Global behavior of solutions for a class of quasi-linear evolution equations.

1. Introduction

We consider the problem

a(x,t)[u.sub.t] - div([[absolute value of [nabla]u].sup.m-2][nabla]u) - [alpha][DELTA][u.sub.t] = f(u), x [member of] [OMEGA}, t > 0 (1.1)

u(x,t) = 0, x [member of] [partial derivative][OMEGA], t > 0 (1.2)

u(x, 0) = [u.sub.0] (x), x [member of] [OMEGA] (1.3)

where [OMEGA] [member of] [R.sup.n] is a bounded domain with smooth boundary [partial derivative][OMEGA], [alpha] > 0, m [greater than or equal to] 2, a(x, t) [greater than or equal to] 0 is properly a value of function and f is a continuous function.

For the case a(x, t) = 1 and [alpha] = 0, research of global behavior and finite time blow-up of solutions for the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has attracted a great deal of people. The obtained results show that global existence and nonexistence depend roughly on m, the degree of nonlinearity in f, the dimension n and the size of the initial data.

In early 70's, Levine [1,2] introduced the concavity method and established the global nonexistence results both for parabolic and hyperbolic equations. The essence of the method lies in the following lemma.

Lemma. (Levine [1,2]) Let [PSI](t) be a positive, twice differentiable function which satisfies for t> 0, the inequality

[PSI] (t)[PSI] (t) - (1 + [epsilon]) [([PSI]' (t)).sup.2] [greater than or equal to] 0

with some [epsilon] > 0. If [PSI](0)> 0 and [PSI]' (0)>0, then there exists a time [t.sub.1] < [PSI] (0)/[epsilon][PSI]] (0) such that V(t) [right arrow] +[infinity] as t [right arrow] [t.sup.-.sub.1].

In , Levine and Payne applied the concavity method to the study of the initialboundary value problems for the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where p > 2, q > 0 and q + 2 > p.

Later, this method has been improved and modified by Kalantarov and Ladyzhenskaya  to accommodate more situations. In  Erdem and Kalantarov has applied the modified concavity method to the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

f (u) (u) [greater than or equal to] 2 ([epsilon] + 1) [[integral].sup.u.sub.0] f ([xi])d[xi], [for all]u [member of] [R.sup.1], [epsilon] > [m - 2]/2, m [greater than or equal to] 2

and

[absolute value of h(u, [nabla]u)] [less than or equal to] C([[absolute value of u].sup.m/2] + [[absolute value of [nabla]u].sup.m/2]).

For the results of the same nature, we refer the reader to [6-12] and the references there in.

Recently Zhou in  by using concavity method obtained global nonexistence result for the quasi-linear parabolic equation

a(x, t)[u.sub.t] - div([[absolute value of [nabla]u].sup.m-2] [nabla]u) = f(u)

with initial datum u(x, 0) possesses suitable positive energy. By the different method, Zhyijian obtained asymptotic behavior and blow-up of solutions for nonlinear wave equations with dissipative terms in .

In this paper, we concern with global in time behavior of solutions and establish a blow-up result for problem (1.1)-(1.3) with initial negative energy. The proof of our technique is similar to the one in .

2. A Global Nonexistence Result

We define the energy function associated with a solution u(x, t) of (1.1)-(1.3) by

E(t) = -[1/m] [[parallel][nabla](.,t)[parallel].sup.m.sub.m] - [[integral].sub.[OMEGA]] F(u)dx, t > 0, (2.1)

where

F(u) := [[integral].sup.u.sub.0] f([xi])d[xi] [less than or equal to] [1/p] uf(u), p > m [greater than or equal to] 2. (2.2)

Theorem. Let a(x, t) [member of] [W.sup.1,[infinity]] (0, [infinity]; [L.sup.[infinity]] ([OMEGA])) and [a.sub.t] (x, t) [greater than or equal to] 0 a.e. for t [greater than or equal to] 0 Assume that the following conditions are valid:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Then the corresponding solution to (1.1)-(1.3) blows up in finite time

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [t.sub.1], [beta] and [PHI] to be determined later.

Proof. Multiplying equation (1.1) by [u.sub.t] in [L.sub.2] ([OMEGA]), we get

E (t) + [alpha][[parallel][nabla][u.sub.t](.,t)[parallel].sup.2.sub.2] + [[integral].sub.[OMEGA]] a(x,t)[u.sup.2.sub.t] dx = 0,(2.4)

E(t) [less than or equal to] E(0) < 0, t [greater than or equal to] 0.(2.5)

Let t

[PSI](t) = [1/2] [[integral].sup.t.sub.0] [[integral].sub.[OMEGA]] a(x, [tau])[u.sup.2] + [alpha][[absolute value of [nabla]u].sup.2])dxd[tau]. (2.6)

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

where

[[integral].sub.[OMEGA]] a (x,t)[uu.sub.t]dx = -[[parallel][nabla]u (.,t)[parallel].sup.m.sub.m] - [alpha] [[integral].sub.[OMEGA]] [nabla]u[nabla][u.sub.t]dx + [[integral].sub.[OMEGA]] f(u)udx

the definition of the energy functional in (2.1) and the assumption (2.3) with (2.5) have been used. Taking the inequality (2.8) and integrating this, we obtain

[PSI]' (t) [greater than or equal to] ([p/m] - 1) [[integral].sup.t.sub.0] [[parallel][nabla]u(.,[tau])[parallel].sup.m.sub.m] d[tau] - pE (0)t + [PSI]' (0), t > 0. (2.9)

Now, addition of the inequalities (2.8) and (2.9) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

According to the Holder's inequality we have

[[parallel][nabla]u(.,t)[parallel].sup.m.sub.m] [greater than or equal to] [[absolute value of Q].sup.1-m/2] [([[parallel][nabla]u(.,t)[parallel].sup.2.sub.2]).sup.m/2] (2.11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

On exploiting (2.11) and (2.12), estimate (2.10) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

where

[PHI](t) := [PSI](t) + [PSI]'(t). (2.14)

Since

-pE(0)(t + 1) + [PSI](0) [right arrow] [infinity] as t [right arrow] [infinity],

there must be a [t.sub.1] [greater than or equal to] 1 such that

-pE(0)(t + 1) + [PSI]' (0) [greater than or equal to] 0 as t [greater than or equal to] [t.sub.1]. (2.15)

Recall (2.14) and using the equalities (2.6) and (2.7), we get [PHI](t) > 0 as t [greater than or equal to] [t.sub.1]. By using the inequality

[([a.sub.1] + [a.sub.2]).sup.[delta]] [less than or equal to] [2.sup.[delta]-1] ([a.sup.[delta].sub.1] + [a.sup.[delta].sub.2])

with real number [delta] [greater than or equal to] 1 and by virtue of (2.15) and using the inequality (2.13), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16)

On the other hand, by Poincare inequality we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

where 0 < [gamma] [less than or equal to] {[[alpha].sup.-1], [([lambda]M).sup.-1]}, [lambda] is the Poincare constant and M [greater than or equal to] a(x, t). Inserting (2.17) in (2.16) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.18)

where [beta] = ([p/m] - 1)[gamma] [[absolute value of 2[OMEGA]].sup.1-[m/2]]. Now taking into account (2.6) and (2.7) together with (2.18), we find

[PHI](t) [greater than or equal to] [[beta].sup.1-[m/2]] [([PHI](t)).sup.m/2], t [greater than or equal to] [t.sub.1]. (2.19)

Therefore, there exists a positive constant

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that

[PHI](t) [right arrow] [infinity] as t [right arrow] [T.sup.-]. (2.20)

By using (2.6), (2.7) and (2.20) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21)

so (2.21) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof.

References

 H.A. Levine, Instability and nonexistence of global solutions to non-linear wave equations of the form [Pu.sub.tt] = -Au + F(u).--Trans. Am. Math. Soc. 192(1974), 1-21.

 H.A. Levine, Some additional remarks on the nonexistence of global solutions to non-linear wave equations.--SIAM J. Math. Anal. 5(1974), 138-146.

 H.A. Levine, L.E. Pyne, Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations.--J. Math. Anal. Appl. 55(1976), 329-334.

 V.K. Kalantarov, O.A. Ladyzhenskaya, Formation of collapses in quasi-linear equations of parabolic and hyperbolic types.--Zap. Nauchn. Semin. LOMI 69(1977), 77-102.

 D. Erdem, V.K. Kalantarov, A remark on nonexistence of global solutions to quasi-linear hyperbolic and parabolic equations.--Appl. Math. Lett. 15, No.5(2002), 521-653.

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 M. Kirane, N. Tatar, A nonexistence result to a Cauchy problem in nonlinear one dimensional thermoelasticity,--J. Math. Anal. Appl. 254(2002), No.1, 71-86.

 Z. Yang, Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms.--J.Math.Anal.Appl. 300(2004), 218-243.

 S.A. Messaoudi, B. Said-Houari, Blow up of solutions of a class of wave equations with nonlinear damping and sourse terms, Math. Methods Appl. Sci. 27(2004), 1687-1696.

 F.Li, Global existence and blow up of solutions for a higher order Kirchhoff type equation with nonlinear dissipation, Appl. Math. Lett. 17(2004), 1409-1414.

 S.A. Messaoudi, B. Said Houari, A blow up result for a higher order nonlinear Kirchhoff type hyperbolic equations, Appl. Math. Lett. 20(2007), 866-871.

 Y. Zhou, Global nonexistence for a quasi-linear evolution equation with a generalized Lewis function, Journal for Anal. and its Applications Vol.24, No.1(2005), 179-187.

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Faramarz Tahamtani

Department of Mathematics, Shiraz University, SHIRAZ(71454),IRAN

tahamtani@susc.ac.ir
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