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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 724815, 36 pages
http://dx.doi.org/10.1155/2011/724815
Research Article

Dynamic Analysis of a Nonlinear Timoshenko Equation

Departamento de Ciencias Básicas, Análisis Matemático y sus Aplicaciones, UAM-Azcapotzalco, Avenida San Pablo 180, Col. Reynosa Tamaulipas, 02200 México, DF, Mexico

Received 8 February 2011; Accepted 28 April 2011

Academic Editor: Norimichi Hirano

Copyright © 2011 Jorge Alfredo Esquivel-Avila. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We characterize the global and nonglobal solutions of the Timoshenko equation in a bounded domain. We consider nonlinear dissipation and a nonlinear source term. We prove blowup of solutions as well as convergence to the zero and nonzero equilibria, and we give rates of decay to the zero equilibrium. In particular, we prove instability of the ground state. We show existence of global solutions without a uniform bound in time for the equation with nonlinear damping. We define and use a potential well and positive invariant sets.

1. Introduction

We consider 𝑢 𝑡 𝑡 + 𝑘 Δ 2 𝑢 𝑀 𝑢 2 2 𝑢 Δ 𝑢 + 𝑔 𝑡 = 𝑓 ( 𝑢 ) i n Ω , ( 1 . 1 ) with initial conditions 𝑢 ( 𝑥 , 0 ) = 𝑢 0 , 𝑢 𝑡 ( 𝑥 , 0 ) = 𝑣 0 , 𝑥 Ω , ( 1 . 2 ) and with one set of the following boundary conditions: 𝑢 = 0 , Δ 𝑢 = 0 o n 𝜕 Ω , ( 1 . 3 ) or 𝑢 = 0 , 𝜕 𝑢 𝜕 𝜈 = 0 o n 𝜕 Ω , ( 1 . 4 ) where Ω 𝑛 is a bounded domain with sufficiently smooth boundary, 2 is the norm in 𝐿 2 ( Ω ) , 𝑀 𝑠 2 = 𝛼 + 𝛽 𝑠 2 𝛾 𝑔 𝑢 , 𝛼 0 , 𝛽 0 , 𝛼 + 𝛽 > 0 , 𝛾 1 , 𝑘 = 1 , ( 1 . 5 ) 𝑡 = 𝛿 𝑢 𝑡 | | 𝑢 𝑡 | | 𝜆 2 , 𝛿 > 0 , 𝜆 2 , ( 1 . 6 ) 𝑓 ( 𝑢 ) = 𝜇 𝑢 | 𝑢 | 𝑟 2 , 𝜇 > 0 , 𝑟 > 0 . ( 1 . 7 ) When the source term 𝑓 0 , there is a considerable set of works studying several properties of equation (1.1), see for instance, the early papers by Ball [1, 2], Haraux and Zuazua [3], and the books by Hale [4], Haraux [5], and references therein. For a destabilizing source term, 𝑠 𝑓 ( 𝑠 ) > 0 , 𝑠 { 0 } , in the works of Payne and Sattinger [6], Georgiev and Todorova [7], and Ikehata [8], qualitative properties of (1.1) are studied, when 𝑘 = 0 = 𝛽 . To understand the dynamics of second-order equations in time, similar to (1.1), active research is reported in Alves and Cavalcanti [9], Barbu et al. [10], Cavalcanti et al. [1116], Rammaha [17] Rammaha and Sakuntasathien [18], and Todorova and Vitillaro [19], Vitillaro [20]. For the Timoshenko equation, with 𝑔 0 , Bainov and Minchev [21] gave sufficient conditions for the nonexistence of smooth solutions of (1.1), with negative initial energy, and gave an upper bound of the maximal time of existence. For positive and sufficiently small initial energy, blowup and globality properties are characterized in Esquivel-Avila [22]. For the Kirchhoff equation, that is, (1.1) with 𝑘 = 0 , the nonexistence of global solutions is studied in [23]. In [24, 25], we characterized properties such as blowup and asymptotic behavior of solutions, for (1.1) with 𝑘 = 0 and 𝛽 = 0 . To the knowledge of the author, such problems are still open for the Timoshenko equation (1.1). Here, we want to give some results about the dynamics of problem (1.1). To do that we will generalize the concept of the depth of the potential well in such manner that our results of the dynamics be as sharp as the ones in [24, 25]. Furthermore, for particular cases, our definition of depth of the potential well will coincide with the one introduced in [6].

2. Preliminaries and Framework

We begin this section with an existence, uniqueness, and continuation theorem for (1.1). The proof is similar to the ones in [7, 8], where semilinear wave equations are studied.

Theorem 2.1. Assume that 𝑟 > 2 and 𝑟 2 ( 𝑛 2 ) / ( 𝑛 4 ) if 𝑛 5 . For every initial data ( 𝑢 0 , 𝑣 0 ) 𝐻 𝐵 × 𝐿 2 ( Ω ) , where 𝐵 is defined either by 𝐵 𝐻 2 ( Ω ) 𝐻 1 0 ( Ω ) , or 𝐵 𝐻 2 0 ( Ω ) , there exists a unique (local) weak solution ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 𝑆 ( 𝑡 ) ( 𝑢 0 , 𝑣 0 ) of problem (1.1), that is, 𝑑 𝑑 𝑡 ( 𝑣 ( 𝑡 ) , 𝑤 ) 2 + ( Δ 𝑢 ( 𝑡 ) , Δ 𝑤 ) 2 + 𝑀 𝑢 ( 𝑡 ) 2 2 ( 𝑢 ( 𝑡 ) , 𝑤 ) 2 + ( 𝑔 ( 𝑣 ( 𝑡 ) ) , 𝑤 ) 2 = ( 𝑓 ( 𝑢 ( 𝑡 ) ) , 𝑤 ) 2 , ( 2 . 1 ) a.e. in ( 0 , 𝑇 ) and for every 𝑤 𝐵 𝐿 𝜆 ( Ω ) , such that [ 𝑢 𝐶 ( 0 , 𝑇 ) ; 𝐵 ) 𝐶 1 [ 0 , 𝑇 ) ; 𝐿 2 ( Ω ) , 𝑣 𝑢 𝑡 𝐿 𝜆 ( ( 0 , 𝑇 ) × Ω ) . ( 2 . 2 )
Here, 𝑆 ( 𝑡 ) denotes the corresponding semigroup on 𝐻 , generated by problem (1.1), and ( , ) 2 is the inner product in 𝐿 2 ( Ω ) .
The following energy equation holds: 𝐸 0 = 𝐸 ( 𝑡 ) + 𝑡 0 𝛿 𝑣 ( 𝜏 ) 𝜆 𝜆 𝑑 𝜏 , ( 2 . 3 ) where 1 𝐸 ( 𝑡 ) 𝐸 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 𝑣 ( 𝑡 ) 2 2 1 + 𝐽 ( 𝑢 ( 𝑡 ) ) , ( 2 . 4 ) 𝐽 ( 𝑢 ) 2 1 𝑎 ( 𝑢 ) + 1 2 ( 𝛾 + 1 ) 𝑐 ( 𝑢 ) 𝑟 𝑏 ( 𝑢 ) , ( 2 . 5 ) with 𝑎 ( 𝑢 ) 𝑢 2 𝐵 , 𝑏 ( 𝑢 ) 𝜇 𝑢 𝑟 𝑟 , 𝑐 ( 𝑢 ) 𝛽 𝑢 2 2 ( 𝛾 + 1 ) . ( 2 . 6 )
Here, 𝐸 0 𝐸 ( 𝑢 0 , 𝑣 0 ) is the initial energy, and 𝑞 denotes the norm in the 𝐿 𝑞 ( Ω ) space.
If the maximal time of existence 𝑇 𝑀 < , then 𝑆 ( 𝑡 ) ( 𝑢 0 , 𝑣 0 ) as 𝑡 𝑇 𝑀 , in the norm of 𝐻 : ( 𝑢 , 𝑣 ) 2 𝐻 𝑢 2 𝐵 + 𝑣 2 2 Δ 𝑢 2 2 + 𝛼 𝑢 2 2 + 𝑣 2 2 . ( 2 . 7 )
In that case, from (2.3)–(2.6), 𝑢 ( 𝑡 ) 𝑟 as 𝑡 𝑇 𝑀 .

Now, we define, respectively, the stable (potential well) and unstable sets: [ ] [ ] , [ 𝐼 ] [ 𝐽 ] , 𝑊 ( 𝐼 ( 𝑢 ) > 0 { 0 } ) 𝐽 ( 𝑢 ) < 𝑑 𝑉 ( 𝑢 ) < 0 ( 𝑢 ) < 𝑑 ( 2 . 8 ) where 𝐼 ( 𝑢 ) 𝑎 ( 𝑢 ) + 𝑐 ( 𝑢 ) 𝑏 ( 𝑢 ) . ( 2 . 9 )

Here, [ 𝐼 ( 𝑢 ) < 0 ] denotes the set of 𝑢 𝐵 with that property, and the depth of the potential well is defined as follows: 𝑑 𝑟 2 𝑆 2 𝑟 𝑟 / ( 𝑟 2 ) , ( 2 . 1 0 ) where 𝑆 i n f 0 𝑢 𝐵 ̂ 𝑏 ( 𝑢 ) > 0 ̂ 𝑎 ( 𝑢 ) 1 / 2 ̂ 𝑏 ( 𝑢 ) 1 / 𝑟 , ( 2 . 1 1 ) ̂ 𝑎 ( 𝑢 ) 𝑎 ( 𝑢 ) + 𝜅 1 𝑐 ̂ 𝑏 ( 𝑢 ) , ( 𝑢 ) 𝑏 ( 𝑢 ) + 𝜅 2 𝑐 ( 𝑢 ) . ( 2 . 1 2 ) with 𝜅 1 𝑟 2 ( 𝛾 + 1 ) ( 𝑟 2 ) ( 𝛾 + 1 ) , 𝜅 2 𝜅 1 1 = 𝑟 𝛾 ( 𝑟 2 ) ( 𝛾 + 1 ) . ( 2 . 1 3 )

We assume that 𝑟 2 ( 𝛾 + 1 ) , and since 𝛾 1 , then 𝜅 1 [ 0 , 1 / 2 ) , and 𝜅 2 [ 1 , 1 / 2 ) . Also note that if 𝑟 = 2 ( 𝛾 + 1 ) , then 𝜅 1 = 0 , 𝜅 2 = 1 , and we have the following characterization of the depth of the potential well (2.10)-(2.11): 𝑑 = i n f 0 𝑢 𝐵 s u p 𝜆 0 𝐽 ( 𝜆 𝑢 ) , ( 2 . 1 4 ) which is the definition given in [6], where a nondissipative nonlinear wave equation is studied.

Consider any 𝑢 𝐵 , 𝑟 > 2 , and 𝑟 2 𝑛 / ( 𝑛 4 ) if 𝑛 5 , then ̂ 𝑎 ( 𝑢 ) 𝐶 ( Ω ) 𝑏 ( 𝑢 ) 2 / 𝑟 + 𝜅 1 𝑐 ( 𝑢 ) 𝐶 ( Ω ) 𝑏 ( 𝑢 ) 2 / 𝑟 , ( 2 . 1 5 ) where 𝐶 ( Ω ) > 0 , is any constant in the Sobolev-Poincaré’s inequality Δ 𝑢 2 2 + 𝛼 𝑢 2 2 1 / 2 𝐶 ( Ω ) 𝜇 1 / 𝑟 𝑢 𝑟 . ( 2 . 1 6 )

Moreover, if ̂ 𝑏 ( 𝑢 ) > 0 , from (2.15) and since ̂ 𝑏 ( 𝑢 ) 𝑏 ( 𝑢 ) , ̂ ̂ 𝑎 ( 𝑢 ) 𝐶 ( Ω ) 𝑏 ( 𝑢 ) 2 / 𝑟 . ( 2 . 1 7 )

Hence, 𝑆 𝐶 ( Ω ) , and 𝑑 𝐷 ( ( 𝑟 2 ) / 2 𝑟 ) 𝐶 ( Ω ) 𝑟 / ( 𝑟 2 ) > 0 .

If 𝑢 𝑒 denotes any nonzero equilibria of equation (1.1), 0 𝑢 𝑒 𝐵 Δ 2 𝑢 𝑒 𝑀 𝑢 2 2 Δ 𝑢 𝑒 𝑢 = 𝑓 𝑒 , ( 2 . 1 8 ) then, by (2.1) in Theorem 2.1 with 𝑢 ( 𝑡 ) = 𝑢 𝑒 = 𝑤 , we get that 𝑢 𝑒 belongs to the Nehari manifold, 𝒩 , that is, 𝒩 0 𝑢 𝐵 𝐼 ( 𝑢 ) = 𝐼 ( 𝑢 ) = 0 , ( 2 . 1 9 ) where ̂ 𝐼 ( 𝑢 ) ̂ 𝑎 ( 𝑢 ) 𝑏 ( 𝑢 ) .

Consequently, ̂ 𝑏 ( 𝑢 𝑒 ) = ̂ 𝑎 ( 𝑢 𝑒 ) > 0 . Furthermore, from (2.17) which is an equality when 𝐶 ( Ω ) = 𝑆 , we conclude that the Nehari manifold can be represented by the line: 𝑦 = 𝑥 , in the plane with axes ̂ 𝑥 = 𝑏 ( 𝑢 ) and 𝑦 = ̂ 𝑎 ( 𝑢 ) , beginning at the point: 𝑦 = 𝑥 = 𝑆 𝑟 / ( 𝑟 2 ) = ( 2 𝑟 / ( 𝑟 2 ) ) 𝑑 . We also note that 1 𝐽 ( 𝑢 ) = 2 1 ̂ 𝑎 ( 𝑢 ) 𝑟 ̂ 𝑏 ( 𝑢 ) . ( 2 . 2 0 )

From these facts it follows that the depth of the potential well (2.10) is characterized by 𝑑 = i n f 𝑢 𝒩 𝐽 ( 𝑢 ) = 𝑟 2 2 𝑟 𝜚 , ( 2 . 2 1 ) where 0 < 𝜚 i n f 𝑢 𝒩 ̂ 𝑎 ( 𝑢 ) = i n f 𝑢 𝒩 ̂ 𝑏 ( 𝑢 ) . ( 2 . 2 2 )

Hence, any equilibrium is such that 𝑢 𝑒 [ 𝐽 ( 𝑢 ) 𝑑 ] . Moreover, like in [6], the set of extremals of (2.21) is characterized by set of equilibria with least energy, that is the ground state 𝒩 𝑢 𝑒 𝑢 𝐽 𝑒 = 𝑢 = 𝑑 𝑒 𝑢 ̂ 𝑎 𝑒 = ̂ 𝑏 𝑢 𝑒 = 𝜚 . ( 2 . 2 3 ) Observe that 𝐽 ( 𝑢 ) = 𝑑 is a tangent line to the curve defined by the equality in (2.17) with 𝐶 ( Ω ) = 𝑆 , at the point 𝒩 , which holds if ̂ 𝑏 ( 𝑢 ) > 0 . On the other hand, we notice that 𝜅 1 ̂ 𝑏 ( 𝑢 ) 𝜅 2 ̂ 𝑎 ( 𝑢 ) = 𝜅 1 𝑏 ( 𝑢 ) 𝜅 2 𝑎 ( 𝑢 ) > 0 , ( 2 . 2 4 ) and is equal to zero if and only if 𝑎 ( 𝑢 ) = 0 = 𝑏 ( 𝑢 ) . Hence, if ̂ 𝑏 ( 𝑢 ) < 0 , then ̂ 𝑎 ( 𝑢 ) > 𝑟 2 ( 𝛾 + 1 ) ̂ 𝑟 𝛾 𝑏 ( 𝑢 ) . ( 2 . 2 5 )

Therefore, next results about the stable and unstable sets follow.

Lemma 2.2. The following properties of 𝑉 and 𝑊 hold: (i) 𝑊 is a neighborhood of 0 𝐵 .(ii) 0 [ 𝐼 ( 𝑢 ) < 0 ] (closure in 𝐵 ), in particular 0 𝑉 .(iii) 𝑊 = 𝑊 + 𝑊 { 0 } , where 𝑊 + ̂ = 𝜚 𝑊 𝑏 ( 𝑢 ) > 0 ( 𝑟 2 ) / 𝑟 ̂ 𝑏 ( 𝑢 ) 2 / 𝑟 2 ̂ 𝑎 ( 𝑢 ) < 𝑟 ̂ 𝑏 ( 𝑢 ) + 𝑟 2 𝑟 ̂ , 𝑊 𝜚 , 0 < 𝑏 ( 𝑢 ) < 𝜚 ̂ 𝑊 𝑏 ( 𝑢 ) < 0 𝑟 2 ( 𝛾 + 1 ) ̂ 2 𝑟 𝛾 𝑏 ( 𝑢 ) < ̂ 𝑎 ( 𝑢 ) < 𝑟 ̂ 𝑏 ( 𝑢 ) + 𝑟 2 𝑟 ̂ . 𝜚 , 𝛾 𝜚 < 𝑏 ( 𝑢 ) < 0 ( 2 . 2 6 ) (iv) 𝜚 𝑉 = ( 𝑟 2 ) / 𝑟 ̂ 𝑏 ( 𝑢 ) 2 / 𝑟 2 ̂ 𝑎 ( 𝑢 ) < 𝑟 ̂ 𝑏 ( 𝑢 ) + 𝑟 2 𝑟 ̂ 𝜚 , 𝑏 ( 𝑢 ) > 𝜚 . (v) 𝒩 = 𝑊 𝑉 = 𝑊 + 𝑉 = [ 𝑢 𝑒 𝒩 , ̂ 𝑎 ( 𝑢 𝑒 ̂ ) = 𝑏 ( 𝑢 𝑒 ) = 𝜚 ] . (vi) 𝑊 = [ 𝐼 ( 𝑢 ) < 0 ] 𝑐 [ 𝐽 ( 𝑢 ) < 𝑑 ] , 𝑉 = ( [ 𝐼 ( 𝑢 ) > 0 ] { 0 } ) 𝑐 [ 𝐽 ( 𝑢 ) < 𝑑 ] .

The following result follows easily like in [23].

Lemma 2.3. One has that 𝐽 ( 𝑢 ) > 𝑟 2 2 𝑟 ̂ 𝑎 ( 𝑢 ) > 𝑟 2 ̂ 2 𝑟 𝑏 ( 𝑢 ) , ( 2 . 2 7 ) 𝐽 ( 𝑢 ) > 𝑟 2 2 𝑟 𝑎 ( 𝑢 ) + 𝑟 2 ( 𝛾 + 1 ) > 𝛾 2 𝑟 ( 𝛾 + 1 ) 𝑐 ( 𝑢 ) 2 ( 𝛾 + 1 ) 𝑎 ( 𝑢 ) + 𝑟 2 ( 𝛾 + 1 ) 2 𝑟 ( 𝛾 + 1 ) 𝑏 ( 𝑢 ) , ( 2 . 2 8 ) for any 𝑢 𝐵 such that 𝐼 ( 𝑢 ) > 0 , in particular if 0 𝑢 𝑊 , and 𝑑 < 𝑟 2 2 𝑟 ̂ 𝑎 ( 𝑢 ) < 𝑟 2 ̂ 2 𝑟 𝑏 ( 𝑢 ) , ( 2 . 2 9 ) 𝑑 < 𝑟 2 2 𝑟 𝑎 ( 𝑢 ) + 𝑟 2 ( 𝛾 + 1 ) < 𝛾 2 𝑟 ( 𝛾 + 1 ) 𝑐 ( 𝑢 ) 2 ( 𝛾 + 1 ) 𝑎 ( 𝑢 ) + 𝑟 2 ( 𝛾 + 1 ) 2 𝑟 ( 𝛾 + 1 ) 𝑏 ( 𝑢 ) , ( 2 . 3 0 ) for any 𝑢 𝐵 , such that 𝐼 ( 𝑢 ) < 0 , in particular if 𝑢 𝑉 .

A set 𝒱 𝐻 is positive invariant, with respect to problem (1.1), if the corresponding generated semigroup 𝑆 ( 𝑡 ) on 𝐻 is such that 𝑆 ( 𝑡 ) 𝒱 𝒱 . ( 2 . 3 1 )

Lemma 2.4. Let ( 𝑢 , 𝑣 ) denote any solution of (1.1), given by Theorem 2.1. Then, the sets [ ] [ ] , [ ] [ ] , 𝒮 𝐸 ( 𝑢 , 𝑣 ) < 𝑑 ( 𝑢 , 𝑣 ) 𝐻 𝑢 𝑊 ( 2 . 3 2 ) 𝒰 𝐸 ( 𝑢 , 𝑣 ) < 𝑑 ( 𝑢 , 𝑣 ) 𝐻 𝑢 𝑉 ( 2 . 3 3 ) are positive invariant.

Proof. First, we show that 𝒮 is positive invariant. In order to do that, we take ( 𝑢 0 , 𝑣 0 ) 𝒮 . Then, by (2.4), 𝐽 ( 𝑢 ( 𝑡 ) ) 𝐸 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 𝐸 0 < 𝑑 , for any 𝑡 0 . Now, if 𝒮 is not positive invariant, there exists some ̂ 𝑡 > 0 , such that ̂ 𝐼 ( 𝑢 ( 𝑡 ) ) = 0 , with ̂ 𝑢 ( 𝑡 ) 0 . Then, by (2.21), ̂ 𝑑 𝐽 ( 𝑢 ( 𝑡 ) ) . But this is impossible because ̂ 𝐽 ( 𝑢 ( 𝑡 ) ) < 𝑑 . The proof of the positive invariance of 𝒰 is quite similar. Indeed, if this is not true there exists some ̂ 𝑡 > 0 , such that ̂ 𝐼 ( 𝑢 ( 𝑡 ) ) = 0 . From (ii) of Lemma 2.2   ̂ 𝑢 ( 𝑡 ) 0 , and this implies the same contradiction as before.

Next result gives an interpretation of sets 𝒮 and 𝒰 and follows from Lemma 2.2.

Lemma 2.5. The sets 𝒮 and 𝒰 have the properties [ ] 𝒮 𝐸 ( 𝑢 , 𝑣 ) < 𝑑 ( 𝑢 , 𝑣 ) 𝐻 𝜚 ( 𝑟 2 ) / 𝑟 ̂ 𝑏 ( 𝑢 ) 2 / 𝑟 2 ̂ 𝑎 ( 𝑢 ) < 𝑟 ̂ 𝑏 ( 𝑢 ) + 𝑟 2 𝑟 ̂ , [ ] 𝜚 , 0 < 𝑏 ( 𝑢 ) < 𝜚 𝐸 ( 𝑢 , 𝑣 ) < 𝑑 ( 𝑢 , 𝑣 ) 𝐻 𝑟 2 ( 𝛾 + 1 ) ̂ 2 𝑟 𝛾 𝑏 ( 𝑢 ) < ̂ 𝑎 ( 𝑢 ) < 𝑟 ̂ 𝑏 ( 𝑢 ) + 𝑟 2 𝑟 ̂ [ ] ( 𝜚 , 𝛾 𝜚 < 𝑏 ( 𝑢 ) < 0 𝒰 = 𝐸 ( 𝑢 , 𝑣 ) < 𝑑 𝑢 , 𝑣 ) 𝐻 𝜚 ( 𝑟 2 ) / 𝑟 ̂ 𝑏 ( 𝑢 ) 2 / 𝑟 2 ̂ 𝑎 ( 𝑢 ) < 𝑟 ̂ 𝑏 ( 𝑢 ) + 𝑟 2 𝑟 ̂ , 𝜚 , 𝑏 ( 𝑢 ) > 𝜚 , ( 2 . 3 4 ) 𝒮 𝑢 𝒰 = 𝑒 , 0 𝐻 𝑢 𝑒 𝒩 = 𝑢 𝑒 , 0 𝐻 𝑢 𝑒 𝑢 𝒩 , ̂ 𝑎 𝑒 = ̂ 𝑏 𝑢 𝑒 . = 𝜚 ( 2 . 3 5 )

The following result is a direct consequence of (vi) in Lemma 2.2 and Lemma 2.4.

Lemma 2.6. For every solution of (1.1), only one of the following holds: (i)there exists some 𝑡 0 0 such that ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) 𝒮 , and remains there for every 𝑡 > 𝑡 0 ,(ii)there exists some 𝑡 0 0 such that ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) 𝒰 , and remains there for every 𝑡 > 𝑡 0 ,(iii) ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) [ 𝐸 ( 𝑢 , 𝑣 ) 𝑑 ] for every 𝑡 0 .

Hence, we notice that the sets 𝒮 and 𝒰 play an important role in the dynamics of (1.1). Moreover, we will prove that any solution eventually contained in 𝒮 converges to the zero equilibrium. If enters in 𝒰 , either blowups in a finite time or it is global but without a uniform bound in 𝐻 for every 𝑡 0 , in the case that 𝜆 > 2 , in (1.6). Also, we will prove that any solution with ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) [ 𝐸 ( 𝑢 , 𝑣 ) 𝑑 ] , for every 𝑡 0 , is bounded and converges to the set of nonzero equilibria .

We will need the following inequalities to show blowup and convergence to the zero equilibrium, respectively, in the dissipative case.

Lemma 2.7. Let 𝑊 1 , 1 l o c ( + ) be a nonnegative function such that ̇ ( 𝑡 ) 𝐶 𝑎 ( 𝑡 ) a . e . f o r 𝑡 0 , ( 2 . 3 6 ) with 𝑎 > 1 and 𝐶 > 0 .
Then, there exists some 𝑇 > 0 such that l i m 𝑡 𝑇 ( 𝑡 ) = .

Proof. Define 𝒢 ( 𝑡 ) 1 𝑎 ( 𝑡 ) , then ̇ 𝒢 ( 𝑡 ) ( 1 𝑎 ) 𝐶 < 0 a . e . f o r 𝑡 0 . ( 2 . 3 7 ) Hence, 0 < 𝐺 ( 0 ) + ( 1 𝑎 ) 𝐶 𝑡 , which is only possible if 𝑡 < 𝑇 ( 1 / 𝐶 ( 𝑎 1 ) ) 1 𝑎 ( 0 ) .

Lemma 2.8. Let 𝑊 1 , 1 l o c ( + ) be a nonnegative function such that ̇ ( 𝑡 ) 𝐶 𝑎 ( 𝑡 ) a . e . f o r 𝑡 0 , ( 2 . 3 8 ) with 𝑎 1 and 𝐶 > 0 .
Then, for 𝑡 0 , if 𝑎 > 1 ( 𝑡 ) 0 1 + 𝑡 𝐶 ( 𝑎 1 ) 0 𝑎 1 1 / ( 𝑎 1 ) , ( 2 . 3 9 ) and, if 𝑎 = 1 ( 𝑡 ) 0 𝑒 𝐶 𝑡 . ( 2 . 4 0 )

Proof. Consider 𝑎 > 1 , and notice that ( ( 1 𝑎 ) ̇ ) ( 𝑡 ) ( 𝑎 1 ) 𝐶 . Then, we integrate and obtain the first inequality. Now, let 𝑎 1 , and the second one follows.

3. Timoshenko Equation

Due to our assumptions on 𝑟 and 𝛾 , we restrict our analysis to dimensions 𝑛 5 . Indeed, since 𝛾 1 , 2 ( 𝛾 + 1 ) < 𝑟 and 𝑟 2 ( 𝑛 2 ) / ( 𝑛 4 ) , if 𝑛 5 , then our analysis considers, 𝑛 = 5 whenever 𝛾 < 2 . We also notice that in any case we do not consider the interval 2 < 𝑟 4 . Moreover, 𝑟 6 whenever 𝑛 = 5 . We begin with a characterization of blowup when 𝛿 > 0 and 𝜆 2 .

Theorem 3.1. Let ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) = 𝑆 ( 𝑡 ) ( 𝑢 0 , 𝑣 0 ) be a solution of problem (1.1), and suppose that 𝑟 > 2 ( 𝛾 + 1 ) . A necessary and sufficient condition for nonglobality, blowup by Theorem 2.1, is that 𝜆 < 𝑟 and there exists 𝑡 0 0 such that ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) 𝒰 .

Proof. Sufficiency
By Lemma 2.4, ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 𝒰 for all 𝑡 > 𝑡 0 .

Now, we consider the function defined, along the solution, by 𝒱 ( 𝑡 ) 𝑑 𝐸 ( 𝑡 ) , ( 3 . 1 ) and notice that because of energy equation (2.3), 𝒱 ( 𝑡 ) 𝑑 𝐸 0 𝒱 0 > 0 , ( 3 . 2 ) where, now 𝐸 0 𝐸 ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) .
Notice that from (2.29) in Lemma 2.3, 𝑟 𝒱 ( 𝑡 ) 𝑑 𝐽 ( 𝑢 ( 𝑡 ) ) 𝑑 1 𝑟 2 𝑑 + 𝑟 ̂ 2 𝑏 ( 𝑢 ( 𝑡 ) ) = 1 𝑟 2 𝑑 + 𝑟 𝑏 𝛾 ( 𝑢 ( 𝑡 ) ) 𝑐 ( 𝑟 2 ) ( 𝛾 + 1 ) ( 𝑢 ( 𝑡 ) ) . ( 3 . 3 )
We will need some estimates. First, we notice that from energy equation in terms of 𝒱 ( 𝑡 ) and (3.3), | | | 𝛿 | | | | 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) 𝑣 ( 𝑡 ) 𝜆 2 2 | | | 𝛿 𝑢 ( 𝑡 ) 𝜆 𝑣 ( 𝑡 ) 𝜆 𝜆 1 𝐶 ( Ω ) 𝛿 𝑢 ( 𝑡 ) 𝑟 𝑣 ( 𝑡 ) 𝜆 𝜆 1 𝐶 ( Ω ) 𝛿 𝑢 ( 𝑡 ) 𝑟 1 𝑘 𝑢 ( 𝑡 ) 𝑘 𝑟 𝑣 ( 𝑡 ) 𝜆 𝜆 1 𝐶 ( Ω ) 𝛿 𝑢 ( 𝑡 ) 𝑟 1 𝑘 𝜈 𝑢 ( 𝑡 ) 𝑟 𝑘 𝜆 + 1 𝐶 ( 𝜈 ) 𝑣 ( 𝑡 ) 𝜆 𝜆 < 𝐶 𝒱 ( 1 𝑘 ) / 𝑟 ( 𝑡 ) 𝜈 𝛿 𝑢 ( 𝑡 ) 𝑟 𝑘 𝜆 + 1 ̇ , 𝐶 ( 𝜈 ) 𝒱 ( 𝑡 ) ( 3 . 4 ) where 𝑘 ( 1 , 𝑟 / 𝜆 ) , 𝐶 𝐶 ( Ω ) ( 𝑟 / 𝜇 ) ( 1 𝑘 ) / 𝑟 , 𝐶 ( Ω ) > 0 is the constant in the continuous embedding 𝐿 𝑟 ( Ω ) 𝐿 𝜆 ( Ω ) , 𝐶 ( 𝜈 ) > 0 , and 𝜈 > 0 will be chosen later.
Consider a positive number 𝑞 to be chosen later, from (3.2)-(3.3), we obtain 𝐼 ( 𝑢 ( 𝑡 ) ) = 𝑞 𝐽 ( 𝑢 ( 𝑡 ) ) + 𝑞 2 2 𝑎 ( 𝑢 ( 𝑡 ) ) + 𝑞 2 ( 𝛾 + 1 ) 2 ( 𝛾 + 1 ) 𝑐 ( 𝑢 ( 𝑡 ) ) + 𝑟 𝑞 𝑟 𝒱 𝑏 ( 𝑢 ( 𝑡 ) ) 𝑞 0 + 𝑑 𝑞 2 2 𝑎 ( 𝑢 ( 𝑡 ) ) + 𝑞 2 ( 𝛾 + 1 ) 2 ( 𝛾 + 1 ) 𝑐 ( 𝑢 ( 𝑡 ) ) + 𝑟 𝑞 𝑟 𝑏 ( 𝑢 ( 𝑡 ) ) . ( 3 . 5 )
If 𝒱 0 𝑑 , we choose 𝑞 2 ( 𝛾 + 1 ) , and from (3.5) we get 𝐼 ( 𝑢 ( 𝑡 ) ) 𝑟 2 ( 𝛾 + 1 ) 𝑟 𝑏 ( 𝑢 ( 𝑡 ) ) . ( 3 . 6 )
If 𝒱 0 < 𝑑 , then we notice that from (3.2)-(3.3), 𝒱 0 𝒱 𝑑 ( 𝑟 2 ) 0 𝑑 ( 𝑟 2 ) 𝒱 0 1 + 2 𝑑 𝑟 𝛾 𝑏 ( 𝑢 ( 𝑡 ) ) ( 𝑟 2 ) ( 𝛾 + 1 ) 𝑐 ( 𝑢 ( 𝑡 ) ) . ( 3 . 7 )
Hence and from (3.5), we have the estimate 𝐼 ( 𝑢 ( 𝑡 ) ) 𝑞 2 2 + 𝑞 𝑎 ( 𝑢 ( 𝑡 ) ) 2 ( 𝛾 + 1 ) 𝑞 2 ( 𝛾 + 1 ) 𝑞 + 2 𝛾 𝑑 𝒱 0 ( 𝑟 2 ) 𝒱 0 + 𝑞 + 2 𝑑 𝑐 ( 𝑢 ( 𝑡 ) ) 𝑟 𝑟 𝑞 𝑞 ( 𝑟 2 ) 𝑑 𝒱 0 ( 𝑟 2 ) 𝒱 0 𝑏 + 2 𝑑 ( 𝑢 ( 𝑡 ) ) . ( 3 . 8 )
In this case, we choose the number 𝑞 so that the coefficient of 𝑐 ( 𝑢 ( 𝑡 ) ) in (3.8) be equal to zero, then 2 𝑞 ( 𝛾 + 1 ) ( 𝑟 2 ) 𝒱 0 + 2 𝑑 ( 𝑟 2 ( 𝛾 + 1 ) ) 𝒱 0 + 2 ( 𝛾 + 1 ) 𝑑 . ( 3 . 9 )
We note that 2 < 𝑞 < 2 ( 𝛾 + 1 ) , and we get 𝐼 ( 𝑢 ( 𝑡 ) ) 𝛾 𝑟 𝑎 ( 𝑢 ( 𝑡 ) ) + ( 𝑟 2 ( 𝛾 + 1 ) ) 𝑏 ( 𝑢 ( 𝑡 ) ) ( 𝑟 2 ( 𝛾 + 1 ) ) 𝒱 0 𝒱 + 2 ( 𝛾 + 1 ) 𝑑 0 . ( 3 . 1 0 )
Therefore, from (3.6) and (3.10), 𝐼 ( 𝑢 ( 𝑡 ) ) 𝐶 𝑏 ( 𝑢 ( 𝑡 ) ) , ( 3 . 1 1 ) where 𝐶 𝑟 2 ( 𝛾 + 1 ) 𝑟 m i n 1 , 𝑟 𝒱 0 ( 𝑟 2 ( 𝛾 + 1 ) ) 𝒱 0 + 2 ( 𝛾 + 1 ) 𝑑 > 0 . ( 3 . 1 2 )
Now, we define the function, along the solution, by ( 𝑡 ) 𝒱 1 / 𝑎 ( 𝑡 ) + 𝜖 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 , ( 3 . 1 3 ) where 𝑎 ( 1 + ( 1 𝑘 ) / 𝑟 ) 1 ( 1 , 2 ) and 𝜖 > 0 will be choosen later.
We intend to apply Lemma 2.7 to functional (3.13). First, we calculate the derivative, along solutions, with respect to 𝑡 . Let us start with the second term of (3.13). From (3.2)–(3.4) and (3.11), one has 𝑑 𝑑 𝑡 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 = 𝑣 ( 𝑡 ) 2 2 | | | | 𝐼 ( 𝑢 ( 𝑡 ) ) 𝛿 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) 𝑣 ( 𝑡 ) 𝜆 2 2 𝑣 ( 𝑡 ) 2 2 + 𝐶 𝜇 𝑢 ( 𝑡 ) 𝑟 𝑟 𝐶 𝒱 ( 1 𝑘 ) / 𝑟 ( 𝑡 ) 𝜈 𝛿 𝑢 ( 𝑡 ) 𝑟 𝑘 𝜆 + 1 ̇ 𝐶 ( 𝜈 ) 𝒱 ( 𝑡 ) 𝑣 ( 𝑡 ) 2 2 + 𝑟 𝐶 𝜇 𝜈 𝐶 𝛿 𝜇 ( 𝑘 𝜆 𝑟 ) / 𝑟 𝒱 𝑏 0 𝑢 ( 𝑡 ) 𝑟 𝑟 𝐶 𝑎 ̇ 𝒱 𝐶 ( 𝜈 ) 1 / 𝑎 ( 𝑡 ) 𝑣 ( 𝑡 ) 2 2 + 𝐶 𝜇 2 𝑢 ( 𝑡 ) 𝑟 𝑟 𝐶 𝑎 ̇ 𝒱 𝐶 ( 𝜈 ) 1 / 𝑎 ( 𝑡 ) , ( 3 . 1 4 ) where 𝑏 ( 𝑘 ( 𝜆 1 ) ( 𝑟 1 ) ) / 𝑟 < 0 , and 𝜈 > 0 is sufficiently small.
Consequently, if 𝜖 > 0 is sufficiently small, ̇ 𝐶 ( ( 𝑡 ) 𝑣 𝑡 ) 2 2 + 𝑢 ( 𝑡 ) 𝑟 𝑟 > 0 , ( 3 . 1 5 ) where 𝐶 𝜖 m i n ( 1 , 𝐶 𝜇 / 2 ) > 0 .
From (3.15) and choosing 𝜖 > 0 small enough, we get ( 𝑡 ) 0 𝒱 0 𝑢 𝑡 + 𝜖 0 𝑡 , 𝑣 0 2 > 0 . ( 3 . 1 6 )
Utilizing two times (3.3), we get 𝑎 ( 𝑡 ) 2 𝑎 1 𝒱 ( 𝑡 ) + 𝜖 𝑎 | | ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 | | 𝑎 2 𝑎 1 𝜇 𝑟 𝑢 ( 𝑡 ) 𝑟 𝑟 + 𝜖 𝑎 𝐶 ( Ω ) 𝑎 𝑢 ( 𝑡 ) 𝑎 𝑟 𝑣 ( 𝑡 ) 𝑎 2 2 𝑎 1 𝜇 𝑟 𝑢 ( 𝑡 ) 𝑟 𝑟 + 𝜖 𝑎 𝐶 ( Ω ) 𝑎 𝑢 ( 𝑡 ) 𝑟 2 𝑎 / ( 2 𝑎 ) + 𝑣 ( 𝑡 ) 2 2 2 𝑎 1 𝜇 𝑟 𝑢 ( 𝑡 ) 𝑟 𝑟 + 𝜖 𝑎 𝐶 ( Ω ) 𝑎 𝜇 ( 𝑟 2 ) 2 𝑟 𝑑 𝑐 𝑢 ( 𝑡 ) 𝑟 𝑟 + 𝑣 ( 𝑡 ) 2 2 𝐶 ( 𝑢 𝑡 ) 𝑟 𝑟 + 𝑣 ( 𝑡 ) 2 2 , ( 3 . 1 7 ) where 𝐶 ( Ω ) > 0 is the imbedding constant of 𝐿 𝑟 ( Ω ) 𝐿 2 ( Ω ) , 𝑐 ( 1 2 𝑎 ) / 𝑟 ( 2 𝑎 ) > 0 , and 𝐶 > 0 .
Hence and from (3.15), we obtain the inequality in order to apply Lemma 2.7. Therefore, the maximal time of existence is finite: 𝑇 < .
Necessity
Suppose that 𝜆 𝑟 . Define the function, along the solution, by 𝒲 ( 𝑡 ) 𝐸 ( 𝑡 ) + 2 𝜇 𝑟 𝑢 ( 𝑡 ) 𝑟 𝑟 . ( 3 . 1 8 )

Then, ̇ ̇ | | | | 𝒲 ( 𝑡 ) = 𝐸 ( 𝑡 ) + 2 𝜇 𝑢 ( 𝑡 ) 𝑢 ( 𝑡 ) 𝑟 2 , 𝑣 ( 𝑡 ) 2 𝛿 𝑣 ( 𝑡 ) 𝜆 𝜆 + 𝛿 2 𝑣 ( 𝑡 ) 𝑟 𝜆 + 𝐶 𝑢 ( 𝑡 ) 𝑟 𝑟 𝐶 ( 𝒲 ( 𝑡 ) + 1 ) , ( 3 . 1 9 ) where 𝐶 m a x ( 𝛿 / 2 , 𝐶 𝑟 / 𝜇 ) , 𝐶 2 𝜇 𝐶 ( Ω ) ( 𝛿 / 2 ) 𝑟 , and 𝐶 ( Ω ) > 0 is the imbedding constant of 𝐿 𝜆 ( Ω ) 𝐿 𝑟 ( Ω ) .
Hence, by Gronwall inequality, it follows that ( 𝑢 , 𝑣 ) is bounded in 𝐻 for any finite time. A contradiction.
Proceeding again by contradiction suppose that, for all 𝑡 0 , ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 𝒰 . Then, by Lemma 2.6, we have either 𝑢 ( 𝑡 ) 𝒮 for all 𝑡 0 , or 𝐸 ( 𝑡 ) 𝑑 for all 𝑡 0 . In the first case, from (2.28) in Lemma 2.3 𝐸 0 1 2 𝑣 ( 𝑡 ) 2 2 + 𝑟 2 2 𝑟 𝑎 ( 𝑢 ( 𝑡 ) ) , ( 3 . 2 0 ) that is, ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) is bounded in 𝐻 . This is not possible. In the second case, 𝛿 𝑡 0 𝑣 ( 𝜏 ) 𝜆 𝜆 𝑑 𝜏 𝐸 0 𝑑 , ( 3 . 2 1 ) where 𝐸 0 𝐸 ( 𝑢 0 , 𝑣 0 ) . Hence, by the Hölder inequality, 𝛿 𝑡 ( 𝜆 1 ) 𝑡 0 𝑣 ( 𝜏 ) 𝑑 𝜏 𝜆 𝜆 𝐸 0 𝑑 , ( 3 . 2 2 ) and consequently 𝑢 ( 𝑡 ) 𝜆 𝐶 ( 𝑇 ) , ( 3 . 2 3 ) for 𝑡 [ 0 , 𝑇 ] , where 𝐶 ( 𝑇 ) 𝑢 0 𝜆 + ( ( 𝐸 0 𝑑 ) / 𝛿 ) 1 / 𝜆 𝑇 ( 𝜆 1 ) / 𝜆 .
From Theorem 2.1, l i m 𝑡 𝑇 M A X 𝑢 ( 𝑡 ) 𝑟 = ; ( 3 . 2 4 ) hence, by Sobolev-Poincaré's inequality (2.16), for every 𝑀 > 𝐸 0 , there exists some ̂ 𝑡 > 0 , such that 𝑀 < 𝑟 2 2 𝑟 𝑎 ( 𝑢 ( 𝑡 ) ) , ( 3 . 2 5 ) for every ̂ 𝑡 𝑡 . This implies the first inequality of (2.29) in Lemma 2.3, replacing 𝑑 by 𝑀 . Now, we consider the function (3.13) ( 𝑡 ) 𝒱 ( 𝑡 ) 1 / 𝑎 + 𝜖 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 , ( 3 . 2 6 ) defined for ̂ 𝑡 𝑡 , where 𝑎 ( 1 , 2 ) , 𝜖 > 0 is sufficiently small, and here 𝒱 ( 𝑡 ) 𝑀 𝐸 ( 𝑡 ) > 0 , ( 3 . 2 7 ) and repeat the sufficiency part of the proof. Then, by Lemma 2.7, ( 𝑡 ) blowups as 𝑡 𝑇 , 𝑇 > ̂ 𝑡 . Moreover, for ̂ 𝑡 𝑡 < 𝑇 , ̂ 𝑡 ( 𝑡 ) ̂ 𝑡 / 𝑇 1 𝑡 ̂ 𝑡 1 / ( 𝑎 1 ) , ( 3 . 2 8 ) hence and from (3.26), (3.27), and since 𝐸 ( 𝑡 ) 𝑑 , 𝑢 ( 𝑡 ) 2 2 𝑢 ̂ 𝑡 2 2 + 2 𝜖 𝑡 ̂ 𝑡 ̂ 𝑡 ̂ 𝑡 / 𝑇 1 𝜏 ̂ 𝑡 1 / ( 𝑎 1 ) 𝒱 1 / 𝑎 𝑢 ̂ 𝑡 ( 𝜏 ) 𝑑 𝜏 2 2 2 ̂ 𝑡 𝑡 𝜖 ( 𝑀 𝑑 ) 1 / 𝑎 + 2 ( 𝑎 1 ) ̂ 𝑡 𝑇 𝜖 ( 2 𝑎 ) ̂ 𝑡 ̂ 𝑡 1 𝑡 𝑇 ̂ 𝑡 ( ( 2 𝑎 ) / ( 𝑎 1 ) ) . 1 ( 3 . 2 9 )
Consequently, l i m 𝑡 𝑇 𝑢 ( 𝑡 ) 2 2 = . ( 3 . 3 0 ) But this contradicts (3.23), since 𝐿 𝜆 ( Ω ) 𝐿 2 ( Ω ) . The proof is complete.

Remark 3.2. From the last result, if 𝜆 = 2 and 𝑟 > 2 ( 𝛾 + 1 ) , any solution of problem (1.1), ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) , is global if and only if either (i) there exists 𝑡 0 0 such that ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) 𝒮 or (ii) ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) [ 𝐸 ( 𝑢 , 𝑣 ) 𝑑 ] , for every 𝑡 0 . On the other hand, if 𝜆 > 2 and 𝑟 > 2 ( 𝛾 + 1 ) , then any solution is global if and only if one of the following holds: (i), (ii), or (iii) 𝜆 𝑟 and there exists 𝑡 0 0 such that ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) 𝒰 .

We next prove a characterization of convergence to the zero equilibrium, and we give rates of decay.

Theorem 3.3. Let ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) = 𝑆 ( 𝑡 ) ( 𝑢 0 , 𝑣 0 ) be a solution of problem (1.1) with 𝜆 2 . Suppose that 𝑟 > 2 ( 𝛾 + 1 ) and that 𝜆 1 0 , if 𝑛 = 5 . A necessary and sufficient condition for ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) ( 0 , 0 ) , strongly in 𝐻 as 𝑡 , is that there exists 𝑡 0 0 such that ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) 𝒮 .
In this case, if ( 𝑡 ) denotes either the energy 𝐸 ( 𝑡 ) 𝐸 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) , ( 3 . 3 1 ) or the norm of the solution in 𝐻 𝜔 ( 𝑡 ) ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 𝐻 𝑎 ( 𝑢 ( 𝑡 ) ) + 𝑣 ( 𝑡 ) 2 2 , ( 3 . 3 2 ) One has the rates of decay, for 𝑡 𝑇 , ( 𝑡 ) 𝐾 0 1 + 𝑡 𝜆 2 2 𝐾 1 𝐾 0 ( 𝜆 2 ) / 2 2 / ( 𝜆 2 ) , ( 3 . 3 3 ) and, for linear dissipation, 𝜆 = 2 , ( 𝑡 ) 𝐾 0 𝑒 𝐾 1 𝑡 , ( 3 . 3 4 ) where 𝑇 > 0 is sufficiently large, and 𝐾 0 > 0 , 𝐾 1 > 0 are constants depending only on initial conditions.

Proof. Necessity
By (ii) in Lemma 2.2, ( 0 , 0 ) 𝒰 , and since the equilibrium ( 0 , 0 ) [ 𝐸 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 𝑑 ] , strong closures in 𝐻 , then, by Lemma 2.6, the solution must eventually enter in 𝒮 .
Sufficiency
By energy equation and (2.27) in Lemma 2.3, the solution must be global and uniformly bounded in the norm of 𝐻 , that is 𝜔 ( 𝑡 ) < 2 𝑟 𝑑 / ( 𝑟 2 ) , for any 𝑡 0 . Hence, there exists a sequence of times, { 𝑡 𝑛 } , such that if 𝑛 then 𝑡 𝑛 , ( 𝑢 ( 𝑡 𝑛 ) , 𝑣 ( 𝑡 𝑛 ̂ ) ) ( ̂ 𝑢 , 𝑣 ) weakly in 𝐻 and, since the embedding 𝐵 𝐿 𝑟 ( Ω ) is compact, 𝑏 ( 𝑢 ( 𝑡 𝑛 ) ) 𝑏 ( ̂ 𝑢 ) . Also, notice that the energy is such that 0 𝐸 l i m 𝑡 𝐸 ( 𝑡 ) = i n f 𝑡 0 𝐸 ( 𝑡 ) < . ( 3 . 3 5 )
Consequently, from the energy equation and the continuous embedding 𝐿 𝜆 ( Ω ) 𝐿 2 ( Ω ) , l i m 𝑡 𝑡 𝑡 + 1 𝑣 ( 𝜏 ) 𝜆 2 𝑑 𝜏 = 0 , ( 3 . 3 6 ) in particular, for any sequence of times { 𝑠 𝑛 } such that 𝑠 𝑛 as 𝑛 , l i m 𝑛 1 0 𝑛 ( 𝜏 ) 𝑑 𝜏 = 0 , ( 3 . 3 7 ) where 𝑛 ( 𝜏 ) 𝑣 ( 𝑠 𝑛 + 𝜏 ) 𝜆 2 , for 𝜏 [ 0 , 1 ] . By Fatou Lemma, l i m i n f 𝑛 𝑣 𝑠 𝑛 + 𝜏 𝜆 2 = l i m i n f 𝑛 𝑛 ( 𝜏 ) = 0 , ( 3 . 3 8 ) for a.e. 𝜏 [ 0 , 1 ] , and by the weak convergence to ̂ 𝑣 , ̂ 𝑣 2 l i m i n f 𝑛 𝑣 𝑡 𝑛 2 = 0 , ( 3 . 3 9 ) where we choose { 𝑠 𝑛 } such that 𝑡 𝑛 = 𝑠 𝑛 + 𝜏 0 , for some 𝜏 0 [ 0 , 1 ] .

It can be shown that the semigroup generated by problem (1.1) is continuous in 𝐻 with the weak topology, and then that the weak limit set is positive invariant, see Ball [26]. Consequently ̂ ( ̂ 𝑢 , 𝑣 ) = ( 𝑢 𝑒 , 0 ) must be an equilibrium of (1.1). Furthermore, by the lower-semicontinuity of the norm in 𝐻 , one has ̂ 𝑏 𝑢 𝑒 𝑢 = ̂ 𝑎 𝑒 l i m i n f 𝑛 𝑣 𝑡 𝑛 2 2 𝑢 𝑡 + ̂ 𝑎 𝑛 = l i m 𝑛 𝑡 2 𝐸 𝑛 + 2 𝑟 ̂ 𝑏 𝑢 𝑡 𝑛 = 2 𝐸 + 2 𝑟 ̂ 𝑏 𝑢 𝑒 . ( 3 . 4 0 )
Hence, 𝑟 2 ̂ 𝑏 𝑢 2 𝑟 𝑒 𝐸 < 𝑑 . ( 3 . 4 1 ) Then, by (2.19) and (2.21), 𝑢 𝑒 = 0 , and l i m 𝑡 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) = ( 0 , 0 ) w e a k l y i n 𝐻 . ( 3 . 4 2 )
Strong convergence follows if we get the rates of decay in our statement. Here, we will adapt the technique used in Haraux and Zuazua [3], to (1.1). That technique is based on the construction of suitable Liapunov functions defined along solutions and the application of Lemma 2.8. One of them is the energy, and we will need one more, defined by 𝑊 ( 𝑡 ) 𝐸 ( 𝑡 ) + 𝜅 𝐸 ( 𝑡 ) 𝜆 / 2 1 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 , ( 3 . 4 3 ) where 𝜅 > 0 is a constant to be chosen later. We next prove that 𝑊 ( 𝑡 ) is equivalent to both, the energy 𝐸 ( 𝑡 ) and the norm 𝜔 ( 𝑡 ) of the solution, in the sense of (2.30) and (2.28) below. First we note that from (2.27) in Lemma 2.3, 𝑟 2 𝐾 2 𝑟 𝜔 ( 𝑡 ) 𝐸 ( 𝑡 ) 2 𝜔 ( 𝑡 ) , ( 3 . 4 4 ) where 𝐾 1 ( 𝛽 / 𝛼 ( 𝛾 + 1 ) ) ( 2 𝑟 𝑑 / 𝛼 ( 𝑟 2 ) ) 𝛾 .
Also, notice that from (3.43), | | | | 𝑊 ( 𝑡 ) 𝐸 ( 𝑡 ) 𝜅 𝐸 0 𝜆 / 2 1 𝑢 ( 𝑡 ) 2 𝑣 ( 𝑡 ) 2 𝜅 𝐸 0 𝜆 / 2 1 𝐶 1 ( Ω ) 𝜔 ( 𝑡 ) , ( 3 . 4 5 ) where 𝐶 1 ( Ω ) > 0 is a constant that depends on the continuous embedding 𝐵 𝐿 2 ( Ω ) . Hence and from (3.44), if 𝜅 is sufficiently small, then 1 2 3 𝐸 ( 𝑡 ) 𝑊 ( 𝑡 ) 2 𝐸 ( 𝑡 ) . ( 3 . 4 6 )
We will need the following estimate: 𝛿 | | | | 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) 𝑣 ( 𝑡 ) 𝜆 1 2 𝛿 𝑢 ( 𝑡 ) 𝜆 𝑣 ( 𝑡 ) 𝜆 𝜆 1 𝛿 𝐶 2 ( Ω ) 𝑎 ( 𝑢 ( 𝑡 ) ) ( 𝜆 2 ) / 2 𝜆 𝑎 ( 𝑢 ( 𝑡 ) ) 1 / 𝜆 𝑣 ( 𝑡 ) 𝜆 𝜆 1 𝛿 𝐶 2 ( Ω ) 2 𝑟 𝐸 𝑟 2 0 ( 𝜆 2 ) / 2 𝜆 𝑎 ( 𝑢 ( 𝑡 ) ) 1 / 𝜆 ̇ 𝐸 ( 𝑡 ) ( 𝜆 1 ) / 𝜆 1 𝜆 𝐶 ̇ 𝑎 ( 𝑢 ( 𝑡 ) ) 𝐸 ( 𝑡 ) , ( 3 . 4 7 ) where we applied (3.44) in the third step and Young inequality in last step, and the constants 𝐶 2 ( Ω ) > 0 , 𝐶 > 0 depend on the continuous embedding 𝐵 𝐿 𝜆 ( Ω ) , and 𝐶 also depends on 𝐸 0 .
It follows that, by (3.42) and since 𝐵 𝐿 𝑟 ( Ω ) is compact, for any 𝜖 > 0 , there exists some 𝑇 > 0 such that for any 𝑡 > 𝑇 𝑏 ( 𝑢 ( 𝑡 ) ) 𝐶 3 ( Ω ) 𝑏 ( 𝑢 ( 𝑡 ) ) ( 𝑟 2 ) / 𝑟 𝑎 ( 𝑢 ( 𝑡 ) ) 𝜖 𝐸 ( 𝑡 ) , ( 3 . 4 8 ) where 𝐶 3 ( Ω ) > 0 is the corresponding embedding constant and we used (3.44) in the last step.
Since we will apply Lemma 2.8, we need to calculate the time derivative of (3.43) and we begin with 𝑑 𝑑 𝑡 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 = 𝑣 ( 𝑡 ) 2 2 | | | | 𝑎 ( 𝑢 ( 𝑡 ) ) 𝑐 ( 𝑢 ( 𝑡 ) ) + 𝑏 ( 𝑢 ( 𝑡 ) ) 𝛿 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) 𝑣 ( 𝑡 ) 𝜆 1 2 𝑣 ( 𝑡 ) 2 2 1 2 𝐶 ̇ 3 𝑎 ( 𝑢 ( 𝑡 ) ) 𝑐 ( 𝑢 ( 𝑡 ) ) 𝐸 ( 𝑡 ) + 𝜖 𝐸 ( 𝑡 ) 2 𝑣 ( 𝑡 ) 2 2 𝐶 ̇ ( 1 𝜖 ) 𝐸 ( 𝑡 ) 𝐸 ( 𝑡 ) , ( 3 . 4 9 ) which holds for any 𝑡 > 𝑇 , and where we used (3.47), (3.48) and definition of 𝐸 ( 𝑡 ) .
We notice that for any small 𝜂 > 0 , and by Young inequality and energy equation 𝐸 𝜆 / 2 1 3 ( 𝑡 ) 2 𝑣 ( 𝑡 ) 2 2 𝐸 𝜆 / 2 1 ( 𝑡 ) 𝐶 4 ( Ω ) 𝑣 ( 𝑡 ) 2 𝜆 𝜂 𝐸 𝜆 / 2 ̇ ( 𝑡 ) 𝐶 ( 𝜂 ) 𝐸 ( 𝑡 ) , ( 3 . 5 0 ) where 𝐶 4 ( Ω ) > 0 , 𝐶 ( 𝜂 ) > 0 depend on the continuous embedding 𝐿 𝜆 ( Ω ) 𝐿 2 ( Ω ) , and 𝐶 ( 𝜂 ) depends on 𝜂 .
Then, for 𝜖 and 𝜂 sufficiently small, (3.49) and (3.50), imply 𝐸 𝜆 / 2 1 𝑑 ( 𝑡 ) 𝑑 𝑡 ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) 2 1 2 𝐸 𝜆 / 2 𝐶 ̇ ( 𝑡 ) 𝐸 ( 𝑡 ) , ( 3 . 5 1 ) for any 𝑡 > 𝑇 , where 𝐶 𝐶 ( 𝜂 ) + 𝐶 𝐸 0 𝜆 / 2 1 .
Consequently, for 𝜅 sufficiently small and any 𝑡 > 𝑇 ̇ ̇ 𝑊 ( 𝑡 ) 𝐸 ( 𝑡 ) 𝜅 𝜆 2 𝑟 2 𝑟 𝐶 1 ( Ω ) 𝐸 0 𝜆 / 2 1 ̇ 1 𝐸 ( 𝑡 ) 𝜅 2 𝐸 𝜆 / 2 𝐶 ̇ 𝜅 ( 𝑡 ) + 𝐸 ( 𝑡 ) 2 𝐸 𝜆 / 2 ( 𝑡 ) 𝜅 0 𝑊 𝜆 / 2 ( 𝑡 ) , ( 3 . 5 2 ) where 𝜅 0 ( 𝜅 / 2 ) ( 2 / 3 ) 𝜆 / 2 and 𝐶 1 ( Ω ) > 0 is the constant in (3.45); also we used (3.44), the fact that the energy is decreasing and (3.46). Then, from (3.52) and Lemma 2.8, we obtain the desired rates of decay for 𝑊 ( 𝑡 ) . The result now follows by (3.46) and (3.44), and the proof is complete.

Remark 3.4. By (2.35), the ground state is: [ ( 𝑢 , 0 ) 𝐻 𝑢 𝒩 ] = 𝒮 𝒰 . Then, in any 𝐻 -neighborhood of that subset of nonzero equilibria, one can choose initial conditions either in 𝒰 or in 𝒮 . Hence, by Theorem 3.1 and (3.3), the ground state is unstable in the sense of Liapunov when the dissipation term 𝑔 ( 𝑢 𝑡 ) is either linear or nonlinear.

Next we will study the behavior of solutions such that ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) [ 𝐸 ( 𝑢 , 𝑣 ) 𝑑 ] for all 𝑡 0 . First, we prove that those solutions are uniformly bounded in time. To that end we will study the cases: 𝜆 = 2 and 𝜆 > 2 separately, First, we consider the case 𝜆 = 2 .

Theorem 3.5. Let ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) = 𝑆 ( 𝑡 ) ( 𝑢 0 , 𝑣 0 ) be a solution of problem (1.1). Assume that 𝑟 > 2 ( 𝛾 + 1 ) , and 𝜆 = 2 . Also, assume that 𝑟 2 ( 6 / 𝑛 + 1 ) if 𝑛 2 . If ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) [ 𝐸 ( 𝑢 , 𝑣 ) 𝑑 ] for all 𝑡 0 , then the solution is global and uniformly bounded in 𝐻 , for all 𝑡 0 .

Proof. Suppose that ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) is not global, then by Theorem 2.1 blowups and by Theorem 3.1, ( 𝑢 ( 𝑡 0 ) , 𝑣 ( 𝑡 0 ) ) 𝒰 for some 𝑡 0 0 . Hence, ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) [ 𝐸 ( 𝑢 , 𝑣 ) < 𝑑 ] for all 𝑡 𝑡 0 . A contradiction.
Next, we will prove that 𝑢 ( 𝑡 ) 2 is uniformly bounded for all 𝑡 0 .
Let ( 𝑡 ) ( 1 / 2 ) 𝑢 ( 𝑡 ) 2 2 𝐶 , where 𝐶 > 0 is the constant given below. Then, we obtain ̈ ̇ ( 𝑡 ) + 𝛿 ( 𝑡 ) = 𝑣 ( 𝑡 ) 2 2 ̂ 𝑏 = ̂ 𝑎 ( 𝑢 ( 𝑡 ) ) + ( 𝑢 ( 𝑡 ) ) 𝑟 + 2 2 𝑣 ( 𝑡 ) 2 2 + 𝑟 2 2 ̂ 𝑎 ( 𝑢 ( 𝑡 ) ) 𝑟 𝐸 ( 𝑡 ) 𝐶 ( Ω ) ( 𝑟 2 ) ( 𝑡 ) , ( 3 . 5 3 ) where 𝐶 ( Ω ) > 0 is the imbedding constant of 𝐵 𝐿 2 ( Ω ) , and 𝐶 𝑟 𝐸 0 / ( 𝑟 2 ) 𝐶 ( Ω ) .
We define 𝒲 ( 𝑡 ) + ( 𝑡 ) s u p { ( 𝑡 ) , 0 } , the positive part of ( 𝑡 ) . We claim that, along solutions of (1.1), the time derivative satisfies ̇ 𝒲 ( 𝑡 ) 0 . Indeed, if this is no the case, there exists some 𝑡 0 > 0 such that 𝑡 0 ̇ 𝑡 > 0 , 0 > 0 . ( 3 . 5 4 )
By a standard comparison result for ordinary differential equations, (3.53) and (3.54) imply that ( 𝑡 ) as 𝑡 . Consequently, for any constant 𝐶 > 𝐸 0 , there exists some 𝑡 0 > 0 , such that for 𝑡 𝑡 0 𝐶 < 𝑟 2 2 𝑟 ̂ 𝑎 ( 𝑢 ( 𝑡 ) ) . ( 3 . 5 5 )
This is (2.29) in Lemma 2.3, replacing 𝑑 by 𝐶 . If we now define, for 𝑡 𝑡 0 , the function 𝒱 ( 𝑡 ) 𝐶 𝐸 ( 𝑡 ) , ( 3 . 5 6 ) we can repeat the sufficiency part of the proof of Theorem 3.1 and show that the solution blowups in a finite time, consequently is nonglobal. A contradiction. Then, 𝑢 ( 𝑡 ) 2 𝐶 < , for all 𝑡 + , and some constant 𝐶 > 0 .
Next, we will prove that uniform boundedness of 𝑢 ( 𝑡 ) 2 implies uniform boundedness of ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) in 𝐻 , for all 𝑡 + . To that end, we consider the functions ( 𝑡 ) 𝒢 ( 𝑡 ) 𝑘 𝐸 0 ̇ ( 𝑡 ) + 𝛿 ( 𝑡 ) 𝑘 𝐸 0 , where now ( 𝑡 ) ( 1 / 2 ) 𝑢 ( 𝑡 ) 2 2 and 𝑘 > 0 is defined below. From the second line in (3.53), ̇ ( 𝑡 ) 𝐾 ( 𝑡 ) , ( 3 . 5 7 ) where 𝐾 m i n { 𝑟 + 2 , 𝛼 𝐶 ( Ω ) ( 𝑟 2 ) / ( 1 + 𝛿 ) } > 0 and 𝑘 𝑟 / 𝐾 .
Hence, for 0 𝑠 𝜏 ,  ( 𝜏 ) ( 𝑠 ) 𝑒 ( 𝐾 ( 𝜏 𝑠 ) ) , ( 3 . 5 8 ) and consequently, from definition of ( 𝑡 ) , for 0 𝑠 𝜏 𝑡 ,  ( 𝑡 ) = ( 𝑠 ) 𝑒 𝛿 ( 𝑡 𝑠 ) + 𝑡 𝑠 ( 𝜏 ) + 𝑘 𝐸 0 𝑒 𝛿 ( 𝜏 𝑡 ) 𝑑 𝜏 ( 𝑠 ) 𝑒 𝛿 ( 𝑡 𝑠 ) + 𝑡 𝑠 ( 𝑠 ) 𝑒 𝐾 ( 𝜏 𝑠 ) + 𝑘 𝐸 0 𝑒 𝛿 ( 𝜏 𝑡 ) 𝑑 𝜏 ( 𝑠 ) 𝑒 𝛿 + 𝐾 𝐾 ( 𝑡 𝑠 ) 𝑒 𝛿 ( 𝑠 𝑡 ) + 𝑘 𝐸 0 𝛿 1 𝑒 𝛿 ( 𝑠 𝑡 ) . ( 3 . 5 9 )
Notice that if ( 𝑠 ) > 0 , for some 𝑠 0 , we obtain from (3.59) that l i m 𝑡 ( 𝑡 ) = . A contradiction. Then, for all 𝑡 0 , 𝒢 ( 𝑡 ) 𝑘 𝐸 0 . ( 3 . 6 0 )
Now, we define ̃ ( 𝑡 ) 𝒢 ( 𝑡 ) + 𝑘 𝐸 0 , and like in (3.57) ̇ ( 𝑡 ) 𝐾 ( 𝑡 ) , ( 3 . 6 1 ) where 𝐾 m i n { 𝑟 + 2 , 𝐶 ( Ω ) ( 𝑟 2 ) } > 0 and ̃ 𝐾 𝑘 𝑟 / . Hence, ( 𝑡 ) ( 0 ) 𝑒 𝐾 𝑡 m i n { ( 0 ) , 0 } , ( 3 . 6 2 ) and consequently, 𝒢 𝒢 ̃ ( 𝑡 ) m i n ( 0 ) , 𝑘 𝐸 0 . ( 3 . 6 3 ) Hence and from (3.60), 𝒢 ( 𝑡 ) is uniformly bounded in time.
We integrate the second line of (3.53) in terms of 𝒢 ( 𝑡 ) and, by the energy equation, we obtain 𝒢 ( 𝑡 + 1 ) 𝒢 ( 𝑡 ) 𝑟 2 ( 𝛾 + 1 ) 2 𝑡 𝑡 + 1 𝑣 ( 𝜏 ) 2 2 1 + 𝑎 ( 𝑢 ( 𝜏 ) ) + 𝛾 + 1 𝑐 ( 𝑢 ( 𝜏 ) ) 𝑑 𝜏 𝑟 𝐸 0 . ( 3 . 6 4 )
Hence and since 𝒢 ( 𝑡 ) is uniformly bounded in time, 𝑡 𝑡 + 1 𝜔 ( 𝜏 ) 𝑑 𝜏 𝐶 , ( 3 . 6 5 ) where 𝐶 > 0 is a constant, and 1 𝜔 ( 𝑡 ) 2 𝑣 ( 𝑡 ) 2 2 + 1 2 1 𝑎 ( 𝑢 ( 𝑡 ) ) + 2 1 ( 𝛾 + 1 ) 𝑐 ( 𝑢 ( 𝑡 ) ) = 𝐸 ( 𝑡 ) + 𝑟 𝑏 ( 𝑢 ( 𝑡 ) ) . ( 3 . 6 6 )
Next, we will show that there exists a constant 𝜅 > 0 , such that 𝜔 ( 𝑡 ) 𝜅 ( 𝜔 ( 𝑠 ) + 1 ) , ( 3 . 6 7 ) for any 0 𝑠 𝑡 𝑠 + 1 .
To this end, we calculate ̇ 𝜔 ( 𝑡 ) = ( 𝑣 ( 𝑡 ) , 𝑓 ( 𝑢 ( 𝑡 ) ) ) 2 𝛿 𝑣 ( 𝑡 ) 2 2 𝜇 𝑣 ( 𝑡 ) 2 𝑢 ( 𝑡 ) 𝑟 1 2 ( 𝑟 1 ) . ( 3 . 6 8 ) If 𝑛 = 1 , we integrate and obtain, for 0 𝑠 𝑡 𝑠 + 1 , that 𝜔 ( 𝑡 ) 𝜔 ( 𝑠 ) + 𝜇 𝑡 𝑠 ( 𝑣 𝜏 ) 2 ( 𝑢 𝜏 ) 𝑟 1 2 ( 𝑟 1 ) 𝜇 𝑑 𝜏 𝜔 ( 𝑠 ) + 2 𝑡 𝑠 ( 𝑣 𝜏 ) 2 2 + 𝑢 ( 𝜏 ) 2 ( 𝑟 1 ) 2 ( 𝑟 1 ) 𝜇 𝑑 𝜏 𝜔 ( 𝑠 ) + 2 𝑠 𝑠 + 1 𝑢 ( 𝜏 ) 2 ( 𝑟 1 ) 2 ( 𝑟 1 ) 𝑑 𝜏 + 𝜇 𝑡 𝑠 𝜔 ( 𝜏 ) 𝑑 𝜏 . ( 3 . 6 9 )
By Gronwall inequality and (3.65), 𝜇 𝜔 ( 𝑡 ) 𝜔 ( 𝑠 ) + 2 𝑢 𝐿 2 ( 𝑟 1 ) 2 ( 𝑟 1 ) ( Ω × ( 𝑠 , 𝑠 + 1 ) ) 𝑒 𝜇 𝐶 𝜔 ( 𝑠 ) + 𝑠 𝑠 + 1 𝑢 ( 𝜏 ) 2 2 + 𝛼 𝑢 ( 𝜏 ) 2 2 + 𝑣 ( 𝜏 ) 2 2 𝑑 𝜏 𝑟 1 𝐶 𝜔 ( 𝑠 ) + 𝑠 𝑠 + 1 𝜔 ( 𝜏 ) 𝑑 𝜏 𝑟 1 𝐶 𝜔 ( 𝑠 ) + 𝐶 𝑟 1 , ( 3 . 7 0 ) where 𝐶 > 0 , 𝐶 > 0 depend on the continuous embeddings 𝐵 𝐿 2 ( Ω ) , and 𝐻 1 ( Ω × ( 𝑠 , 𝑠 + 1 ) ) 𝐿 2 ( 𝑟 1 ) ( Ω × ( 𝑠 , 𝑠 + 1 ) ) .
If 2 𝑛 5 , we use Galiardo-Niremberg's inequality, 𝑢 𝑟 1 2 ( 𝑟 1 ) 𝐶 ( Ω ) 𝑢 𝐵 ( 𝑟 1 ) 𝑎 𝑢 2 ( 𝑟 1 ) ( 1 𝑎 ) , ( 3 . 7 1 ) where 𝐶 ( Ω ) > 0 , and 𝑎 = 𝑛 ( 𝑟 2 ) / 4 ( 𝑟 1 ) . Notice that 𝑎 < 1 if 2 𝑛 4 , and 𝑎 1 if 𝑛 = 5 , because 𝑟 2 ( 𝑛 2 ) / ( 𝑛 4 ) = 6 . Then, from ̇ 𝜔 ( 𝑡 ) 𝜇 𝑣 ( 𝑡 ) 2 𝑢 ( 𝑡 ) 𝑟 1 2 ( 𝑟 1 ) , ( 3 . 7 2 ) we integrate and apply Gronwall inequality, for 0 𝑠 𝑡 𝑠 + 1 , 𝜔 ( 𝑡 ) 𝜔 ( 𝑠 ) + 𝜇 𝑡 𝑠 𝑣 ( 𝜏 ) 2 𝑢 ( 𝜏 ) 𝑟 1 2 ( 𝑟 1 ) 𝑑 𝜏 𝜔 ( 𝑠 ) + 𝜇 𝐶 ( Ω ) 𝑡 𝑠 𝑣 ( 𝜏 ) 2 ( 𝑢 𝜏 ) 𝐵 ( 𝑟 1 ) 𝑎 𝑢 ( 𝜏 ) 2 ( 𝑟 1 ) ( 1 𝑎 ) 𝑑 𝜏 𝜔 ( 𝑠 ) + 𝜇 𝐶 ( Ω ) 𝑡 𝑠 𝜔 ( 𝜏 ) 𝑢 ( 𝜏 ) 𝐵 ( 𝑟 1 ) 𝑎 1 𝑢 ( 𝜏 ) 2 ( 𝑟 1 ) ( 1 𝑎 ) 𝑑 𝜏 𝜔 ( 𝑠 ) e x p 𝜇 𝐶 ( Ω ) 𝑡 𝑠 𝑢 ( 𝜏 ) 𝐵 ( 𝑟 1 ) 𝑎 1 𝑢 ( 𝜏 ) 2 ( 𝑟 1 ) ( 1 𝑎 ) 𝐶 𝑑 𝜏 𝜔 ( 𝑠 ) e x p 𝑡 𝑠 𝑢 ( 𝜏 ) 𝐵 ( 𝑟 1 ) 𝑎 1 , 𝑑 𝜏 ( 3 . 7 3 ) where 𝐶 𝜇 𝐶 ( Ω ) s u p 𝑡 0 𝑢 ( 𝑡 ) 2 ( 𝑟 1 ) ( 1 𝑎 ) .
Notice that ( 𝑟 1 ) 𝑎 1 2 because by hypothesis 𝑟 2 ( 6 / 𝑛 + 1 ) , then we use the Hölder inequality, and from (3.65) we get 𝐶 𝜔 ( 𝑡 ) 𝜔 ( 𝑠 ) e x p 𝑡 𝑠 𝑢 ( 𝜏 ) 2 𝐵 𝑑 𝜏 { ( 𝑟 1 ) 𝑎 1 } / 2 𝐶 𝜔 ( 𝑠 ) e x p 𝑠 𝑠 + 1 𝜔 ( 𝜏 ) 𝑑 𝜏 { ( 𝑟 1 ) 𝑎 1 } / 2 𝜔 ( 𝑠 ) e x p 𝐶 𝐶 { ( 𝑟 1 ) 𝑎 1 } / 2 . ( 3 . 7 4 )
Then (3.67) holds for any 𝑛 1 , under our assumptions on 𝑟 .
Consequently, (3.65) and (3.67) imply that 𝑣 ( 𝑡 ) 2 2 + 𝑢 ( 𝑡 ) 2 𝐵 𝑡 𝑡 1 2 𝜔 ( 𝑡 ) 𝑑 𝑠 2 𝜅 𝑡 𝑡 1 ( 𝜔 ( 𝑠 ) + 1 ) 𝑑 𝑠 2 𝜅 ( 𝐶 + 1 ) , ( 3 . 7 5 ) and the proof is complete.

Next, we consider the case 𝜆 > 2 . Due to our assumptions on 𝑟 , we restrict our analysis to 𝑛 𝛾 < 4 . Since 𝛾 1 , our analysis considers, at most, dimensions 𝑛 3 , whenever 𝛾 < 4 / 3 .

Theorem 3.6. Let ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) = 𝑆 ( 𝑡 ) ( 𝑢 0 , 𝑣 0 ) be a solution of problem (1.1). Assume that 𝑟 > 2 ( 𝛾 + 1 ) , and 𝜆 > 2 with 𝜆 2 ( 𝑛 + 1 ) / ( 𝑛 1 ) if 𝑛 2 . Also assume that 𝑟 < 2 ( 4 / 𝑛 + 1 ) for 𝑛 1 . If ( 𝑢