Research ArticleBlowup of Solution for a Class of DoublyNonlinear Parabolic Systems
Jun Lu1 Qingying Hu2 and Hongwei Zhang2
1 Department of Mathematics Zhengzhou Normal University Zhengzhou 450044 China2Department of Mathematics Henan University of Technology Zhengzhou 450001 China
Correspondence should be addressed to Hongwei Zhang whz661163com
Received 5 August 2014 Accepted 3 November 2014 Published 18 November 2014
Academic Editor Ajda Fosner
Copyright copy 2014 Jun Lu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
An initial boundary value problem for a class of doubly parabolic equations is studied We obtain sufficient conditions for theblowup of solutions under suitable initial data using differential inequalities
1 Introduction
In this paper we study the following initial boundary valueproblem of a class of reaction diffusion equations withmultiple nonlinearities
where 119896 gt 2 119901 gt 2 are real numbers and Ω isbounded domain in 119877
119899 with smooth boundary 120597Ω so thatthe divergence theorem can be applied Here Δ denotes theLaplace operator inΩ
This type of problems not only is important from thetheoretical point of view but also arises in many physicalapplications and describes a great deal of models in appliedscience It appears in the models of chemical reactions heattransfer and population dynamics (see [1] and referencestherein) Equation (1) can describe an electric breakdownin crystalline semiconductors with allowance for the lineardissipation of bound- and free-charge sources [2 3]
In the absence of the nonlinear diffusion term |119906|119896minus2
119906119905 (1)
reduced to the following equation
119906119905minus Δ119906 minus Δ119906
119905= |119906|119901minus2
119906 (4)
which is called pseudoparabolic equation (see [4] and thereferences) A related problem to (4) without term minusΔ119906
119905has
attracted a great deal of attention in the last two decadesand many results appeared on the existence blowup andasymptotic behavior of solution It is well known that thenonlinear |119906|119901minus2119906 reaction term drives the solution of (4) toblowup in finite time The diffusion term is known to yieldexistence of global solution if the reaction term is removedfrom [5] The more general equation
119906119905minus div (|nabla119906|119898minus2 nabla119906) = 119891 (119906) (5)
has also attracted a great deal of people and the known resultsshow that global existence and nonexistence depend roughlyon 119898 the degree of nonlinearity in 119891 the dimension 119899 andthe size of the initial data See in this regard the worksof Levine [6] Kalantarov and Ladyzhenskaya [7] Levineet al [8] Messaoudi [9] Liu and Wang [10] and referencestherein Pucci and Serrin [11] have discussed the stability ofthe following equation
1003816100381610038161003816119906119905
1003816100381610038161003816
119897minus2
119906119905minus div (|nabla119906|119898minus2 nabla119906) = 119891 (119906) (6)
Levine et al [8] got the global existence and nonexistence ofsolution for (6) Pang et al [12 13] andBerrimi andMessaoudi[14] gave the sufficient condition of blowup result for certainsolutions of (6) with positive or negative initial energy
Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 924596 5 pageshttpdxdoiorg1011552014924596
2 Journal of Function Spaces
Equation (1) without term minusΔ119906119905can also be a special case
of doubly nonlinear parabolic-type equations (or the porousmedium equation) [8 15 16]
120573 (119906)119905minus Δ119906 = |119906|
119901minus2119906 (7)
if we take 120573(119906) = 119906 + |119906|119898minus2
119906 The authors of [15 16] took(7) as dynamical systems and studied their attractors Levineand Sacks [17 18] and Blanchard and Francfort [19] provedthe existence of the solution
Polat [20] established a blowup result for the solutionwith vanishing initial energy of the following initial boundaryvalue problem
They also gave detailed results of the necessary and sufficientblowup conditions together with blowup rate estimates forthe positive solution of the problem
subject to various boundary conditions Korpusov and Svesh-nikov [2 3] gave the local strong solution and the sufficientclose-to-necessary conditions for the blowup of solutions tothe problem
119896minus2119906119905= 119906 (119906 + 120572) (119906 minus 120573) (10)
with initial boundary values (2) and (3) in 1198773 for 120572 120573 gt 0 bythe convex method [6 7]
In this paper we will investigate the problem (1)ndash(3) andthere are few results of the problem to our knowledge Wewill give sufficient conditions for the blowup of solutionsin a finite time interval under suitable initial data usingdifferential inequalities An essential tool of the proof is anidea used in [21 22] whichwas based on an auxiliary function(which is a small perturbation of the total energy) usingdifferential inequalities and obtaining the result It is differentwith the result of [2 3] This paper is organized as followsSection 2 is concerned with some notations and statement ofassumptions In Section 3 we give and prove that the result ifthe initial energy119864(0) of our solutions is negative (this meansthat our initial data are large enough) or the initial energy is119864(0) gt 0
2 Preliminaries
In this section we will give some notations and statement ofassumptions for 119898 119901 119892 We denote 119871119901(Ω) by 119871119901 1198671
0(Ω) by
1198671
0 the usual Soblev space The norm and inner of 119871119901(Ω) are
denoted by sdot 119901= sdot
119871119901(Ω)
and (119906 V) = intΩ119906(119909)V(119909)119889119909
respectively Particularly sdot = sdot 1198712(Ω)
for 119901 = 2For the numbers 119896 and 119901 we assume that
Since 119901 gt 119896 gt 2 and by embedding theorem taking intoaccount (22) we obtain for some positive constants 119862
1and
1198622
119867120590(119905) 119906
119896
119896le 1198621119906119901120590
119901119906119896
119896le 1198622119906119901120590+119896
119901 (29)
Since 0 lt 119896119901 lt 1 now applying the inequality 119909119897 le (119909+1) le(1 + (1119911))(119909 + 119911) which holds for all 119909 ge 0 0 le 119897 le 1119911 gt 0 in particular by the choice of 120590 taking 119909 = 119906
119901
119901
119897 = (119901120590 + 119896)119901 119911 = 119867(0) and by using (22) we have
119906119901120590+119896
119901le (1 +
1
119867 (0)
) (119906119901
119901+ 119867 (0)) le 119862
3119906119901
119901 (30)
Taking into account (28) and (30) we have
1198711015840(119905) ge (1 minus 120590 minus119872120598)119867
For large119872 such that 1minus (2119901)minus1198623119872minus1= 1198624gt 0 once119872 is
fixed we pick 120598 small enough such that 1 minus 120590 minus119872120598 gt 0 thenthere exist 119862
5gt 0 such that (31) become
1198711015840(119905) ge 119862
5(119867 (119905) + 119906
119901
119901) (32)
Then we have
119871 (119905) ge 119871 (0) ge 0 (33)
On the other hand by the definition of 119871(119905) and (21) we have
119871 (119905) = 1198671minus120590
(119905) minus 120598 (119867 (119905) minus
1
119901
119906119901
119901) +
120598
2
1199062
le (1 minus 120598)1198671minus120590
(119905) +
120598
119901
119906119901
119901+
120598
2
1199062
(34)
where we have used the fact 119867(119905) ge 1198671minus120590
(119905) (this can beensured by (19) (20) 0 lt 120590 lt 1 and 119864(0) is sufficientnegative) Now by inequality 119909119897 le (1 + (1119911))(119909 + 119911) againby taking 119909 = 119906
119871 (119905) le (1 minus 120598)1198671minus120590
(119905) + 1198626119906119901(1minus120590)
119901+
120598
2
1199062 (36)
Then by embedding theorem since 119901 gt 2 we have for fixed120598 sufficient small
1198711(1minus120590)
(119905) le 1198627[119867 (119905) + 119906
119901
119901+ 1199062(1minus120590)
119901] (37)
Now by inequality 119909119897 le (1 + (1119911))(119909 + 119911) again by taking119909 = 119906
119901
119901 119897 = 2119901(1 minus 120590) lt 1 since 120590 lt (119901 minus 119896)119901 lt (119901 minus
2)119901 119911 = 119867(0) we have
1199062(1minus120590)
119901= (119906
119901
119901)
2119901(1minus120590)
le (1 +
1
119867 (0)
) (119906119901
119901+ 119867 (0))
le 1198628119906119901
119901
(38)
4 Journal of Function Spaces
From (37) and (38) we obtain
1198711(1minus120590)
(119905) le 1198629[119867 (119905) + 119906
119901
119901] (39)
Combining with (32) and (39) we arrive to
1198711015840(119905) ge 119862
101198711(1minus120590)
(119905) (40)
Integration of (40) between 0 and 119905 gives the desired resultsIn the following we will prove that the energy will
grow up as an exponential function as time goes to infinityprovided that the initial energy 119864(0) gt 0
The following lemma will play an essential role in theproof of our main result and it is similar to a Lemma usedfirstly by Vitillaro [23] In order to give the result and for thesake of simplicity we set
1205821= 119862minus119901(119901minus2)
lowast 119864
1= (
1
2
minus
1
119901
)1205822
1 (41)
where 119862lowastis the best Poincarersquos constant
Lemma 2 (see [22]) Let 119906 be a solution of (1)ndash(3) Supposethat the assumption of 119896 119901 hold Assume further that119864(0) lt 119864
1
and nabla1199060 gt 1205821Then there exists a constant 120582
2gt 1205821such that
nabla119906 gt 1205822
Theorem3 Suppose that the assumption about 119896 119901 hold 1199060isin
1198671
0and 119906 is a local solution of the system (1)ndash(3) nabla119906
0 gt 120582
1
and 119864(0) lt 1198641 Then the solution of the system (1)ndash(3) blows
where 1198642is a constant and 119864(0) lt 119864
2lt 1198641 By the definition
of119867(119905) and (15)
1198671015840(119905) = minus119864
1015840(119905) ge 0 (43)
Consequently
119867(0) = 1198642minus 119864 (0) gt 0 (44)
It is clear that by (43) and (44)
0 lt 119867 (0) le 119867 (119905) (45)
By (42) the expression of 119864(119905) and Lemma 2
119867(119905) = 1198642minus
1
2
nabla1199062+
1
119901
119906119901
119901
le 1198641minus
1
2
1205822
1+
1
119901
119906119901
119901
= minus
1
119901
1205822
1+
1
119901
119906119901
119901lt
1
119901
119906119901
119901
(46)
One implies
0 lt 119867 (0) le 119867 (119905) le
1
119901
119906119901
119901 (47)
Then we can prove the theorem similar to the proof ofTheorem 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Foun-dation of China (no 11171311) and partially by the NaturalScience Foundation of Henan Province (1323004100360)
References
[1] Z Jiang S Zheng and X Song ldquoBlow-up analysis for a non-linear diffusion equation with nonlinear boundary conditionsrdquoApplied Mathematics Letters vol 17 no 2 pp 193ndash199 2004
[2] M O Korpusov and A G Sveshnikov ldquoSufficient close-to-necessary conditions for the blowup of solutions to a stronglynonlinear generalized Boussinesq equationrdquo ComputationalMathematics and Mathematical Physics vol 48 no 9 pp 1591ndash1599 2008
[3] A B Alrsquoshin M O Korpusov and A G Sveshnikov Blowup inNonlinear Soblev Type Equation De Gruyter Berlin Germany2011
[4] C Yang Y Cao and S Zheng ldquoSecond critical exponent andlife span for pseudo-parabolic equationrdquo Journal of DifferentialEquations vol 253 no 12 pp 3286ndash3303 2012
[5] K Deng and H A Levine ldquoThe role of critical exponents inblow-up theorems the sequelrdquo Journal ofMathematical Analysisand Applications vol 243 no 1 pp 85ndash126 2000
[6] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119865(119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[7] V K Kalantarov and O A Ladyzhenskaya ldquoThe occurrence ofcollapse for quasilinear equations of parabolic and hyperbolictypesrdquo Journal of Soviet Mathematics vol 10 no 1 pp 53ndash701978
[8] H A Levine S R Park and J Serrin ldquoGlobal existence andnonexistence theorems for quasilinear evolution equations offormally parabolic typerdquo Journal of Differential Equations vol142 no 1 pp 212ndash229 1998
[9] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[10] W Liu andMWang ldquoBlow-up of the solution for a119901-Laplacianequation with positive initial energyrdquo Acta Applicandae Mathe-maticae vol 103 no 2 pp 141ndash146 2008
[11] P Pucci and J Serrin ldquoAsymptotic stability for nonlinearparabolic systemsrdquo in EnergyMethods in ContinuumMechanicsKluwer Academic Publishers Dordrecht The Netherlands1996
[12] J-S Pang and H-W Zhang ldquoExistence and nonexistence of theglobal solution on the quasilinear parabolic equationrdquo ChineseQuarterly Journal of Mathematics vol 22 no 3 pp 444ndash4502007
[13] J S Pang and Q Y Hu ldquoGlobal nonexistence for a class ofquasilinear parabolic equation with source term and positiveinitial energyrdquo Journal of Henan University (Natural Science)vol 37 no 5 pp 448ndash451 2007 (Chinese)
Journal of Function Spaces 5
[14] S Berrimi and S A Messaoudi ldquoA decay result for a quasilinearparabolic systemrdquo in Elliptic and Parabolic Problems vol 63 ofProgress in Nonlinear Differential Equations and Their Applica-tions pp 43ndash50 2005
[15] A Eden B Michaux and J-M Rakotoson ldquoDoubly nonlinearparabolic-type equations as dynamical systemsrdquo Journal ofDynamics and Differential Equations vol 3 no 1 pp 87ndash1311991
[16] H E Ouardi and A E Hachimi ldquoAttractors for a class of doublynonlinear parabolic systemsrdquo Electronic Journal of QualitativeDifferential Equations vol 2006 no 1 pp 1ndash15 2006
[17] H A Levine and P E Sacks ldquoSome existence and nonexistencetheorems for solutions of degenerate parabolic equationsrdquoJournal of Differential Equations vol 52 no 2 pp 135ndash161 1984
[18] P E Sacks ldquoContinuity of solutions of a singular parabolicequationrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 7 no 4 pp 387ndash409 1983
[19] D Blanchard and G A Francfort ldquoStudy of a doubly nonlinearheat equation with no growth assumptions on the parabolictermrdquo SIAM Journal onMathematical Analysis vol 19 no 5 pp1032ndash1056 1988
[20] N Polat ldquoBlow up of solution for a nonlinear reaction diffusionequation with multiple nonlinearitiesrdquo International Journal ofScience and Technology vol 2 no 2 pp 123ndash128 2007
[21] S Gerbi and B Said-Houari ldquoLocal existence and exponentialgrowth for a semilinear damped wave equation with dynamicboundary conditionsrdquo Advances in Differential Equations vol13 no 11-12 pp 1051ndash1074 2008
[22] B Said-Houari ldquoGlobal nonexistence of positive initial-energysolutions of a system of nonlinear wave equations with dampingand source termsrdquo Differential and Integral Equations vol 23no 1-2 pp 79ndash92 2010
[23] E Vitillaro ldquoGlobal nonexistence theorems for a class of evolu-tion equations with dissipationrdquo Archive for Rational Mechanicsand Analysis vol 149 no 2 pp 155ndash182 1999
Equation (1) without term minusΔ119906119905can also be a special case
of doubly nonlinear parabolic-type equations (or the porousmedium equation) [8 15 16]
120573 (119906)119905minus Δ119906 = |119906|
119901minus2119906 (7)
if we take 120573(119906) = 119906 + |119906|119898minus2
119906 The authors of [15 16] took(7) as dynamical systems and studied their attractors Levineand Sacks [17 18] and Blanchard and Francfort [19] provedthe existence of the solution
Polat [20] established a blowup result for the solutionwith vanishing initial energy of the following initial boundaryvalue problem
They also gave detailed results of the necessary and sufficientblowup conditions together with blowup rate estimates forthe positive solution of the problem
subject to various boundary conditions Korpusov and Svesh-nikov [2 3] gave the local strong solution and the sufficientclose-to-necessary conditions for the blowup of solutions tothe problem
119896minus2119906119905= 119906 (119906 + 120572) (119906 minus 120573) (10)
with initial boundary values (2) and (3) in 1198773 for 120572 120573 gt 0 bythe convex method [6 7]
In this paper we will investigate the problem (1)ndash(3) andthere are few results of the problem to our knowledge Wewill give sufficient conditions for the blowup of solutionsin a finite time interval under suitable initial data usingdifferential inequalities An essential tool of the proof is anidea used in [21 22] whichwas based on an auxiliary function(which is a small perturbation of the total energy) usingdifferential inequalities and obtaining the result It is differentwith the result of [2 3] This paper is organized as followsSection 2 is concerned with some notations and statement ofassumptions In Section 3 we give and prove that the result ifthe initial energy119864(0) of our solutions is negative (this meansthat our initial data are large enough) or the initial energy is119864(0) gt 0
2 Preliminaries
In this section we will give some notations and statement ofassumptions for 119898 119901 119892 We denote 119871119901(Ω) by 119871119901 1198671
0(Ω) by
1198671
0 the usual Soblev space The norm and inner of 119871119901(Ω) are
denoted by sdot 119901= sdot
119871119901(Ω)
and (119906 V) = intΩ119906(119909)V(119909)119889119909
respectively Particularly sdot = sdot 1198712(Ω)
for 119901 = 2For the numbers 119896 and 119901 we assume that
Since 119901 gt 119896 gt 2 and by embedding theorem taking intoaccount (22) we obtain for some positive constants 119862
1and
1198622
119867120590(119905) 119906
119896
119896le 1198621119906119901120590
119901119906119896
119896le 1198622119906119901120590+119896
119901 (29)
Since 0 lt 119896119901 lt 1 now applying the inequality 119909119897 le (119909+1) le(1 + (1119911))(119909 + 119911) which holds for all 119909 ge 0 0 le 119897 le 1119911 gt 0 in particular by the choice of 120590 taking 119909 = 119906
119901
119901
119897 = (119901120590 + 119896)119901 119911 = 119867(0) and by using (22) we have
119906119901120590+119896
119901le (1 +
1
119867 (0)
) (119906119901
119901+ 119867 (0)) le 119862
3119906119901
119901 (30)
Taking into account (28) and (30) we have
1198711015840(119905) ge (1 minus 120590 minus119872120598)119867
For large119872 such that 1minus (2119901)minus1198623119872minus1= 1198624gt 0 once119872 is
fixed we pick 120598 small enough such that 1 minus 120590 minus119872120598 gt 0 thenthere exist 119862
5gt 0 such that (31) become
1198711015840(119905) ge 119862
5(119867 (119905) + 119906
119901
119901) (32)
Then we have
119871 (119905) ge 119871 (0) ge 0 (33)
On the other hand by the definition of 119871(119905) and (21) we have
119871 (119905) = 1198671minus120590
(119905) minus 120598 (119867 (119905) minus
1
119901
119906119901
119901) +
120598
2
1199062
le (1 minus 120598)1198671minus120590
(119905) +
120598
119901
119906119901
119901+
120598
2
1199062
(34)
where we have used the fact 119867(119905) ge 1198671minus120590
(119905) (this can beensured by (19) (20) 0 lt 120590 lt 1 and 119864(0) is sufficientnegative) Now by inequality 119909119897 le (1 + (1119911))(119909 + 119911) againby taking 119909 = 119906
119871 (119905) le (1 minus 120598)1198671minus120590
(119905) + 1198626119906119901(1minus120590)
119901+
120598
2
1199062 (36)
Then by embedding theorem since 119901 gt 2 we have for fixed120598 sufficient small
1198711(1minus120590)
(119905) le 1198627[119867 (119905) + 119906
119901
119901+ 1199062(1minus120590)
119901] (37)
Now by inequality 119909119897 le (1 + (1119911))(119909 + 119911) again by taking119909 = 119906
119901
119901 119897 = 2119901(1 minus 120590) lt 1 since 120590 lt (119901 minus 119896)119901 lt (119901 minus
2)119901 119911 = 119867(0) we have
1199062(1minus120590)
119901= (119906
119901
119901)
2119901(1minus120590)
le (1 +
1
119867 (0)
) (119906119901
119901+ 119867 (0))
le 1198628119906119901
119901
(38)
4 Journal of Function Spaces
From (37) and (38) we obtain
1198711(1minus120590)
(119905) le 1198629[119867 (119905) + 119906
119901
119901] (39)
Combining with (32) and (39) we arrive to
1198711015840(119905) ge 119862
101198711(1minus120590)
(119905) (40)
Integration of (40) between 0 and 119905 gives the desired resultsIn the following we will prove that the energy will
grow up as an exponential function as time goes to infinityprovided that the initial energy 119864(0) gt 0
The following lemma will play an essential role in theproof of our main result and it is similar to a Lemma usedfirstly by Vitillaro [23] In order to give the result and for thesake of simplicity we set
1205821= 119862minus119901(119901minus2)
lowast 119864
1= (
1
2
minus
1
119901
)1205822
1 (41)
where 119862lowastis the best Poincarersquos constant
Lemma 2 (see [22]) Let 119906 be a solution of (1)ndash(3) Supposethat the assumption of 119896 119901 hold Assume further that119864(0) lt 119864
1
and nabla1199060 gt 1205821Then there exists a constant 120582
2gt 1205821such that
nabla119906 gt 1205822
Theorem3 Suppose that the assumption about 119896 119901 hold 1199060isin
1198671
0and 119906 is a local solution of the system (1)ndash(3) nabla119906
0 gt 120582
1
and 119864(0) lt 1198641 Then the solution of the system (1)ndash(3) blows
where 1198642is a constant and 119864(0) lt 119864
2lt 1198641 By the definition
of119867(119905) and (15)
1198671015840(119905) = minus119864
1015840(119905) ge 0 (43)
Consequently
119867(0) = 1198642minus 119864 (0) gt 0 (44)
It is clear that by (43) and (44)
0 lt 119867 (0) le 119867 (119905) (45)
By (42) the expression of 119864(119905) and Lemma 2
119867(119905) = 1198642minus
1
2
nabla1199062+
1
119901
119906119901
119901
le 1198641minus
1
2
1205822
1+
1
119901
119906119901
119901
= minus
1
119901
1205822
1+
1
119901
119906119901
119901lt
1
119901
119906119901
119901
(46)
One implies
0 lt 119867 (0) le 119867 (119905) le
1
119901
119906119901
119901 (47)
Then we can prove the theorem similar to the proof ofTheorem 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Foun-dation of China (no 11171311) and partially by the NaturalScience Foundation of Henan Province (1323004100360)
References
[1] Z Jiang S Zheng and X Song ldquoBlow-up analysis for a non-linear diffusion equation with nonlinear boundary conditionsrdquoApplied Mathematics Letters vol 17 no 2 pp 193ndash199 2004
[2] M O Korpusov and A G Sveshnikov ldquoSufficient close-to-necessary conditions for the blowup of solutions to a stronglynonlinear generalized Boussinesq equationrdquo ComputationalMathematics and Mathematical Physics vol 48 no 9 pp 1591ndash1599 2008
[3] A B Alrsquoshin M O Korpusov and A G Sveshnikov Blowup inNonlinear Soblev Type Equation De Gruyter Berlin Germany2011
[4] C Yang Y Cao and S Zheng ldquoSecond critical exponent andlife span for pseudo-parabolic equationrdquo Journal of DifferentialEquations vol 253 no 12 pp 3286ndash3303 2012
[5] K Deng and H A Levine ldquoThe role of critical exponents inblow-up theorems the sequelrdquo Journal ofMathematical Analysisand Applications vol 243 no 1 pp 85ndash126 2000
[6] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119865(119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[7] V K Kalantarov and O A Ladyzhenskaya ldquoThe occurrence ofcollapse for quasilinear equations of parabolic and hyperbolictypesrdquo Journal of Soviet Mathematics vol 10 no 1 pp 53ndash701978
[8] H A Levine S R Park and J Serrin ldquoGlobal existence andnonexistence theorems for quasilinear evolution equations offormally parabolic typerdquo Journal of Differential Equations vol142 no 1 pp 212ndash229 1998
[9] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[10] W Liu andMWang ldquoBlow-up of the solution for a119901-Laplacianequation with positive initial energyrdquo Acta Applicandae Mathe-maticae vol 103 no 2 pp 141ndash146 2008
[11] P Pucci and J Serrin ldquoAsymptotic stability for nonlinearparabolic systemsrdquo in EnergyMethods in ContinuumMechanicsKluwer Academic Publishers Dordrecht The Netherlands1996
[12] J-S Pang and H-W Zhang ldquoExistence and nonexistence of theglobal solution on the quasilinear parabolic equationrdquo ChineseQuarterly Journal of Mathematics vol 22 no 3 pp 444ndash4502007
[13] J S Pang and Q Y Hu ldquoGlobal nonexistence for a class ofquasilinear parabolic equation with source term and positiveinitial energyrdquo Journal of Henan University (Natural Science)vol 37 no 5 pp 448ndash451 2007 (Chinese)
Journal of Function Spaces 5
[14] S Berrimi and S A Messaoudi ldquoA decay result for a quasilinearparabolic systemrdquo in Elliptic and Parabolic Problems vol 63 ofProgress in Nonlinear Differential Equations and Their Applica-tions pp 43ndash50 2005
[15] A Eden B Michaux and J-M Rakotoson ldquoDoubly nonlinearparabolic-type equations as dynamical systemsrdquo Journal ofDynamics and Differential Equations vol 3 no 1 pp 87ndash1311991
[16] H E Ouardi and A E Hachimi ldquoAttractors for a class of doublynonlinear parabolic systemsrdquo Electronic Journal of QualitativeDifferential Equations vol 2006 no 1 pp 1ndash15 2006
[17] H A Levine and P E Sacks ldquoSome existence and nonexistencetheorems for solutions of degenerate parabolic equationsrdquoJournal of Differential Equations vol 52 no 2 pp 135ndash161 1984
[18] P E Sacks ldquoContinuity of solutions of a singular parabolicequationrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 7 no 4 pp 387ndash409 1983
[19] D Blanchard and G A Francfort ldquoStudy of a doubly nonlinearheat equation with no growth assumptions on the parabolictermrdquo SIAM Journal onMathematical Analysis vol 19 no 5 pp1032ndash1056 1988
[20] N Polat ldquoBlow up of solution for a nonlinear reaction diffusionequation with multiple nonlinearitiesrdquo International Journal ofScience and Technology vol 2 no 2 pp 123ndash128 2007
[21] S Gerbi and B Said-Houari ldquoLocal existence and exponentialgrowth for a semilinear damped wave equation with dynamicboundary conditionsrdquo Advances in Differential Equations vol13 no 11-12 pp 1051ndash1074 2008
[22] B Said-Houari ldquoGlobal nonexistence of positive initial-energysolutions of a system of nonlinear wave equations with dampingand source termsrdquo Differential and Integral Equations vol 23no 1-2 pp 79ndash92 2010
[23] E Vitillaro ldquoGlobal nonexistence theorems for a class of evolu-tion equations with dissipationrdquo Archive for Rational Mechanicsand Analysis vol 149 no 2 pp 155ndash182 1999
Since 119901 gt 119896 gt 2 and by embedding theorem taking intoaccount (22) we obtain for some positive constants 119862
1and
1198622
119867120590(119905) 119906
119896
119896le 1198621119906119901120590
119901119906119896
119896le 1198622119906119901120590+119896
119901 (29)
Since 0 lt 119896119901 lt 1 now applying the inequality 119909119897 le (119909+1) le(1 + (1119911))(119909 + 119911) which holds for all 119909 ge 0 0 le 119897 le 1119911 gt 0 in particular by the choice of 120590 taking 119909 = 119906
119901
119901
119897 = (119901120590 + 119896)119901 119911 = 119867(0) and by using (22) we have
119906119901120590+119896
119901le (1 +
1
119867 (0)
) (119906119901
119901+ 119867 (0)) le 119862
3119906119901
119901 (30)
Taking into account (28) and (30) we have
1198711015840(119905) ge (1 minus 120590 minus119872120598)119867
For large119872 such that 1minus (2119901)minus1198623119872minus1= 1198624gt 0 once119872 is
fixed we pick 120598 small enough such that 1 minus 120590 minus119872120598 gt 0 thenthere exist 119862
5gt 0 such that (31) become
1198711015840(119905) ge 119862
5(119867 (119905) + 119906
119901
119901) (32)
Then we have
119871 (119905) ge 119871 (0) ge 0 (33)
On the other hand by the definition of 119871(119905) and (21) we have
119871 (119905) = 1198671minus120590
(119905) minus 120598 (119867 (119905) minus
1
119901
119906119901
119901) +
120598
2
1199062
le (1 minus 120598)1198671minus120590
(119905) +
120598
119901
119906119901
119901+
120598
2
1199062
(34)
where we have used the fact 119867(119905) ge 1198671minus120590
(119905) (this can beensured by (19) (20) 0 lt 120590 lt 1 and 119864(0) is sufficientnegative) Now by inequality 119909119897 le (1 + (1119911))(119909 + 119911) againby taking 119909 = 119906
119871 (119905) le (1 minus 120598)1198671minus120590
(119905) + 1198626119906119901(1minus120590)
119901+
120598
2
1199062 (36)
Then by embedding theorem since 119901 gt 2 we have for fixed120598 sufficient small
1198711(1minus120590)
(119905) le 1198627[119867 (119905) + 119906
119901
119901+ 1199062(1minus120590)
119901] (37)
Now by inequality 119909119897 le (1 + (1119911))(119909 + 119911) again by taking119909 = 119906
119901
119901 119897 = 2119901(1 minus 120590) lt 1 since 120590 lt (119901 minus 119896)119901 lt (119901 minus
2)119901 119911 = 119867(0) we have
1199062(1minus120590)
119901= (119906
119901
119901)
2119901(1minus120590)
le (1 +
1
119867 (0)
) (119906119901
119901+ 119867 (0))
le 1198628119906119901
119901
(38)
4 Journal of Function Spaces
From (37) and (38) we obtain
1198711(1minus120590)
(119905) le 1198629[119867 (119905) + 119906
119901
119901] (39)
Combining with (32) and (39) we arrive to
1198711015840(119905) ge 119862
101198711(1minus120590)
(119905) (40)
Integration of (40) between 0 and 119905 gives the desired resultsIn the following we will prove that the energy will
grow up as an exponential function as time goes to infinityprovided that the initial energy 119864(0) gt 0
The following lemma will play an essential role in theproof of our main result and it is similar to a Lemma usedfirstly by Vitillaro [23] In order to give the result and for thesake of simplicity we set
1205821= 119862minus119901(119901minus2)
lowast 119864
1= (
1
2
minus
1
119901
)1205822
1 (41)
where 119862lowastis the best Poincarersquos constant
Lemma 2 (see [22]) Let 119906 be a solution of (1)ndash(3) Supposethat the assumption of 119896 119901 hold Assume further that119864(0) lt 119864
1
and nabla1199060 gt 1205821Then there exists a constant 120582
2gt 1205821such that
nabla119906 gt 1205822
Theorem3 Suppose that the assumption about 119896 119901 hold 1199060isin
1198671
0and 119906 is a local solution of the system (1)ndash(3) nabla119906
0 gt 120582
1
and 119864(0) lt 1198641 Then the solution of the system (1)ndash(3) blows
where 1198642is a constant and 119864(0) lt 119864
2lt 1198641 By the definition
of119867(119905) and (15)
1198671015840(119905) = minus119864
1015840(119905) ge 0 (43)
Consequently
119867(0) = 1198642minus 119864 (0) gt 0 (44)
It is clear that by (43) and (44)
0 lt 119867 (0) le 119867 (119905) (45)
By (42) the expression of 119864(119905) and Lemma 2
119867(119905) = 1198642minus
1
2
nabla1199062+
1
119901
119906119901
119901
le 1198641minus
1
2
1205822
1+
1
119901
119906119901
119901
= minus
1
119901
1205822
1+
1
119901
119906119901
119901lt
1
119901
119906119901
119901
(46)
One implies
0 lt 119867 (0) le 119867 (119905) le
1
119901
119906119901
119901 (47)
Then we can prove the theorem similar to the proof ofTheorem 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Foun-dation of China (no 11171311) and partially by the NaturalScience Foundation of Henan Province (1323004100360)
References
[1] Z Jiang S Zheng and X Song ldquoBlow-up analysis for a non-linear diffusion equation with nonlinear boundary conditionsrdquoApplied Mathematics Letters vol 17 no 2 pp 193ndash199 2004
[2] M O Korpusov and A G Sveshnikov ldquoSufficient close-to-necessary conditions for the blowup of solutions to a stronglynonlinear generalized Boussinesq equationrdquo ComputationalMathematics and Mathematical Physics vol 48 no 9 pp 1591ndash1599 2008
[3] A B Alrsquoshin M O Korpusov and A G Sveshnikov Blowup inNonlinear Soblev Type Equation De Gruyter Berlin Germany2011
[4] C Yang Y Cao and S Zheng ldquoSecond critical exponent andlife span for pseudo-parabolic equationrdquo Journal of DifferentialEquations vol 253 no 12 pp 3286ndash3303 2012
[5] K Deng and H A Levine ldquoThe role of critical exponents inblow-up theorems the sequelrdquo Journal ofMathematical Analysisand Applications vol 243 no 1 pp 85ndash126 2000
[6] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119865(119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[7] V K Kalantarov and O A Ladyzhenskaya ldquoThe occurrence ofcollapse for quasilinear equations of parabolic and hyperbolictypesrdquo Journal of Soviet Mathematics vol 10 no 1 pp 53ndash701978
[8] H A Levine S R Park and J Serrin ldquoGlobal existence andnonexistence theorems for quasilinear evolution equations offormally parabolic typerdquo Journal of Differential Equations vol142 no 1 pp 212ndash229 1998
[9] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[10] W Liu andMWang ldquoBlow-up of the solution for a119901-Laplacianequation with positive initial energyrdquo Acta Applicandae Mathe-maticae vol 103 no 2 pp 141ndash146 2008
[11] P Pucci and J Serrin ldquoAsymptotic stability for nonlinearparabolic systemsrdquo in EnergyMethods in ContinuumMechanicsKluwer Academic Publishers Dordrecht The Netherlands1996
[12] J-S Pang and H-W Zhang ldquoExistence and nonexistence of theglobal solution on the quasilinear parabolic equationrdquo ChineseQuarterly Journal of Mathematics vol 22 no 3 pp 444ndash4502007
[13] J S Pang and Q Y Hu ldquoGlobal nonexistence for a class ofquasilinear parabolic equation with source term and positiveinitial energyrdquo Journal of Henan University (Natural Science)vol 37 no 5 pp 448ndash451 2007 (Chinese)
Journal of Function Spaces 5
[14] S Berrimi and S A Messaoudi ldquoA decay result for a quasilinearparabolic systemrdquo in Elliptic and Parabolic Problems vol 63 ofProgress in Nonlinear Differential Equations and Their Applica-tions pp 43ndash50 2005
[15] A Eden B Michaux and J-M Rakotoson ldquoDoubly nonlinearparabolic-type equations as dynamical systemsrdquo Journal ofDynamics and Differential Equations vol 3 no 1 pp 87ndash1311991
[16] H E Ouardi and A E Hachimi ldquoAttractors for a class of doublynonlinear parabolic systemsrdquo Electronic Journal of QualitativeDifferential Equations vol 2006 no 1 pp 1ndash15 2006
[17] H A Levine and P E Sacks ldquoSome existence and nonexistencetheorems for solutions of degenerate parabolic equationsrdquoJournal of Differential Equations vol 52 no 2 pp 135ndash161 1984
[18] P E Sacks ldquoContinuity of solutions of a singular parabolicequationrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 7 no 4 pp 387ndash409 1983
[19] D Blanchard and G A Francfort ldquoStudy of a doubly nonlinearheat equation with no growth assumptions on the parabolictermrdquo SIAM Journal onMathematical Analysis vol 19 no 5 pp1032ndash1056 1988
[20] N Polat ldquoBlow up of solution for a nonlinear reaction diffusionequation with multiple nonlinearitiesrdquo International Journal ofScience and Technology vol 2 no 2 pp 123ndash128 2007
[21] S Gerbi and B Said-Houari ldquoLocal existence and exponentialgrowth for a semilinear damped wave equation with dynamicboundary conditionsrdquo Advances in Differential Equations vol13 no 11-12 pp 1051ndash1074 2008
[22] B Said-Houari ldquoGlobal nonexistence of positive initial-energysolutions of a system of nonlinear wave equations with dampingand source termsrdquo Differential and Integral Equations vol 23no 1-2 pp 79ndash92 2010
[23] E Vitillaro ldquoGlobal nonexistence theorems for a class of evolu-tion equations with dissipationrdquo Archive for Rational Mechanicsand Analysis vol 149 no 2 pp 155ndash182 1999
Integration of (40) between 0 and 119905 gives the desired resultsIn the following we will prove that the energy will
grow up as an exponential function as time goes to infinityprovided that the initial energy 119864(0) gt 0
The following lemma will play an essential role in theproof of our main result and it is similar to a Lemma usedfirstly by Vitillaro [23] In order to give the result and for thesake of simplicity we set
1205821= 119862minus119901(119901minus2)
lowast 119864
1= (
1
2
minus
1
119901
)1205822
1 (41)
where 119862lowastis the best Poincarersquos constant
Lemma 2 (see [22]) Let 119906 be a solution of (1)ndash(3) Supposethat the assumption of 119896 119901 hold Assume further that119864(0) lt 119864
1
and nabla1199060 gt 1205821Then there exists a constant 120582
2gt 1205821such that
nabla119906 gt 1205822
Theorem3 Suppose that the assumption about 119896 119901 hold 1199060isin
1198671
0and 119906 is a local solution of the system (1)ndash(3) nabla119906
0 gt 120582
1
and 119864(0) lt 1198641 Then the solution of the system (1)ndash(3) blows
where 1198642is a constant and 119864(0) lt 119864
2lt 1198641 By the definition
of119867(119905) and (15)
1198671015840(119905) = minus119864
1015840(119905) ge 0 (43)
Consequently
119867(0) = 1198642minus 119864 (0) gt 0 (44)
It is clear that by (43) and (44)
0 lt 119867 (0) le 119867 (119905) (45)
By (42) the expression of 119864(119905) and Lemma 2
119867(119905) = 1198642minus
1
2
nabla1199062+
1
119901
119906119901
119901
le 1198641minus
1
2
1205822
1+
1
119901
119906119901
119901
= minus
1
119901
1205822
1+
1
119901
119906119901
119901lt
1
119901
119906119901
119901
(46)
One implies
0 lt 119867 (0) le 119867 (119905) le
1
119901
119906119901
119901 (47)
Then we can prove the theorem similar to the proof ofTheorem 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Foun-dation of China (no 11171311) and partially by the NaturalScience Foundation of Henan Province (1323004100360)
References
[1] Z Jiang S Zheng and X Song ldquoBlow-up analysis for a non-linear diffusion equation with nonlinear boundary conditionsrdquoApplied Mathematics Letters vol 17 no 2 pp 193ndash199 2004
[2] M O Korpusov and A G Sveshnikov ldquoSufficient close-to-necessary conditions for the blowup of solutions to a stronglynonlinear generalized Boussinesq equationrdquo ComputationalMathematics and Mathematical Physics vol 48 no 9 pp 1591ndash1599 2008
[3] A B Alrsquoshin M O Korpusov and A G Sveshnikov Blowup inNonlinear Soblev Type Equation De Gruyter Berlin Germany2011
[4] C Yang Y Cao and S Zheng ldquoSecond critical exponent andlife span for pseudo-parabolic equationrdquo Journal of DifferentialEquations vol 253 no 12 pp 3286ndash3303 2012
[5] K Deng and H A Levine ldquoThe role of critical exponents inblow-up theorems the sequelrdquo Journal ofMathematical Analysisand Applications vol 243 no 1 pp 85ndash126 2000
[6] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119865(119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[7] V K Kalantarov and O A Ladyzhenskaya ldquoThe occurrence ofcollapse for quasilinear equations of parabolic and hyperbolictypesrdquo Journal of Soviet Mathematics vol 10 no 1 pp 53ndash701978
[8] H A Levine S R Park and J Serrin ldquoGlobal existence andnonexistence theorems for quasilinear evolution equations offormally parabolic typerdquo Journal of Differential Equations vol142 no 1 pp 212ndash229 1998
[9] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[10] W Liu andMWang ldquoBlow-up of the solution for a119901-Laplacianequation with positive initial energyrdquo Acta Applicandae Mathe-maticae vol 103 no 2 pp 141ndash146 2008
[11] P Pucci and J Serrin ldquoAsymptotic stability for nonlinearparabolic systemsrdquo in EnergyMethods in ContinuumMechanicsKluwer Academic Publishers Dordrecht The Netherlands1996
[12] J-S Pang and H-W Zhang ldquoExistence and nonexistence of theglobal solution on the quasilinear parabolic equationrdquo ChineseQuarterly Journal of Mathematics vol 22 no 3 pp 444ndash4502007
[13] J S Pang and Q Y Hu ldquoGlobal nonexistence for a class ofquasilinear parabolic equation with source term and positiveinitial energyrdquo Journal of Henan University (Natural Science)vol 37 no 5 pp 448ndash451 2007 (Chinese)
Journal of Function Spaces 5
[14] S Berrimi and S A Messaoudi ldquoA decay result for a quasilinearparabolic systemrdquo in Elliptic and Parabolic Problems vol 63 ofProgress in Nonlinear Differential Equations and Their Applica-tions pp 43ndash50 2005
[15] A Eden B Michaux and J-M Rakotoson ldquoDoubly nonlinearparabolic-type equations as dynamical systemsrdquo Journal ofDynamics and Differential Equations vol 3 no 1 pp 87ndash1311991
[16] H E Ouardi and A E Hachimi ldquoAttractors for a class of doublynonlinear parabolic systemsrdquo Electronic Journal of QualitativeDifferential Equations vol 2006 no 1 pp 1ndash15 2006
[17] H A Levine and P E Sacks ldquoSome existence and nonexistencetheorems for solutions of degenerate parabolic equationsrdquoJournal of Differential Equations vol 52 no 2 pp 135ndash161 1984
[18] P E Sacks ldquoContinuity of solutions of a singular parabolicequationrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 7 no 4 pp 387ndash409 1983
[19] D Blanchard and G A Francfort ldquoStudy of a doubly nonlinearheat equation with no growth assumptions on the parabolictermrdquo SIAM Journal onMathematical Analysis vol 19 no 5 pp1032ndash1056 1988
[20] N Polat ldquoBlow up of solution for a nonlinear reaction diffusionequation with multiple nonlinearitiesrdquo International Journal ofScience and Technology vol 2 no 2 pp 123ndash128 2007
[21] S Gerbi and B Said-Houari ldquoLocal existence and exponentialgrowth for a semilinear damped wave equation with dynamicboundary conditionsrdquo Advances in Differential Equations vol13 no 11-12 pp 1051ndash1074 2008
[22] B Said-Houari ldquoGlobal nonexistence of positive initial-energysolutions of a system of nonlinear wave equations with dampingand source termsrdquo Differential and Integral Equations vol 23no 1-2 pp 79ndash92 2010
[23] E Vitillaro ldquoGlobal nonexistence theorems for a class of evolu-tion equations with dissipationrdquo Archive for Rational Mechanicsand Analysis vol 149 no 2 pp 155ndash182 1999
[14] S Berrimi and S A Messaoudi ldquoA decay result for a quasilinearparabolic systemrdquo in Elliptic and Parabolic Problems vol 63 ofProgress in Nonlinear Differential Equations and Their Applica-tions pp 43ndash50 2005
[15] A Eden B Michaux and J-M Rakotoson ldquoDoubly nonlinearparabolic-type equations as dynamical systemsrdquo Journal ofDynamics and Differential Equations vol 3 no 1 pp 87ndash1311991
[16] H E Ouardi and A E Hachimi ldquoAttractors for a class of doublynonlinear parabolic systemsrdquo Electronic Journal of QualitativeDifferential Equations vol 2006 no 1 pp 1ndash15 2006
[17] H A Levine and P E Sacks ldquoSome existence and nonexistencetheorems for solutions of degenerate parabolic equationsrdquoJournal of Differential Equations vol 52 no 2 pp 135ndash161 1984
[18] P E Sacks ldquoContinuity of solutions of a singular parabolicequationrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 7 no 4 pp 387ndash409 1983
[19] D Blanchard and G A Francfort ldquoStudy of a doubly nonlinearheat equation with no growth assumptions on the parabolictermrdquo SIAM Journal onMathematical Analysis vol 19 no 5 pp1032ndash1056 1988
[20] N Polat ldquoBlow up of solution for a nonlinear reaction diffusionequation with multiple nonlinearitiesrdquo International Journal ofScience and Technology vol 2 no 2 pp 123ndash128 2007
[21] S Gerbi and B Said-Houari ldquoLocal existence and exponentialgrowth for a semilinear damped wave equation with dynamicboundary conditionsrdquo Advances in Differential Equations vol13 no 11-12 pp 1051ndash1074 2008
[22] B Said-Houari ldquoGlobal nonexistence of positive initial-energysolutions of a system of nonlinear wave equations with dampingand source termsrdquo Differential and Integral Equations vol 23no 1-2 pp 79ndash92 2010
[23] E Vitillaro ldquoGlobal nonexistence theorems for a class of evolu-tion equations with dissipationrdquo Archive for Rational Mechanicsand Analysis vol 149 no 2 pp 155ndash182 1999
