Blow-up phenomena for a reaction diffusion equation with special diffusion process
aa r X i v : . [ m a t h . A P ] J un Blow-up phenomena for a reaction diffusionequation with special diffusion process
Yuzhu Han
School of Mathematics, Jilin University, Changchun 130012, P.R. China
Abstract
This paper is concerned with the blow-up property of solutions to an initialboundary value problem for a reaction diffusion equation with special diffusion processes. Itis shown, under certain conditions on the initial data, that the solutions to this problem blowup in finite time, by combining Hardy inequality, “moving” potential well methods with somedifferential inequalities. Moreover, the upper and lower bounds for the blow-up time are alsoderived when blow-up occurs.
Keywords blow-up; blow-up time; reaction diffusion equation; special diffusion processes.
AMS Mathematics Subject Classification 2010:
It is well known, by the conservation law, that many diffusion processes with reaction canbe described by the following equation (see [7]) u t − ∇ · ( D ∇ u ) = f ( x, t, u, ∇ u ) , (1.1)where u ( x, t ) stands for the mass concentration in chemical reaction processes or temperaturein heat conduction, at position x in the diffusion medium and time t , the function D is calledthe diffusion coefficient or the thermal diffusivity, the term ∇ · ( D ∇ u ) represents the rate ofchange due to diffusion and f ( x, t, u, ∇ u ) is the rate of change due to reaction.In this paper, we consider the finite time blow-up properties of solutions to an initial bound-ary value problem of (1.1) with the special diffusion coefficient D = | x | , i.e., to the followingproblem u t | x | − ∆ u = k ( t ) u p , ( x, t ) ∈ Ω × (0 , T ) ,u ( x, t ) = 0 , ( x, t ) ∈ ∂ Ω × (0 , T ) ,u ( x,
0) = u ( x ) , x ∈ Ω , (1.2)where Ω is a bounded domain in R n ( n ≥
3) containing the origin 0 with smooth boundary ∂ Ω,1 < p < n +2 n − , and the initial datum u ∈ H (Ω) is nonnegative and nontrivial. Moreover, for x = ( x , x , · · · , x n ) ∈ R n , | x | = p x + x + · · · + x n . The weight function k ( t ) satisfies(A) k ( t ) ∈ C [0 , + ∞ ) , k (0) > , k ′ ( t ) ≥ , ∀ t ∈ [0 , + ∞ ) . † Corresponding author.Email addresses: [email protected](Y. Han). ∗ Supported by NSFC (11401252) and by The Education Department of Jilin Province (JJKH20190018KJ).
1n many practical situations one would like to know whether the solutions to some evolutionproblems blow up, and if so, at which time T blow-up occurs. Since T can not be determinedexplicitly in most cases, it is an important issue to establish the lower and (or) upper bounds for T . There is an abounding literature on blow-up properties of solutions to nonlinear evolutionequations and systems, of which we only refer the reader to the monograph of Hu [4], and to thesurvey papers of Levine [5] and of Galaktionov and V´azquez [1]. Clearly this list of referencesis far from complete and could be enlarged by the numerous papers cited in [1, 4, 5].For the case D ≡ f ( x, t, u, ∇ u ) = u p ( p > k ( t ) ≡
1, Tan [12] considered theexistence and asymptotic estimates of global solutions and finite time blow-up of local solutionsto problem (1.2). By using the potential well method proposed by Sattinger and Payne [8, 10]and Hardy inequality, he gave some sufficient conditions for the solutions to exist globally orto blow up in finite time, when the initial energy is subcritical, i.e., initial energy smaller thanthe mountain pass level. These results were later extended by Tan to p -Laplace equation withsubcritical initial energy [13], by Han to p -Laplace equation with supercritical initial energy [2]and by Zhou to porous medium equation and polytropic filtration equation [14, 15].Motivated by the works mentioned above, we shall consider the blow-up phenomena forproblem (1.2) and investigate what role the weight function k ( t ) plays in determining the blow-up condition and blow-up time of solutions to problem (1.2). To be a little more precise, weshall show, under the assumption (A) on k ( t ), that the solutions to problem (1.2) blow up infinite time if one of the following three assumptions holds:(i) the initial energy is negative, i.e., J ( u ; 0) < I ( u ; 0) < J ( u ; 0) ≤ d ( ∞ );(iii) 0 < J ( u ; 0) < C k u | x | k for some positive constant C .Moreover, the upper and lower bounds for the blow-up time are also derived, with the helpof Gagliardo-Nirenberg’s inequality. The main difficulties are of course caused by the weightfunction k ( t ) and the singular coefficient | x | − . Since k ( t ) may not be a constant, we have toconsider the “moving” potential wells, i.e., potential wells vary with time t , when proving case(ii). To overcome the difficulty caused by | x | − , we apply Hardy inequality when dealing withcase (iii).The rest of this paper is organized as follows. In Section 2, we shall introduce some defi-nitions and auxiliary lemmas as preliminaries. In Section 3, we give three sufficient conditionsfor the solutions to problem (1.2) to blow up in finite time, and derive the upper bounds forblow-up time for each case. The lower bound for blow-up time will be derived in Section 4. Throughout this paper, we denote by k · k r the norm on L r (Ω)(1 ≤ r ≤ ∞ ), and by ( · , · )the inner product in L (Ω). By H (Ω) we denote the Sobolev space such that both u and |∇ u | belong to L (Ω) for any u ∈ H (Ω), which will be endowed with the equivalent norm k u k H (Ω) = k∇ u k .We first recall a classical result essentially due to Hardy (see [3]).2 emma 2.1. Assume that u ∈ H ( R n ) , n ≥ . Then u | x | ∈ L ( R n ) , and Z R n | u | | x | d x ≤ H n Z R n |∇ u | d x, (2.1) where H n = 4 / ( n − . Remark 2.1.
For any u ∈ H (Ω) , extend u ( x ) to be for x ∈ R n \ Ω . Then u ∈ H ( R n ) and therefore (2.1) also holds for u ∈ H (Ω) . For any u ∈ H (Ω) and t ≥
0, define the time-dependent energy functional and Neharifunctional, respectively, by J ( u ; t ) = 12 k∇ u k − k ( t ) p + 1 k u k p +1 p +1 , (2.2)and I ( u ; t ) = k∇ u k − k ( t ) k u k p +1 p +1 . (2.3)Since p + 1 < nn − , both J ( · ; t ) and I ( · ; t ) are well defined and continuous in H (Ω) for any t ≥
0. We also define, for any t ≥
0, the “moving” Nahari’s manifold by N ( t ) = { v ∈ H (Ω) \ { } : I ( v ; t ) = 0 } . It is not hard to verify that N ( t ) is non-empty and the potential well depth d ( t ) = inf v ∈ H v =0 sup λ ≥ J ( λv ; t ) = inf v ∈ N ( t ) J ( v ; t )is positive for any t ≥ Lemma 2.2.
Suppose that (A) holds. Then for any t ∈ [0 , ∞ ) , (1) d ( t ) = p − p + 1) ( k ( t )) − p S p +1) p − p > , (2.4) where S p = inf v ∈ H v =0 k∇ v k k v k p +1 . (2.5)(2) d ( t ) is non-increasing and d ( ∞ ) ∈ [0 , d (0)] , where d ( ∞ ) := lim t →∞ d ( t ) .Proof. (1) Fix 0 = v ∈ H (Ω) and t ≥
0. Set F ( λ ) := J ( λv ; t ) = λ k∇ v k − k ( t ) p + 1 λ p +1 k v k p +1 p +1 , λ ≥ . Then it is easy to see that F ( λ ) has only one critical point λ = (cid:18) k∇ v k k ( t ) k v k p +1 p +1 (cid:19) p − , F ( λ ) isincreasing on (0 , λ ), decreasing on ( λ , ∞ ) and takes its maximum at λ = λ . Therefore, d ( t ) = inf v ∈ H
10 (Ω) v =0 sup λ ≥ J ( λv ; t ) = inf v ∈ H v =0 F ( λ )3 inf v ∈ H
10 (Ω) v =0 (cid:26) λ k∇ v k − k ( t ) p + 1 λ p +10 k v k p +1 p +1 (cid:27) = p − p + 1) ( k ( t )) − p inf v ∈ H v =0 k∇ v k k v k p +1 p +1) p − = p − p + 1) ( k ( t )) − p S p +1) p − p . (2) From assumption (A) and (2.4) we know that the conclusions in (2) are valid. The proofis complete.In this paper, we consider weak solutions to problem (1.2), which is defined as follows. Definition 2.1. (See [12]) A function u is called a (weak) solution to problem (1.2) in Ω × (0 , T ) if u ∈ L ∞ (0 , T ; H (Ω)) , Z T (cid:13)(cid:13)(cid:13) u t ( t ) | x | (cid:13)(cid:13)(cid:13) dt < ∞ , and u ( x, t ) satisfies u ( x,
0) = u ( x ) and (cid:16) u t | x | , v (cid:17) + ( ∇ u, ∇ v ) = k ( t )( u p , v ) , ∀ v ∈ H (Ω) , t ∈ (0 , T ) . (2.6)Local existence of weak solutions to problem (1.2) can be obtained by using the standardregularization method. Interested reader may refer to [12, 13] for a similar proof. Moreover, itfollows from the weak maximum principle that u ( x, t ) is nonnegative since u ( x ) ≥ u ( t ) to denote the weak solution u ( x, t ) to problem (1.2).From now on, we shall denote by T ∗ ∈ [0 , + ∞ ) the maximal existence time of u ( t ), which isdefined as follows. Definition 2.2.
Let u ( t ) be a weak solution to problem (1.2) . We say that u ( t ) blows up ata finite time T provided that u ( t ) exists for all t ∈ [0 , T ) and lim t → T (cid:13)(cid:13)(cid:13) u ( t ) | x | (cid:13)(cid:13)(cid:13) = + ∞ . (2.7) In this case, we say that the maximal existence time of u ( t ) is T . If (2.7) does not happen forany finite T , then u ( t ) is said to be a global solution and the maximal existence time of u ( t ) is + ∞ . Let the assumption (A) hold and assume that u ( t ) is a weak solution to problem (1.2). Thenthe following energy identity follows from a quite standard argument. Lemma 2.3. ([11]) Let the assumption (A) hold and u ( t ) be a weak solution to problem (1.2) . Then J ( u ( t ); t ) is non-increasing in t and it holds, for any t ∈ (0 , T ∗ ) , that J ( u ( t ); t ) + Z t (cid:16)(cid:13)(cid:13)(cid:13) u τ ( τ ) | x | (cid:13)(cid:13)(cid:13) + k ′ ( τ ) p + 1 k u ( τ ) k p +1 p +1 (cid:17) d τ = J ( u ; 0) . (2.8)Denote by S the set of weak solutions to the following elliptic problem ( − ∆ w = k (0) | w | p − w, x ∈ Ω ,w ( x ) = 0 , x ∈ ∂ Ω . (2.9)4o show the finite time blow-up of solutions to problem (1.2) for subcritical initial energy, weneed some basic properties of S which are summarized into the following lemma. Interestedreader may refer to [11] for a similar proof. Lemma 2.4.
Assume that (A) holds and u is a weak solution to problem (1.2) with initialdatum u ∈ H (Ω) . Then (1) u
6∈ S provided that k∇ u k = λ k u k > , where λ > is the first eigenvalue of − ∆ in Ω under homogeneous Dirichlet boundary condition. (2) S 6 = ∅ , S ⊂ N (0) and N (0) \ S 6 = ∅ . (3) (cid:13)(cid:13)(cid:13) u t (0) | x | (cid:13)(cid:13)(cid:13) > provided that u
6∈ S . We shall end up this section with the next two lemmas. The first one is a special formof Gagliardo-Nirenberg’s inequality (see [2]) and the second one is the starting point whenapplying concavity argument [6].
Lemma 2.5.
Let < p < n +2 n − . Then for any u ∈ H (Ω) we have k u k p +1 p +1 ≤ G k∇ u k α ( p +1)2 k u k (1 − α )( p +1)2 , (2.10) where α = n ( p − p +1) ∈ (0 , and G > is a constant depending only on Ω , n and p . Lemma 2.6. (See [2, 6]) Suppose that a positive, twice-differentiable function ψ ( t ) satisfiesthe inequality ψ ′′ ( t ) ψ ( t ) − (1 + θ )( ψ ′ ( t )) ≥ , ∀ t ≥ t ≥ , where θ > . If ψ ( t ) > , ψ ′ ( t ) > , then ψ ( t ) → ∞ as t → t ∗ ≤ t ∗ = t + ψ ( t ) θψ ′ ( t ) . With the preliminaries given in Section 2 at hand, we can now state and prove the mainresults in this paper. For simplicity, we shall write L ( t ) = 12 (cid:13)(cid:13)(cid:13) u ( t ) | x | (cid:13)(cid:13)(cid:13) in the sequel. We firstprove a finite time blow-up result for problem (1.2) with negative initial energy. Theorem 3.1.
Let (A) hold and u ( t ) be a weak solution to problem (1.2) . If J ( u ; 0) < ,then u ( t ) blows up in finite time. Moreover, T ∗ ≤ L (0)(1 − p ) J ( u ; 0) = (cid:13)(cid:13)(cid:13) u | x | (cid:13)(cid:13)(cid:13) (1 − p ) J ( u ; 0) .Proof. We shall apply the first order differential inequality technique from Philippin [9] toshow the finite time blow-up result for problem (1.2) with negative initial energy. For this, set K ( t ) = − J ( u ( t ); t ). Then L (0) > K (0) >
0. From (A) and (2.8) it follows that K ′ ( t ) = − ddt J ( u ( t ); t ) = (cid:13)(cid:13)(cid:13) u t ( t ) | x | (cid:13)(cid:13)(cid:13) + k ′ ( t ) p + 1 k u ( t ) k p +1 p +1 ≥ , which implies K ( t ) ≥ K (0) > t ∈ [0 , T ∗ ). Recalling (2.2), (2.3) and (2.6), we obtain,for any t ∈ [0 , T ∗ ), that L ′ ( t ) = (cid:16) u t ( t ) | x | , u ( t ) (cid:17) = − I ( u ( t ); t ) = p − k∇ u ( t ) k − ( p + 1) J ( u ( t ); t ) ≥ ( p + 1) K ( t ) . (3.1)5ecalling (2.8) and making use of Cauchy-Schwarz inequality, we arrive at L ( t ) K ′ ( t ) ≥ (cid:13)(cid:13)(cid:13) u ( t ) | x | (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) u t ( t ) | x | (cid:13)(cid:13)(cid:13) ≥ (cid:16) u ( t ) | x | , u t ( t ) (cid:17) = 12 ( L ′ ( t )) ≥ p + 12 L ′ ( t ) K ( t ) , (3.2)which then implies (cid:16) K ( t ) L − p +12 ( t ) (cid:17) ′ = L − p +32 ( t ) (cid:16) K ′ ( t ) L ( t ) − p + 12 K ( t ) L ′ ( t ) (cid:17) ≥ . Therefore,0 < κ := K (0) L − p +12 (0) ≤ K ( t ) L − p +12 ( t ) ≤ p + 1 L ′ ( t ) L − p +12 ( t ) = 21 − p (cid:16) L − p ( t ) (cid:17) ′ . (3.3)Integrating (3.3) over [0 , t ] for any t ∈ (0 , T ∗ ) and noticing that p >
1, one has κt ≤ − p (cid:16) L − p ( t ) − L − p (0) (cid:17) , or equivalently 0 ≤ L − p ( t ) ≤ L − p (0) − p − κt, t ∈ (0 , T ∗ ) . (3.4)It is obvious that (3.4) can not hold for all t >
0. Therefore, T ∗ < + ∞ . Moreover, it can beinferred from (3.4) that T ∗ ≤ p − κ L − p (0) = 2 L (0)(1 − p ) J ( u ; 0) . The proof is complete.
Remark 3.1.
According to Theorem 3.1, if the weak solution u ( t ) to problem (1.2) existsglobally, then J ( u ( t ); t ) ≥ for all t ∈ [0 , + ∞ ) . For the case of J ( u ; 0) ≥
0, we obtain a blow-up results when the initial energy is “subcrit-ical” and when the initial Nehari functional is negative. More precisely, we have the followingtheorem.
Theorem 3.2.
Assume that (A) holds and that u ∈ H (Ω) satisfies J ( u ; 0) ≤ d ( ∞ ) and I ( u ; 0) < . (3.5) Then the weak solution u ( t ) to problem (1.2) blows up in finite time. Furthermore, T ∗ can beestimated from above as follows T ∗ ≤ t + 8 pL ( t )( p + 1)( p − [ d ( ∞ ) − J ( u ( t ); t )] , where t ≥ is any finite time such that J ( u ( t ); t ) < d ( ∞ ) . In particular, if J ( u ; 0) < d ( ∞ ) ,then T ∗ ≤ p (cid:13)(cid:13)(cid:13) u | x | (cid:13)(cid:13)(cid:13) ( p + 1)( p − [ d ( ∞ ) − J ( u ; 0)] . (3.6)6 roof. We will divide the proof into three steps.
Step I.
Set V ( t ) = { v ∈ H (Ω) : J ( v ; t ) < d ( t ) , I ( v ; t ) < } t ∈ [0 , T ∗ ) . We claim that there exists a t ∈ [0 , T ∗ ) such that u ( t ) ∈ V ( t ) for all t ∈ [ t , T ∗ ) provided that J ( u ; 0) ≤ d ( ∞ ) and I ( u ; 0) < J ( u ; 0) < d ( ∞ ), take t = 0, then it follows from (2.4) and (2.8)that J ( u ( t ); t ) ≤ J ( u ; 0) < d ( ∞ ) ≤ d ( t ) , t ∈ [0 , T ∗ ) . (3.7)It remains to show that I ( u ( t ); t ) < t ∈ [0 , T ∗ ). Since I ( u ; 0) <
0, by continuity, thereexists a suitably small t > I ( u ( t ); t ) < t ∈ [0 , t ). Suppose on the contrarythat there exists a t > t such that I ( u ( t ); t ) = 0 and I ( u ( t ); t ) < t ∈ [0 , t ). Thenby the definition of d ( t ), one obtains J ( u ( t ); t ) ≥ inf v ∈N ( t ) J ( v ; t ) = d ( t ) , which contradicts (3.7). Therefore, I ( u ( t ); t ) < t ∈ [0 , T ∗ ).When J ( u ; 0) = d ( ∞ ), by continuity and I ( u ; 0) < t > I ( u ( t ); t ) < t ∈ [0 , t ). In addition, Lemma 2.4 says that u is not a weak solutionto problem (2.9) and (cid:13)(cid:13)(cid:13) u t (0) | x | (cid:13)(cid:13)(cid:13) >
0. By continuity again, there exists a t ∈ (0 , t ) such that I ( u ( t ); t ) < (cid:13)(cid:13)(cid:13) u t ( t ) | x | (cid:13)(cid:13)(cid:13) > t ∈ [0 , t ). Therefore, by recalling (2.8) and Lemma2.2, one obtains J ( u ( t ); t ) ≤ J ( u ( t ); t ) ≤ J ( u ; 0) − Z t (cid:13)(cid:13)(cid:13) u τ ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ < J ( u ; 0) = d ( ∞ ) ≤ d ( t ) . (3.8)By applying the argument similar to the case of J ( u ; 0) < d ( ∞ ), we can show that I ( u ( t ); t ) < t ∈ [ t , T ∗ ), and therefore u ( t ) ∈ V ( t ) for all t ∈ [ t , T ∗ ). Moreover, since L ′ ( t ) = − I ( u ( t ); t ), L ( t ) is strictly increasing on [ t , T ∗ ). Step II.
We show that k∇ u ( t ) k ≥ p + 1) d ( t ) p − , t ∈ [ t , T ∗ ) . (3.9)From Step I we know that I ( u ( t ); t ) < t ∈ [ t , T ∗ ). Therefore, k∇ u ( t ) k < k ( t ) k u ( t ) k p +1 p +1 ≤ k ( t ) S p +1 p k∇ u ( t ) k p +12 , t ∈ [ t , T ∗ ) , which implies (3.9), by the definition of d ( t ). Step III.
We show that T ∗ < + ∞ . For any T ∈ ( t , T ∗ ), define the positive function F ( t ) = Z tt L ( τ )d τ + ( T − t ) L ( t ) + β t + σ ) , t ∈ [ t , T ] , (3.10)where β > σ > − t . 7y direct computations F ′ ( t ) = L ( t ) − L ( t ) + β ( t + σ ) = Z tt ddτ L ( τ )d τ + β ( t + σ )= Z tt (cid:16) u ( τ ) , u τ ( τ ) | x | (cid:17) d τ + β ( t + σ ) , (3.11) F ′′ ( t ) = L ′ ( t ) + β = (cid:16) u ( t ) , u t ( t ) | x | (cid:17) + β = − I ( u ( t ); t ) + β = p − k∇ u ( t ) k − ( p + 1) J ( u ( t ); t ) + β = p − k∇ u ( t ) k − ( p + 1) h J ( u ( t ); t ) − Z tt (cid:16)(cid:13)(cid:13)(cid:13) u τ ( τ ) | x | (cid:13)(cid:13)(cid:13) + k ′ ( τ ) p + 1 k u ( τ ) k p +1 p +1 (cid:17) d τ i + β ≥ p − k∇ u ( t ) k − ( p + 1) J ( u ( t ); t ) + ( p + 1) Z tt (cid:13)(cid:13)(cid:13) u τ ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ + β. (3.12)Applying Cauchy-Schwarz inequality and H¨older’s inequality to yield f ( t ) := h Z tt (cid:13)(cid:13)(cid:13) u ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ + β ( t + σ ) ih Z tt (cid:13)(cid:13)(cid:13) u τ ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ + β i − h Z tt (cid:16) u, u τ | x | (cid:17) d τ + β ( t + σ ) i = h Z tt (cid:13)(cid:13)(cid:13) u ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ Z tt (cid:13)(cid:13)(cid:13) u τ ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ − (cid:16) Z tt (cid:16) u, u τ | x | (cid:17) d τ (cid:17) i + β h ( t + σ ) Z tt (cid:13)(cid:13)(cid:13) u τ ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ + Z tt (cid:13)(cid:13)(cid:13) u ( τ ) | x | (cid:13)(cid:13)(cid:13) d τ − t + σ ) Z tt (cid:16) u, u τ | x | (cid:17) d τ i ≥ . Therefore, by recalling (3.11), (3.12) and noticing the nonnegativity of f ( t ), we arrive at F ( t ) F ′′ ( t ) − p + 12 ( F ′ ( t )) = F ( t ) F ′′ ( t ) + p + 12 h f ( t ) − [2 F ( t ) − T − t ) L ( t )] (cid:0) Z tt (cid:13)(cid:13)(cid:13) u τ | x | (cid:13)(cid:13)(cid:13) d τ + β (cid:1)i ≥ F ( t ) F ′′ ( t ) − ( p + 1) F ( t ) (cid:16) Z tt (cid:13)(cid:13)(cid:13) u τ | x | (cid:13)(cid:13)(cid:13) d τ + β (cid:17) ≥ F ( t ) h p − k∇ u ( t ) k − ( p + 1) J ( u ( t ); t ) + ( p + 1) Z tt (cid:13)(cid:13)(cid:13) u τ | x | (cid:13)(cid:13)(cid:13) d τ + β − ( p + 1) Z tt (cid:13)(cid:13)(cid:13) u τ | x | (cid:13)(cid:13)(cid:13) d τ − ( p + 1) β i = F ( t ) h p − k∇ u ( t ) k − ( p + 1) J ( u ( t ); t ) − pβ i . (3.13)In view of (3.8), (3.9) and (3.13), we get, for any t ∈ [ t , T ] and β ∈ (cid:16) , ( p + 1)( d ( ∞ ) − J ( u ( t ); t )) p i that F ( t ) F ′′ ( t ) − p + 12 ( F ′ ( t )) ≥ , t ∈ [ t , T ] . < T − t ≤ F ( t )( p − F ′ ( t ) = 2 L ( t )( p − β ( t + σ ) ( T − t ) + t + σp − , or ( T − t ) (cid:16) − L ( t )( p − β ( t + σ ) (cid:17) ≤ t + σp − . (3.14)Fix a β ∈ (cid:16) , ( p + 1)( d ( ∞ ) − J ( u ( t ); t )) p i . Then for any σ ∈ (cid:16) L ( t )( p − β − t , + ∞ (cid:17) , wehave 0 < L ( t )( p − β ( t + σ ) <
1, which, together with (3.14), implies that T ≤ t + t + σp − (cid:16) − L ( t )( p − β ( t + σ ) (cid:17) − = t + β ( t + σ ) ( p − β ( t + σ ) − L ( t ) . (3.15)Minimizing the right hand side in (3.15) for σ ∈ (cid:16) L ( t )( p − β − t , + ∞ (cid:17) to yield T ≤ inf σ ∈ ( L ( t p − β − t , + ∞ ) h t + β ( t + σ ) ( p − β ( t + σ ) − L ( t ) i = t + 8 L ( t )( p − β . (3.16)Minimizing the right hand side of (3.16) with respect to β ∈ (cid:16) , ( p + 1)( d ( ∞ ) − J ( u ( t ); t )) p i one obtains T ≤ t + 8 pL ( t )( p + 1)( p − [ d ( ∞ ) − J ( u ( t ); t )] . By the arbitrariness of
T < T ∗ we finally get T ∗ ≤ t + 8 pL ( t )( p + 1)( p − [ d ( ∞ ) − J ( u ( t ); t )] . In particular, if J ( u ; 0) < d ( ∞ ), then, by taking t = 0, we have T ∗ ≤ p (cid:13)(cid:13)(cid:13) u | x | (cid:13)(cid:13)(cid:13) ( p + 1)( p − [ d ( ∞ ) − J ( u ; 0)] . The proof is complete.
Remark 3.2.
It is easily seen from (2.2) and (2.3) that J ( u ; 0) < implies I ( u ; 0) < .Therefore, Theorem 3.1 can be viewed as a special case of Theorem 3.2. But we obtained theupper bounds for T ∗ by using different techniques. By comparing the two upper bounds directlyone can see that the one in Theorem 3.1 is more accurate when ≤ d ( ∞ ) ≤ p +1 p − ( − J ( u ; 0)) ,while the one in Theorem 3.2 is more accurate when < p +1 p − ( − J ( u ; 0)) ≤ d ( ∞ ) . At the end of this section, we give another blow-up condition for problem (1.2), whichensures that problem (1.2) admits blow-up solutions at arbitrarily high initial energy level.The result in this direction is the following theorem.9 heorem 3.3.
Assume that (A) holds and that u ( t ) is a weak solution to problem (1.2) . If < J ( u ; 0) < L (0) C , (3.17) then u ( t ) blows up at some finite time T ∗ . Moreover, the upper bound for T ∗ has the followingform T ∗ ≤ pH n L (0)( p − [ L (0) − C J ( u ; 0)] , where C = ( p +1) H n p − and H n is the positive constant given in Hardy inequality.Proof. This theorem will be proved by using some ideas from [2, 11] and an application ofHardy inequality.First, by using (3.12) and Hardy inequality (2.1) we have ddt L ( t ) = p − k∇ u ( t ) k − ( p + 1) J ( u ( t ); t ) ≥ p − H n (cid:13)(cid:13)(cid:13) u ( t ) | x | (cid:13)(cid:13)(cid:13) − ( p + 1) J ( u ( t ); t )= p − H n h L ( t ) − C J ( u ( t ); t ) i . (3.18)Set M ( t ) = L ( t ) − C J ( u ( t ); t ) , t ∈ [0 , T ∗ ) , then M (0) = L (0) − C J ( u ; 0) > ddt M ( t ) = ddt L ( t ) − C ddt J ( u ( t ); t ) ≥ ddt L ( t ) ≥ p − H n M ( t ) . (3.19)Therefore, an application of Gronwall’s inequality implies that M ( t ) ≥ M (0) e p − Hn t > , (3.20)which, together with (3.18), shows that L ( t ) is strictly increasing on [0 , T ∗ ).For any T ∈ (0 , T ∗ ), β > σ >
0, define F ( t ) = Z t L ( τ )d τ + ( T − t ) L (0) + β t + σ ) , t ∈ [0 , T ] . Similarly to the derivation of (3.13) we get F ( t ) F ′′ ( t ) − p + 12 ( F ′ ( t )) ≥ F ( t ) h p − k∇ u ( t ) k − ( p + 1) J ( u ; 0) − pβ i . (3.21)Applying Hardy inequality (2.1) again and noticing the monotonicity of L ( t ), we further obtain F ( t ) F ′′ ( t ) − p + 12 ( F ′ ( t )) ≥ F ( t ) h p − H n (cid:13)(cid:13)(cid:13) u ( t ) | x | (cid:13)(cid:13)(cid:13) − ( p + 1) J ( u ; 0) − pβ i ≥ F ( t ) h p − H n (cid:13)(cid:13)(cid:13) u | x | (cid:13)(cid:13)(cid:13) − ( p + 1) J ( u ; 0) − pβ i = p − H n F ( t ) h M (0) − pH n p − β i ≥ , (3.22)10or all β ∈ (0 , ( p − M (0) pH n ].Starting with (3.22), recalling Lemma 2.6 and applying similar arguments to that in theproof of Theorem 3.2 we get T ∗ ≤ pH n L (0)( p − M (0) = 8 pH n L (0)( p − [ L (0) − C J ( u ; 0)] . The proof is complete.
Remark 3.3.
Theorem 3.3 implies that for any
R > , there exists a u such that J ( u ; 0) = R < L (0) /C , while the corresponding solution u ( x, t ) to problem (1.2) with u as initial datumblows up in finite time. We refer the interested reader to [2, 11] for the standard proof of thisstatement. In this section, we shall derive a lower bound for the blow-up time T ∗ , by combining thefamous Gagliardo-Nirenberg’s inequality with the first order differential inequalities. Theorem 4.1.
Assume that (A) holds and < p < n . Let u ( t ) be a weak solution toproblem (1.2) that blows up at T ∗ . Then T ∗ ≥ L − γ (0) C ∗ ( γ − , where γ > and C ∗ > are twoconstants that will be determined in the proof.Proof. Combining (3.12) with Gagliardo-Nirenberg’s inequality and recalling the monotonicityof k ( t ), we have L ′ ( t ) = − I ( u ( t ); t ) = k ( t ) k u ( t ) k p +1 p +1 − k∇ u ( t ) k ≤ k G k∇ u ( t ) k α ( p +1)2 k u ( t ) k (1 − α )( p +1)2 − k∇ u ( t ) k , (4.1)where k is an arbitrary upper bound for k ( T ∗ ), G and α are the positive constants given inLemma 2.5. Since 1 < p < n , it is directly verified that0 < α ( p + 1) = n ( p − < . Applying Young’s inequality to the first term on the right hand side of (4.1), we obtain, for any ε >
0, that k∇ u ( t ) k α ( p +1)2 k u ( t ) k (1 − α )( p +1)2 ≤ α ( p + 1)2 ε k∇ u ( t ) k + 2 − α ( p + 1)2 ε − α ( p +1)2 − α ( p +1) k u ( t ) k γ , (4.2)where γ = (1 − α )( p + 1)2 − α ( p + 1) >
1. Taking ε = 2 k Gα ( p + 1) and substituting (4.2) into (4.1) toyield L ′ ( t ) ≤ C k u ( t ) k γ ≤ γ ( diam (Ω)) γ C L γ ( t ) := C ∗ L γ ( t ) , (4.3)where C is a positive constant depending on n , p , k and G , diam (Ω) > C ∗ = 2 γ ( diam (Ω)) γ C . Integrating (4.3) over [0 , t ), we get11 − γ n L − γ ( t ) − L − γ (0) o ≤ C ∗ t. γ >
1, letting t → T ∗ in the above inequality and recalling that lim t → T ∗ L ( t ) = + ∞ , weobtain T ∗ ≥ L − γ (0) C ∗ ( γ − . The proof is complete.
Remark 4.1.
In [11], the authors investigated the blow-up properties of solutions to aclass of semilinear parabolic or pseudo-parabolic equations, and obtained, among many otherinteresting results, the lower bounds for the blow-up time only for the pseudo-parabolic case. Inour paper, by applying the famous Gagliardo-Nirenberg’s inequality, we derived a lower boundfor the blow-up time for the parabolic problem (1.2) . Moreover, our treatment can also be appliedto the parabolic problem considered in [11] to obtain the lower bound for the blow-up time.
Acknowledgements
The author would like to express his sincere gratitude to Professor Wenjie Gao in JilinUniversity for his enthusiastic guidance and constant encouragement. He would also like tothank the referees for their valuable comments and suggestions, especially for pointing out themistake of the blow-up time in Theorem 3.3 in the original manuscript.
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