strong solution to (1.1), see [14]. The local existence of such a strong solution can be obtained by applying the Galerkin method as in [3] and the details are omitted.Definition 2.1. A strong solution to Problem (1.1) is a function u ∈ C([0, T ); H1()) ∩ C1([0, T ); L2()), satisfyingt∫0∫(utφ + ∇u∇φ −s∫0g(s − τ )∇u(τ )∇φ(s)dτ)dx ds =t∫0∫∂|u|p−2uφ dσ dsfor all t ∈ [0, T ) and all φ ∈ C([0, T ); H1()).By the Sobolev trace imbedding theorem (Theorem 5.36 in [2]), we know that condition (2.3) implies |u|p−2u ∈ L2(∂) and hence ∫∂|u|p−2uφ dσ makes sense. Condition (2.2) is necessary to guarantee the parabolicity of (1.1).By multiplying the equation in (1.1) by ut , integrating by parts over , one obtains that∫u2t dx =∫uut dx −∫ut( t∫0g(t − s)u(x, s)ds)dx=∫∂ut(∂u∂ν−t∫0g(t − s) ∂u∂νds)dσ −∫∇u∇ut dx+∫( t∫0g(t − s)∇u(s)∇ut(t)ds)dx= − ddt(12∫|∇u|2dx)+ ddt( 1p∫∂|u|pdσ)+t∫0g(t − s)∫∇u(s)∇ut(t)dx ds.The last term of the right-hand side of the above equality can be rewritten ast∫0g(t − s)∫∇u(s)∇ut(t)dx ds= −12ddt( t∫0g(t − s)∫|∇u(s) − ∇u(t)|2 dx ds)+ 12ddt( t∫0g(s)ds∫|∇u(t)|2dx)+ 12t∫g′(t − s)∫|∇u(s) − ∇u(t)|2 dx ds − 12g(t)∫|∇u(t)|2 dx.0  828 Y. Han et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 825–830Combining the above two equalities with (2.1), we see thatddtE(t) = −∫u2t dx −12g(t)∫|∇u(t)|2 dx + 12t∫0g′(t − s)∫|∇u(s) − ∇u(t)|2 dx ds ≤ 0 (2.4)for smooth solutions. The same result can be established for strong solutions and for almost every t , by a standard density argument. The main result of this paper is the following.Theorem 2.1. Suppose that (2.2)–(2.3) hold and that u0 ∈ H1() satisfying E(0) < 0. If∞∫0g(s)ds <p − 2p − 3/2 , (2.5)then any strong solution of (1.1) blows up in finite time.Proof. The theorem will be proved by using a concavity argument introduced by Levine. For this, setI(t) =t∫0∫u2(x, τ )dx dτ + A, (2.6)where A > 0 is a constant to be fixed. ThenI ′(t) =∫u2(x, t)dx,andI ′′(t) = 2∫uut dx= 2∫∂|u|pdσ − 2(1 −t∫0g(s)ds)∫|∇u|2dx+ 2t∫0g(t − s)∫∇u(t)(∇u(s) − ∇u(t))dx ds. (2.7)By using Cauchy–Schwarz’s inequality first and Young’s inequality next, we get from (2.7) thatI ′′(t) ≥ 2∫∂|u|pdσ − 2(1 −t∫0g(s)ds)∫|∇u|2dx− 2t∫0g(t − s)‖∇u(t)‖2‖∇u(t) − ∇u(s)‖2 ds≥ 2∫∂|u|p dσ − 2(1 − 34t∫0g(s)ds)∫|∇u|2dx − 2(g  ∇u)(t). (2.8)By comparing the terms in (2.8) with (2.1) and noticing (2.2) and (2.5), we have, for some δ > 0, thatI ′′(t) ≥ −4(1 + δ)E(t)= 4(1 + δ)(− E(0) +t∫ ∫u2τ dx dτ +12t∫(g(τ )∫|∇u(τ )|2dx)dτ0  0 Y. Han et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 825–830 829− 12t∫0τ∫0g′(τ − s)∫|∇u(s) − ∇u(τ )|2 dx ds dτ)≥ 4(1 + δ)(− E(0) +t∫0∫u2τ dx dτ).Clearly,I ′(t) =∫u2(x, t)dx = 2t∫0∫uuτ dx dτ +∫u20(x)dx. (2.9)It follows that, for any ε > 0,I ′(t)2 ≤ 4(1 + ε)t∫0∫u2 dx dτt∫0∫u2τ dx dτ + (1 +1ε)(∫u20(x)dx)2. (2.10)Combining the above estimates, we find that for α > 0,I ′′(t)I(t) − (1 + α)I ′(t)2≥ 4(1 + δ)(− E(0) +t∫0∫u2τ dx dτ)( t∫0∫u2(x, τ )dx dτ + A)− 4(1 + α)(1 + ε)t∫0∫u2 dx dτt∫0∫u2τ dx dτ− (1 + α)(1 + 1ε)(∫u20(x)dx)2. (2.11)Now we choose ε and α small enough such that1 + δ > (1 + α)(1 + ε). (2.12)Thus, it is clear from (2.11), (2.12) and the assumption E(0) < 0 thatI ′′(t)I(t) − (1 + α)I ′(t)2 > 0, (2.13)for A > 0 large enough, which implies thatddt( I ′(t)I1+α(t))> 0,orI ′(t)I1+α(t)≥ I′(0)I1+α(0),as long as u exists. It follows from the above inequality that I(t) = ∫ t0∫u2(x, τ ) dx dτ + A cannot remain finite for all t > 0. The proof is complete. AcknowledgementsThe authors are grateful to the referees for their valuable comments and suggestions that helped improve the original manuscript.830 Y. Han et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 825–830References[1] E. Acerbi, G. Mingione, Regularity results for stationary electrorheological fluids, Arch. Ration. Mech. Anal. 164 (2002) 213–259.[2] R.A. Admas, J.J.F. Fournier, Sobolev Spaces, access online via Elsevier, 2003.[3] Z.B. Fang, L.R. Liu, Global existence and uniform energy decay rates for the semilinear parabolic equation with a memory term and mixed boundary condition, Abstr. Appl. Anal. 2013 (2013), Article ID 532935.[4] Z.B. Fang, L. Sun, Blow up of solutions with positive initial energy for the nonlocal semilinear heat equation, J. Korean Soc. Ind. Appl. Math. 16 (2012) 235–242.[5] M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comment. Math. Univ. Carol. 30 (1989) 479–484.[6] M. Fila, P. 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Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux

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Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux

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