2000 Mathematics Subject Classification: 35K55, 35K60.We investigate the blow-up of the solutions to a nonlinear parabolic system with Robin boundary conditions and time dependent coefficients. We derive sufficient conditions on the nonlinearities and the initial data in order to obtain explicit lower and upper bounds for the blow up time t*

Marras, M.

We investigate the blow-up of the solutions to a nonlinear parabolic system with Robin boundary conditions and time dependent coefficients. We derive sufficient conditions on the nonlinearities and the initial data in order to obtain explicit lower and upper bounds for the blow up time t

MARRAS, MONICA

Archivio istituzionale della ricerca - Università di Cagliari

Remarks on blow up time for solutions of a nonlinear diffusion systems  with time dependent coefficients

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 37 (2011), 227–236REMARKS ON BLOW UP TIME FOR SOLUTIONS OF ANONLINEAR DIFFUSION SYSTEM WITH TIMEDEPENDENT COEFFICIENTSM. MarrasCommunicated by T. GramchevAbstract. We investigate the blow-up of the solutions to a nonlinearparabolic system with Robin boundary conditions and time dependent coef-ficients. We derive sufficient conditions on the nonlinearities and the initialdata in order to obtain explicit lower and upper bounds for the blow uptime t∗.1. Introduction. The blow-up phenomena for solutions to linear andnonlinear parabolic equations and systems have been widely studied and differentmethods have been introduced in order to find bounds for blow-up time t∗. Werefer to the books of Straughan [11] and Quittner-Souplet [10], and the papers ofVazquez [12] and Weissler [13], [14]. Recently Payne, Philippin, Schaefer in [5]–[6] and Payne, Philippin and Vernier-Piro in [7]–[8] (see also references therein)have considered nonlinear boundary value problems under different boundary2010 Mathematics Subject Classification: 35K55, 35K60.Key words: Parabolic problems, blow-up.228 M. Marrasconditions, deriving lower and upper bounds for the blow-up time t∗ as well assufficient conditions for global existence of the solutions.In this paper we investigate the blow-up phenomena of the classical solu-tion (u(x, t), v(x, t)) of the following Robin system(1.1)∆u+K1(t)f1(v) = ut in Ω× (0, t∗),∆v +K2(t)f2(u) = vt in Ω× (0, t∗),∂u∂ν(x, t) = −αu, on ∂Ω× (0, t∗),∂v∂ν(x, t) = −βv, on ∂Ω× (0, t∗),u(x, 0) = u0(x) ≥ 0, on Ω,v(x, 0) = v0(x) ≥ 0, on Ω,where Ω is a bounded domain in RN , N ≥ 2 with ∂Ω sufficiently smooth, thecoefficients K1(t), K2(t) and f1, f2 positive functions and∂u∂ν,∂v∂νare the out-ward normal derivative of the vector–valued solution (u, v) on the boundary ∂Ω,α and β are two positive constants satisfying(1.2) 0 ≤ α ≤ β.The initial data u0(x), v0(x) are nonnegative smooth functions, such that∂u0∂ν= −αu0, ∂v0∂ν= −βv0, x ∈ ∂Ω, while t∗ stands for the blow-up time if blowup occurs, otherwise t∗ =∞. It is well known that the solution can fail to existonly by blowing up at finite time ([1], [2]) and the geometry of the domain Ω,the nonlinearities, the boundary data and the initial conditions greatly affect theevolution in time of the solution.We note that u and v are non negative in x and t ∈ (0, t∗) by the parabolicmaximum principle applied to the the system (1.1).The main aim of this paper is to derive explicit upper and lower boundsfor the blow-up time for the solution of (1.1). We remark that if K1, K2 areconstant, bounds for the blow-up time are derived in [3], where on ∂Ω × (0, t∗)∂u∂ν= g1(u),∂v∂ν= g2(v), g1, g2 positive functions.For systems with Dirichlet and Neumann boundary conditions we referto [3] and [9].We consider another class of parabolic systems with Robin type boundaryconditions in [4], where sufficient conditions are introduced in order to obtainRemarks on blow up time 229global existence of the solution as well as upper and lower bounds for the blow-up, if blow-up occurs.The paper is organized as follows. In Section 2 we derive a lower boundfor the blow up time t∗ under suitable conditions on data and for convex domainin R3. In Section 3 we remove the restriction on the domain and under alternativeconditions on data an upper bound is obtained.Throughout the paper the notations ui =∂u∂xi, i = 1, . . . , N and (·)′ forthe derivative of the coefficients will be used and the summation convention overrepeated indexes will be assumed.2. Lower bound. Under suitable conditions on non linearities and Ωwe get a differential inequality and we derive a lower bound for t∗.To this end we will use a Sobolev-type inequality introduced in [3] whichplays an important role in deriving the lower bound. However, it must be notedthat the inequality holds only if we consider our domain Ω ⊂ R3.Let us suppose K1 ≤ K2 and we define the auxiliary function(2.1) Φ(t) = K22 (t)∫Ω(u2 + v2)2pdx, p > 1We say that (u, v) blows up in Φ-measure whenlimt→t∗Φ(t) =∞.We now prove the following resultTheorem 2.1. Let (u, v) be the solution of (1.1), in a convex domainΩ ⊂ R3, blowing up at time t∗ in Φ-measure. Assume that there exists a positiveconstant C0 such that(2.2) uf1(v) + vf2(u) ≤ C0(u2 + v2)p+1.Moreover assume that K2(t) satisfies(2.3)K ′2(t)K2(t)≤ δ, δ > 0.Then the solution blows up in the Φ-measure and(2.4) t∗ ≥ T :=∫∞Φ(0)dηφ(η),230 M. Marraswhere φ(Φ) =3∑i=1(ξiΦhi) with some if not all hi greater than one.P r o o f. We computeΦ′(t) = 2(K ′2K2)K22∫Ω(u2 + v2)2p + 4pK22∫Ω(u2 + v2)2p−1(u∆u+ v∆v)dx+ 4pK22∫Ω(u2 + v2)2p−1[K1uf1(v) +K2vf2(u)]dx.Now by (2.2), (2.3) and the assumption K1 ≤ K2 we obtainΦ′(t) ≤ 2δΦ + 4pK22∫Ωdiv[(u2 + v2)2p−1(u∇u+ v∇v)]dx− 8p(2p − 1)K22∫Ω(u2 + v2)2p−2(u∇u+ v∇v)(u∇u+ v∇v)dx− 4pK22∫Ω(u2 + v2)2p−1(|∇u|2 + |∇v|2)dx+ 4pC0K32∫Ω(u2 + v2)3pdx.Now by divergence theorem and boundary conditions in (1.1) and as-sumption (1.2) we get(2.5) Φ′(t) ≤ 2δΦ − 4pK22α∫∂Ω(u2 + v2)2pds− 8p(2p − 1)K22∫Ω(u2 + v2)2p−2(u∇u+ v∇v)(u∇u+ v∇v)dx− 4pK22∫Ω(u2 + v2)2p−1(|∇u|2 + |∇v|2)dx+ 4pC0K32∫Ω(u2 + v2)3pdx.Now we neglect the second (negative) term in (2.5), and we use the in-equality (u2+ v2)(|∇u|2+ |∇v|2) ≥ (u∇u+ v∇v)(u∇u+ v∇v) in the fourth termto have(2.6) Φ′(t) ≤ 2δΦ − 4p(4p − 1)K22∫Ω(u2 + v2)2p−2(u∇u+ v∇v)(u∇u+ v∇v)dx+ 4pC0K32∫Ω(u2 + v2)3pdx.For simplicity we define(2.7) V = (u2 + v2)p,Remarks on blow up time 231then(2.8) ViVi = 4p2(u2 + v2)2p−2(u∇u+ v∇v)(u∇u+ v∇v).By plugging (2.7) and (2.8) in (2.6) we have(2.9) Φ′(t) ≤ 2δΦ − (4− 1p)K22∫ΩViVi + 4pC0K32∫ΩV 3dx.In the third term in (2.9) we use the inequality∫ΩV 3dx ≤{32ρ0∫ΩV 2dx+(1 +dρ0)(∫ΩV 2dx) 12(∫Ω|∇V |2dx) 12} 32withρ0 = min∂Ω(xi · νi), d2 = maxΩ¯(xi · xi),obtained by application of a Sobolev type inequality derived in [8] Lemma A.2,valid only in a convex domain ⊂ R3 we have(2.10) K32 (t)∫ΩV 3dx ≤{32ρ0K22∫ΩV 2dx+(1 +dρ0)(1τK22∫ΩV 2dx) 12(τK22∫Ω|∇V |2dx) 12} 32τ > 0 a suitable constant to be chosen later on.In (2.10) we use the basic inequality (a + b)32 ≤ √2(a32 + b32)and weobtain(2.11) K32 (t)∫ΩV 3dx ≤√2(32ρ0) 32Φ32+√2(1 +dρ0) 32(1τΦ)34(τK22∫Ω|∇V |2dx) 34.Now we use the inequality arbs ≤ ra+ sb, r + s = 1, we have(2.12) K32∫ΩV 3dx ≤ 3322ρ320Φ32 +1232 τ3(1 +dρ0) 32Φ3+(3232)(1 +dρ0) 32τK22∫ΩViVidx = C1Φ32 + C2Φ3 + C3τK22∫ΩViVidx232 M. Marraswith(2.13) C1 =3322ρ320, C2 =1232 τ3(1 +dρ0) 32, C3 =3232(1 +dρ0) 32.Inserting (2.12) in (2.9) we obtain(2.14) Φ′(t) ≤ 2δΦ−(4− 1p− 4pC3C0τ)K22∫ΩViVi+4pC1C0Φ32 +4pC2C0Φ3.By choosing τ =4− 1p4pC3C0, we obtain(2.15) Φ′(t) ≤ ξ1Φ+ ξ2Φ32 + ξ3Φ3 = φ(Φ),with ξ1 = 2δ, ξ2 = 4pC1C0, ξ3 = 4pC2C0.Then if Φ blows up at time t∗, there exists a time t0 such that Φ(t) > Φ(0)and by integration from 0 to t∗, we obtain the following lower bound for t∗∫∞Φ(0)dηφ(η)≤ t∗ − t0 ≤ t∗which is the desired lower bound (2.4).3. Upper bound. We seek in this section an upper bound for the blowup time t∗ by defining the following auxiliary function(3.1) χ(t) := χ1(t) + χ2(t) = K1(t)1q−1∫Ωuϕ1dx+K2(t)1q−1∫Ωvϕ1dx,where ϕ1 is the first eigenfunction of the following membrane problem(3.2)∆ϕ1 + λ1ϕ1 = 0, ϕ1 > 0, in Ω,∂ϕ1∂ν= −βϕ1, on ∂Ωwith(3.3)∫Ωϕ1dx = 1Remarks on blow up time 233with β in (1.1) satisfying (1.2). We prove the followingTheorem 3.1. Let (u, v) be the solution of (1.1), in a bounded domainΩ in RN . Let K(t) = min{K1(t),K2(t)}. Assume that(3.4) fi(s) ≥ sq q > 1, i = 1, 2,and(3.5)K ′iKi≥ γ˜,with γ˜ a positive constant. Assume moreover that there exists a constant γ¯ with0 < γ¯ ≤ 1 such that(3.6)KKi≥ γ¯.If (u,v) becomes unbounded in χ measure at some finite time t∗, then t∗is bounded from above by T with(3.7) T =∫∞χ(0)dηγ2ηq − γ1η ,whereγ1 = λ1 − γ˜q − 1 , γ2 = 21−q γ¯qq−1 ,provided(3.8) γ2χq−1(0) > γ1.P r o o f. By derivative of (3.1) we have(3.9) χ′(t) =1q − 1(K ′1K1)K1q−11∫Ωuϕ1dx+1q − 1(K ′2K2)K1q−12∫Ωvϕ1dx+K1q−11∫Ωutϕ1dx+K1q−12∫Ωvtϕ1dx.By applying in (3.9) the condition (3.5) we obtain(3.10) χ′(t) ≥ γ˜q − 1χ+K1q−11∫Ω∆uϕ1dx+K1q−12∫Ω∆vϕ1dx+Kqq−11∫Ωf1ϕ1dx+Kqq−12∫Ωf2ϕ1dx,234 M. MarrasWe compute∫Ω∆uϕ1dx =∫Ωdiv[∇(uϕ1)]dx−∫Ωu∆ϕ1dx− 2∫Ω∇u∇ϕ1dx.By the equation in (3.2), the divergence theorem and boundary conditionin (3.2) we obtain(3.11)∫Ω∆uϕ1dx = (β − α)∫∂Ωuϕ1ds− λ1∫Ωuϕ1dx ≥ −λ1∫Ωuϕ1dxanalogously, we obtain(3.12)∫Ω∆vϕ1dx = −λ1∫Ωvϕ1dx.By inserting (3.11) and (3.12) in (3.10) we have(3.13) χ′(t) ≥ γ˜q − 1χ− λ1[K1q−11∫Ωuϕ1dx+K1q−12∫Ωvϕ1dx]+Kqq−11∫Ωf1ϕ1dx+Kqq−12∫Ωf2ϕ1dx= −(λ1 − γ˜q − 1)χ+Kqq−11∫Ωvqϕ1dx+Kqq−12∫Ωuqϕ1dx.In the last two terms of (3.13) we used (3.4), and by expression of K(t) we obtain(3.14) χ′(t) ≥ −γ1χ+Kqq−1∫Ω(uq + vq)ϕ1dx,with γ1 = λ1 − γ˜q−1 .To estimate∫Ω(uq + vq)ϕ1dx we use Ho¨lder inequality and (3.3)(3.15)∫Ωuqϕ1dx ≥(∫Ωuϕ1dx)q∫Ωvqϕ1dx ≥(∫Ωvϕ1dx)qBy substitution of (3.15) in (3.14) we find(3.16) χ′(t) ≥ −γ1χ+Kqq−1{(∫Ωuϕ1dx)q+(∫Ωvϕ1dx)q}Remarks on blow up time 235= −γ1χ+(KK1) qq−1χq1 +(KK2) qq−1χq2.By inserting in the last term of (3.16) the assumption (3.6), we obtain(3.17) χ′(t) ≥ −γ1χ+ γ¯qq−1 (χq1 + χq2).Using in (3.17) the inequality χq1 + χq2 ≥ 21−q(χ1 + χ2)q we obtain(3.18) χ′(t) ≥ −γ1χ+ γ2χq.with γ2 = 21−qγ¯qq−1 .Since (3.8) holds we conclude that χ(t) is an increasing function of t ∈(0, t∗) and that by integration of (3.18) from 0 to t we find(3.19)∫ χ(t)χ(0)dηγ2ηq − γ1η ≥ t.and we obtain the upper bound for t∗:t∗ ≤∫∞χ(0)dηη(γ2ηq−1 − γ1) .Acknowledgments. The author thanks G.Philippin and S.VernierPiro for useful discussions on the subject.REFERENCES[1] J. M. Ball. Remarks on blow-up and nonexistence theorems for nonlinearevolution equations. Quart. J. Math. Oxford 28 (1977), 473–486.[2] H. Kielho¨fer. Halbgruppen und semilineare Anfangs-randwert-probleme.Manuscripta Math. 12 (1974), 121–152.[3] M. Marras. Bounds for blow-up time in nonlinear parabolic systems undervarious boundary conditions. Numer. Funct. Anal. Optim. 32, (2011), 453–468.236 M. Marras[4] M. Marras, S. Vernier Piro. Blow-up phenomena in reaction-diffusionsystems. In: Special Issue of Discrete and Continuous Dynamical Systems onNonlinear Elliptic and Parabolic Problems, 2011 (to appear).[5] L. E. Payne, G. A. Philippin, P. W. Schaefer. Bounds for blow-up timein nonlinear parabolic problems. J. Math. Anal. Appl. 338 (2008), 438–447.[6] L. E. Payne, G. A. Philippin, P. W. Schaefer. Blow-up phenomena forsome nonlinear parabolic problems. Nonlinear Anal. 69 (2008), 3495–3502.[7] L. E. Payne, G. A. Philippin, S. Vernier-Piro. Blow-up phenomena fora semilinear heat equation with nonlinear boundary condition I. Z. Angew.Math. Phys. 61 (2010), 971–978.[8] L. E. Payne, G. A. Philippin, S. Vernier-Piro. Blow-up phenomena fora semilinear heat equation with nonlinear boundary condition II. NonlinearAnal. 73 (2010), 971–978.[9] L. E. Payne, P. W. Schaefer. Blow-up phenomena for some nonlinearparabolic systems. Int. J. Pure Appl. Math. 42 (2008), 193–202.[10] P. Quittner, P. Souplet. Superlinear Parabolic Problems. Blow-up,Global Existence and Steady States, Birkha¨user Advanced Texts, Basel,2007.[11] B. Straughan. Explosive Instabilities in Mechanics. Springer, Berlin, 1998.[12] J. L. Vazquez. The problems of blow-up for nonlinear heat equations. Com-plete blow-up and avalanche formation. Rend. Mat. Acc. Lincei ser. 9 15(2004), 281–300.[13] F. B. Weissler. Local existence and nonexistence for semilinear parabolicequations in Lp. Indiana Univ. Math. J. 29 (1980), 79–102.[14] F. B. Weissler. Existence and nonexistence of global solutions for a heatequation. Israel J. Math. 38, 1–2 (1981), 29–40.Dipartimento di Matematica e InformaticaUniversita’ di CagliariItalye-mail: mmarras@unica.it Received July 4, 2011

Remarks on blow up time for solutions of a nonlinear diffusion system with time dependent coefficients

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Remarks on blow up time for solutions of a nonlinear diffusion system with time dependent coefficients

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