Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities
aa r X i v : . [ m a t h . A P ] N ov BLOW-UP AND LIFESPAN ESTIMATE FOR THE GENERALIZEDTRICOMI EQUATION WITH MIXED NONLINEARITIES
MAKRAM HAMOUDA AND MOHAMED ALI HAMZA Abstract.
We study in this article the blow-up of the solution of the generalizedTricomi equation in the presence of two mixed nonlinearities, namely we consider(
T r ) u tt − t m ∆ u = | u t | p + | u | q , in R N × [0 , ∞ ) , with small initial data, where m ≥ T r ) with m = 0, which corresponds to the uniform wave speed ofpropagation, it is known that the presence of mixed nonlinearities generates a newblow-up region in comparison with the case of a one nonlinearity ( | u t | p or | u | q ). Weshow in the present work that the competition between the two nonlinearities stillyields a new blow region for the Tricomi equation ( T r ) with m ≥
0, and we derive anestimate of the lifespan in terms of the Tricomi parameter m . As an application ofthe method developed for the study of the equation ( T r ) we obtain with a differentapproach the same blow-up result as in [18] when we consider only one time-derivativenonlinearity, namely we keep only | u t | p in the right-hand side of ( T r ). Introduction
We consider the following semilinear damped wave equation characterized by a speedof propagation of polynomial type in time, namely the well-known
Generalized Tricomi wave equation with mixed nonlinearities reads as follows:(1.1) ( u tt − t m ∆ u = | u t | p + | u | q , in R N × [0 , ∞ ) ,u ( x,
0) = εf ( x ) , u t ( x,
0) = εg ( x ) , x ∈ R N , where m ≥ N ≥ ε > f, g are positive functions chosen in theenergy space with compact support. Moreover, the functions f and g are supposed tobe positive and compactly supported on B R N (0 , R ) , R > m ≥ p, q > q ≤ NN − if N ≥ Mathematics Subject Classification.
Key words and phrases. blow-up, generalized Tricomi equation, lifespan, critical curve, nonlinear waveequations, time-derivative nonlinearity. he system (1.1) in a more general context. More precisely, we consider the followingfamily of semilinear wave equations:(1.2) ( u tt − t m ∆ u = a | u t | p + b | u | q , in R N × [0 , ∞ ) ,u ( x,
0) = εf ( x ) , u t ( x,
0) = εg ( x ) , x ∈ R N , where a and b are two nonnegative constants.Let ( a, b ) = (0 ,
1) and m = 0 in (1.2). It is worth mentioning that the obtainedsystem was subject to several works and the related blow-up phenomenon is more orless well understood and is linked to the Strauss conjecture. More precisely, this caseinvolves the critical power, q S , which is the positive root of the quadratic equation( N − q − ( N + 1) q − q S = q S ( N ) := N + 1 + √ N + 10 N − N − . Indeed, if q ≤ q S then no global solution for (1.2) does exist (under some assumptionson the initial data), and for q > q S a global solution exists for small initial data; see e.g.[17, 22, 26, 27].Now, the case m = 0 and ( a, b ) = (1 ,
0) is connected with the Glassey conjecturesaying that the critical power p G should be given by(1.4) p G = p G ( N ) := 1 + 2 N − . The critical value p G generates two regions for the power p ; one stating the global exis-tence (for p > p G ) and another one for the nonexistence (for p ≤ p G ) of a global smalldata solution; see e.g. [12, 14, 16, 21, 24, 28].The case m = 0 and a, b = 0 (we may assume that ( a, b ) = (1 , p ≤ p G or q ≤ q S , the blow-up of thesolution of (1.2) is easy. However, compared to a one single nonlinearity, the presence ofmixed nonlinearities produces a supplementary blow-up region. In fact, for p > p G and q > q S , the new blow-up frontier is characterized by the following relationship between p and q :(1.5) λ ( p, q, N ) := ( q − (cid:2) ( N − p − (cid:3) < m ≥ a, b ) = (0 ,
1) in (1.2). This case corresponds to thesemilinear generalized Tricomi equation with power nonlinearity which has been widelystudied in several works. In [9, 19], the authors proved the blow-up results for both the ubcritical and critical cases. However, the question of global existence for the system(1.2) with small data in the supercritical case has been proved only for m = 1 and fordimensions N = 1 ,
2; see e.g. [8, 9, 10, 11]. Hence, the critical exponent for (1.2), thatwe denote by q C ( N, m ), should be given by the greatest root of the following quadraticequation Q tr ( q ) := (( m + 1) N − q − (( m + 1) N + 1 − m ) q − m + 1) = 0 . (1.6)Naturally, for m = 0 we find again the well-known Strauss exponent q S ( N ), namely q C ( N,
0) = q S ( N ), where q S ( N ) is given by (1.3).For m ≥ a, b ) = (1 , ( u tt − t m ∆ u = | u t | p , in R N × [0 , ∞ ) ,u ( x,
0) = εf ( x ) , u t ( x,
0) = εg ( x ) , x ∈ R N . Concerning the blow-up results and lifespan estimate of the solution of (1.7), it wasproven in [20] that for p ≤ p G ( N ( m + 1)) we have the blow-up. To prove this result,the authors used, among other tools, the integral representation formula in one spacedimension. However, this result was recently improved in [18]. The main tool in thelatter work is the construction of adequate test functions using the properties of Besselfunctions. Hence, the new obtained region involves an almost sure candidate for thecritical exponent, that is(1.8) p ≤ p tr ( N, m ) := 1 + 2( m + 1)( N − − m . The authors in [18] conjecture that p tr ( N, m ) is in fact the critical exponent for theblow-up of the solution of (1.7). Of course this has to be confirmed by a global existenceresult for (1.7).We are interested in this article in studying the blow-up result of the solution of (1.1)for m ≥
0. First, let us recall that the case m = 0 in (1.1) corresponds in fact to theclassical wave equation with mixed nonlinearities which, compared to the case with onenonlinearity | u t | p or | u | q , for p > p G ( N ) or q > q S ( N ), generates a new blow-up region,given by (1.5), due to the interaction between the two mixed nonlinearities. In thiswork we will examine the influence of the parameter m on the blow-up result and thelifespan estimate for the Cauchy problem (1.1) for m ≥
0. Naturally, the emphasis willbe here for the exponents p > p tr ( N, m ) or q > q C ( N, m ). Our target is to give the curvedelimiting the new blow-up region for the Tricomi equation given by (1.1), and to lookinto its change compared to (1.5).The Tricomi equation constitutes somehow a time-dependent wave speed of propaga-tion. As shown below (see (1.10)), the effect of this Tricomi term is more or less similar o a scale-invariant damping. For instance, let v ( x, τ ) = u ( x, t ) where(1.9) τ = ξ ( t ) := t m +1 m + 1 . Hence, the solution v ( x, τ ) verifies the following equation: v ττ − ∆ v + µ m τ ∂ τ v = C m,p τ µ m ( p − | ∂ τ v | p + K m τ − µ m | v | q , in R N × [0 , ∞ ) , (1.10)where µ m := mm +1 , K m = ( m + 1) − µ m and C m,p = ( m + 1) µ m ( p − . Note that the equation(1.10) is somehow related to the Euler-Darboux-Poisson equation. Moreover, thanks tothis transformation, which implies a kind of similarity with the scale-invariant dampingcase, we can inherit the methods used in our previous works [4, 5, 6] to build the proofsof our main results which are related, as a first target, to the blow-up of the solution of(1.1) and, as a secondary aim, to the blow-up of (1.7).The article is organized as follows. The beginning of Section 2 is devoted to thesetting of the weak formulation of (1.1) in the energy space. Then, we state the maintheorems of our work, and we end this section by some remarks. To develop the proofsof our results, we prove some technical lemmas in Section 3. Finally, Sections 4 and 5are committed to the proofs of Theorems 2.2 and 2.3.2. Main Results
In this section, we start by giving a sense to the energy solution of (1.1) which readsas follows:
Definition 2.1.
Let f, g ∈ C ∞ ( R N ) . We say that u is an energy solution of (1.1) on [0 , T ) if ( u ∈ C ([0 , T ) , H ( R N )) ∩ C ([0 , T ) , L ( R N )) ,u ∈ L qloc ((0 , T ) × R N ) and u t ∈ L ploc ((0 , T ) × R N ) , satisfies, for all Φ ∈ C ∞ ( R N × [0 , T )) and all t ∈ [0 , T ) , the following equation: (2.1) Z R N u t ( x, t )Φ( x, t ) dx − Z R N u t ( x, x, dx − Z t Z R N u t ( x, s )Φ t ( x, s ) dx ds + Z t s m Z R N ∇ u ( x, s ) · ∇ Φ( x, s ) dx ds = Z t Z R N {| u t ( x, s ) | p + | u ( x, s ) | q } Φ( x, s ) dx ds. It is clear that the weak formulation corresponding to (1.7) is also given by an anal-ogous identity to (2.1) with omitting the nonlinear term | u | q .Now, we state the main results of this article which are related to the blow-up regionand the lifespan estimate of the solutions of (1.1) and (1.7), respectively. heorem 2.2. Let p, q > and m ≥ such that (2.2) Λ( p, q, N, m ) < , where Λ( p, q, N, m ) is defined by (2.3) Λ( p, q, N, m ) := ( q − (cid:2) ( m + 1)( N − p − m ( p − − (cid:3) , and p > p tr ( N, m ) and q > q C ( N, m ) .Suppose that f, g ∈ C ∞ ( R N ) are non-negative functions which are compactly supportedon B R N (0 , R ) , and do not vanish everywhere. Let u be an energy solution of (1.1) on [0 , T ε ) (in the sense of (2.1) ) such that supp( u ) ⊂ { ( x, t ) ∈ R N × [0 , ∞ ) : | x | ≤ ξ ( t )+ R } .Then, there exists a constant ε = ε ( f, g, N, R, p, q, m ) > such that T ε verifies T ε ≤ C ε − p ( q − − Λ( p,q,N,m ) , where C is a positive constant independent of ε and < ε ≤ ε . Theorem 2.3.
Let p > and m ≥ . Assume that the initial data f and g verify thesame hypotheses as in Theorem 2.2. Then, for u an energy solution of (1.7) on [0 , T ε ) which satisfies supp( u ) ⊂ { ( x, t ) ∈ R N × [0 , ∞ ) : | x | ≤ ξ ( t ) + R } , there exists a constant ε = ε ( f, g, N, R, p, m ) > such that T ε verifies T ε ≤ C ε − p − − (( m +1)( N − − m )( p − for < p < p tr ( N, m ) , exp (cid:0) Cε − ( p − (cid:1) for p = p tr ( N, m ) = 1 + m +1)( N − − m , where C is a positive constant independent of ε and < ε ≤ ε .Remark . Note that for m = 0 in (1.1), we have Λ( p, q, N,
0) = λ ( p, q, N ), which arerespectively given by (2.3) and (1.5), and this gives raise to the classical wave equationwith combined nonlinearities. Remark . Let p := p tr ( N, m ) + β = 1 + 2( m + 1)( N − − m + β , and q := 1 + 4( m + 1)( N − p − m ( p − − − β > m + 1) N − − β , where β i is a positive constant taken small enough for all i = 1 , , Q tr (cid:16) m +1) N − (cid:17) >
0. Then, for β and β small enough, wehave Q tr ( q ) >
0. Consequently, we have q > q C ( N, m ) for β and β small enough.Hence, the pair ( p , q ) satisfies (2.2), p > p tr ( N, m ) and q > q C ( N, m ). Hence, thehypothesis on p and q in Theorem 2.2 makes sense. emark . We note that, for p > p G ( N ) and q > q S ( N ), the equality in (1.5) yieldsthe global existence of the solution of (1.2) (with m = 0 and ( a, b ) = (1 , m >
0, namely when Λ( p, q, N, m ) = 4 and for p > p tr ( N, m ) and q > q C ( N, m ). To complete the whole picture, the global existencein-time of the solution of (1.1) will be studied in a subsequent work.3.
Auxiliary results
First, we start by introducing a test function which somehow follows the dynamicsof the linear equation associated with (1.1) and that we will use subsequently to derivethe behavior of the solution of (1.1). More precisely, we have(3.1) ψ ( x, t ) := ρ ( t ) φ ( x ); φ ( x ) := Z S N − e x · ω dω for N ≥ ,e x + e − x for N = 1 , where φ ( x ) is introduced in [26], and ρ ( t ) verifies(3.2) d ρ ( t ) dt − t m ρ ( t ) = 0 , ∀ t ≥ , The expression of ρ ( t ) is given by ([9, 15]):(3.3) ρ ( t ) = α m t K m +2 (cid:18) t m +1 m + 1 (cid:19) , ∀ t > , , at t = 0 , where K ν ( t ) = Z ∞ exp( − t cosh ζ ) cosh( νζ ) dζ , ν ∈ R , and α m = 2(2 m + 2) m +2 Γ( m +2 ) . Thanks to some properties of the modified Bessel function of second kind K ν ( t ) near t = 0, we infer that ρ ( t ) satisfies(3.4) ρ ′ (0) = − (2 m + 2) mm +1 (cid:0) Γ (cid:0) m +12 m +2 (cid:1)(cid:1) m +12 m +2 Γ (cid:0) m +2 (cid:1) . On the other hand, it is easy to see that φ ( x ) satisfies(3.5) ∆ φ = φ. Consequently, the function ψ ( x, t ) = φ ( x ) ρ ( t ) is solution of the following equation:(3.6) ∂ t ψ ( x, t ) − t m ∆ ψ ( x, t ) = 0 . or a later use, we list here some properties of the function ρ ( t ) ([2, 3, 15, 23]): (i) The functions ρ ( t ) and − ρ ′ ( t ) are decreasing on [0 , ∞ ) and verify lim t → + ∞ ρ ( t ) =lim t → + ∞ ρ ′ ( t ) = 0. (ii) For all t >
1, there exist constants C and C such that the function ρ ( t ) verifies(3.7) C − t − m exp( − ξ ( t )) ≤ ρ ( t ) ≤ C t − m exp( − ξ ( t )) , ( ξ ( t ) is given by (1.9)) , and(3.8) C − t m exp( − ξ ( t )) ≤ | ρ ′ ( t ) | ≤ C t m exp( − ξ ( t )) . (iii) We have(3.9) lim t → + ∞ (cid:18) ρ ′ ( t ) t m ρ ( t ) (cid:19) = − . Note that the proof of the properties (i) and (ii) can be found in [3], and the one of (iii) is shown in the appendix (Section 6).Along this work, we will denote by C any positive constant which may depend onthe data ( p, q, m, N, R, f, g ) but not on ε and whose value may change from line to line.However, in some occurrences, we will make precise the dependence of the constant C on the parameters of the problem.The following lemma holds true for the function ψ ( x, t ). Lemma 3.1 ([26]) . Let r > . There exists a constant C = C ( N, R, p, r ) > such that (3.10) Z | x |≤ ξ ( t )+ R (cid:16) ψ ( x, t ) (cid:17) r dx ≤ Cρ r ( t ) e rξ ( t ) (1 + ξ ( t )) (2 − r )( N − , ∀ t ≥ . We now introduce the following functionals:(3.11) G ( t ) := Z R N u ( x, t ) ψ ( x, t ) dx, and(3.12) G ( t ) := Z R N u t ( x, t ) ψ ( x, t ) dx. The next two lemmas give the first lower bounds for G ( t ) and G ( t ), respectively. Moreprecisely, we will prove that t m G ( t ) and G ( t ) are two coercive functions for t largeenough.We recall that the proof of Lemma 3.2 is known in the literature; see [9]. However,in order to make the presentation complete, we include here all the details about theproof of this lemma. Nevertheless, Lemma 3.3 constitutes a novelty in this work and itsutilization in the proofs of Theorems 2.2 and 2.3 is crucial. emma 3.2. Let u be an energy solution of the system (1.1) with initial data satisfyingthe assumptions in Theorem 2.2. Then, there exists T = T ( m ) > such that (3.13) G ( t ) ≥ C G ε t − m , for all t ≥ T , where C G is a positive constant which may depend on f , g , N, R and m .Proof. Let t ∈ (0 , T ). Using Definition 2.1, performing an integration by parts in spacein the fourth term in the left-hand side of (2.1) and then substituting Φ( x, t ) by ψ ( x, t ),we obtain(3.14) Z R N u t ( x, t ) ψ ( x, t ) dx − ε Z R N g ( x ) ψ ( x, dx − Z t Z R N (cid:8) u t ( x, s ) ψ t ( x, s ) + s m u ( x, s )∆ ψ ( x, s ) (cid:9) dx ds = Z t N ( s ) ds, where(3.15) N ( t ) = Z R N {| u t ( x, t ) | p + | u ( x, t ) | q } ψ ( x, t ) dx. Performing integration by parts for the first and third terms in the second line of (3.14)and utilizing (3.1) and (3.6), we infer that(3.16) Z R N (cid:2) u t ( x, t ) ψ ( x, t ) − u ( x, t ) ψ t ( x, t ) (cid:3) dx = Z t N ( s ) ds + εC m ( f, g ) , where(3.17) C m ( f, g ) := Z R N (cid:2) a ( m ) f ( x ) + g ( x ) (cid:3) φ ( x ) dx, with a ( m ) := − ρ ′ (0); and ρ ′ (0) is given by (3.4).It is clear that the constant C m ( f, g ) is positive thanks to (3.4) and the non negativityof the initial data which do not vanish everywhere. Hence, using the definition of G ,as in (3.11), and (3.1), the equation (3.16) yields(3.18) G ′ ( t ) + Γ( t ) G ( t ) = Z t N ( s ) ds + ε C m ( f, g ) , where(3.19) Γ( t ) := − ρ ′ ( t ) ρ ( t ) . Multiplying (3.18) by ρ ( t ) and integrating over (0 , t ), we deduce that G ( t ) ≥ G (0) ρ ( t ) + εC m ( f, g ) ρ ( t ) Z t dsρ ( s ) . (3.20) hanks to (3.3) and the fact that G (0) >
0, the estimate (3.20) yields G ( t ) ≥ εC m ( f, g ) tK m +2 ( ξ ( t )) Z tt/ sK m +2 ( ξ ( s )) ds, ∀ t > , (3.21)where ξ ( t ) is given by (1.9).From (6.4), we have the existence of T = T ( m ) > ξ ( t ) K m +2 ( ξ ( t )) > π e − ξ ( t ) and ξ ( t ) − K − m +2 ( ξ ( t )) > π e ξ ( t ) , ∀ t ≥ T / . (3.22)Hence, combining (3.22) in (3.21) and using (1.9), we deduce that G ( t ) ≥ εCC m ( f, g ) t − m , ∀ t ≥ T . (3.23)This ends the proof of Lemma 3.2. (cid:3) Now we are in a position to prove the following lemma.
Lemma 3.3.
Let u be an energy solution of the system (1.1) with initial data fulfillingthe assumptions in Theorem 2.2. Then, there exists T = T ( m ) > such that (3.24) G ( t ) ≥ C G ε, for all t ≥ T , where C G is a positive constant depending possibly on f , g , N and m .Proof. Let t ∈ [0 , T ) and recall the definitions of G and G , given respectively by (3.11)and (3.12), (3.1) and the identity(3.25) G ′ ( t ) − ρ ′ ( t ) ρ ( t ) G ( t ) = G ( t ) , then the equation (3.18) gives(3.26) G ( t ) − ρ ′ ( t ) ρ ( t ) G ( t ) = Z t N ( s ) ds + ε C m ( f, g ) , where N ( t ) is given by (3.15).Differentiating the equation (3.26) in time, we get G ′ ( t ) − ρ ′ ( t ) ρ ( t ) G ′ ( t ) − (cid:18) ρ ′′ ( t ) ρ ( t ) − ( ρ ′ ( t )) ρ ( t ) (cid:19) G ( t ) = N ( t ) . (3.27)Using (3.2) and (3.25), the equation (3.27) yields G ′ ( t ) − ρ ′ ( t ) ρ ( t ) G ( t ) − t m G ( t ) = N ( t ) . (3.28)Recall the definition of Γ( t ), given by (3.19), we infer that(3.29) G ′ ( t ) + 3Γ( t )4 G ( t ) = N ( t ) + Σ ( t ) + Σ ( t ) , here(3.30) Σ ( t ) = − ρ ′ ( t )2 ρ ( t ) (cid:18) G ( t ) − ρ ′ ( t ) ρ ( t ) G ( t ) (cid:19) , and(3.31) Σ ( t ) = t m − (cid:18) ρ ′ ( t ) t m ρ ( t ) (cid:19) ! G ( t ) . Now, from (3.9), we deduce the existence of ˜ T = ˜ T ( m ) ≥ T such that(3.32) 12 ≤ − ρ ′ ( t ) t m ρ ( t ) ≤ , ∀ t > ˜ T . Consequently, using (3.26), we deduce that(3.33) Σ ( t ) ≥ ε C m ( f, g ) t m t m Z t N ( s ) ds, ∀ t ≥ ˜ T . Furthermore, using Lemma 3.2 and (3.32), we have(3.34) Σ ( t ) ≥ , ∀ t ≥ ˜ T . Combining (3.29), (3.33) and (3.34), we infer that(3.35) G ′ ( t ) + 3Γ( t )4 G ( t ) ≥ ε C m ( f, g ) t m N ( t ) + t m Z t N ( s ) ds, ∀ t ≥ ˜ T . At this level, we can ignore the nonnegative nonlinear term N ( t ) .Now, multiplying (3.35) by ρ / ( t ) and integrating over ( ˜ T , t ), we deduce that G ( t ) ≥ G ( ˜ T ) ρ / ( t ) ρ / ( ˜ T ) + ε C m ( f, g )4 ρ / ( t ) Z t ˜ T s m ρ / ( s ) ds, ∀ t ≥ ˜ T . (3.36)Now, in order to prove that G ( t ) is coercive starting from a relatively large time, wewill show in the following that G ( t ) ≥ t ≥
0. For that purpose, we use (3.28)and we easily obtain (cid:18) G ( t ) ρ ( t ) (cid:19) ′ = ρ − ( t ) (cid:0) t m G ( t ) + N ( t ) (cid:1) , ∀ t ≥ . (3.37)Hence, employing the above identity together with (3.20) and using the fact that theinitial data are nonnegative, we infer that(3.38) G ( t ) ≥ , ∀ t ≥ . In fact, the conservation of the term N ( t ) up to this step is only useful for proving Theorem 2.3 lateron. So, we could get rid of it earlier in the proof without any change. herefore, using (3.38) and the positivity of ρ ( t ), we deduce that G ( ˜ T ) ρ / ( t ) ρ / ( ˜ T ) ≥ , ∀ t ≥ . (3.39)Employing (3.39), the estimate (3.36) yields, for all t ≥ ˜ T ,(3.40) G ( t ) ≥ C ερ / ( t ) Z t ˜ T s m ρ / ( s ) ds. We rewrite (3.22) as follows:( ξ ( t )) K
32 12 m +2 ( ξ ( t )) > (cid:16) π (cid:17) e − ξ ( t ) and ( ξ ( t )) − K − m +2 ( ξ ( t )) > (cid:18) π (cid:19) e ξ ( t ) , ∀ t ≥ T / . (3.41)Hence, using the definitions of the expression of ρ ( t ) and ξ ( t ), given respectively by (3.3)and (1.9), we have G ( t ) ≥ εCt − m e − ξ ( t ) Z tt/ s m ξ ′ ( s ) e ξ ( s ) ds (3.42) ≥ εCe − ξ ( t ) Z tt/ ξ ′ ( s ) e ξ ( s ) ds, ∀ t ≥ T . Finally, using e ξ ( t ) > e ξ ( t/ , ∀ t ≥ T := T ( m ) ≥ T , we deduce that G ( t ) ≥ C ε, ∀ t ≥ T . (3.43)This concludes the proof of Lemma 3.3. (cid:3) Remark . Note that the same conclusions as in Lemmas 3.2 and 3.3 can be obtainedfor any positive nonlinearity of the form F ( u, u t ) instead of | u t | p + | u | q . Indeed, inthe proofs of these lemmas we only used the non negativity of the nonlinearities | u t | p and | u | q . Moreover, the results of Lemmas 3.2 and 3.3 naturally hold true for a onenonlinearity | u t | p or | u | q as it is the case for (1.7).4. Proof of Theorem 2.2
This section is aimed to detail the proof of the first main blow-up result which isrelated to the solution of the Cauchy problem (1.1). To this end, we will make use ofthe lemmas proven in Section 3 together with a Kato’s lemma type.Following the hypotheses in Theorem 2.2, we recall that supp( u ) ⊂ { ( x, t ) ∈ R N × [0 , ∞ ) : | x | ≤ ξ ( t ) + R } . For t ∈ [0 , T ), we set(4.1) F ( t ) := Z R N u ( x, t ) dx. ecall here the weak formulation (2.1) in which we set a test function Φ such that Φ ≡ { ( x, s ) ∈ R N × [0 , t ] : | x | ≤ ξ ( s ) + R } , and we obtain(4.2) Z R N u t ( x, t ) dx − Z R N u t ( x, dx = Z t Z R N {| u t ( x, s ) | p + | u ( x, s ) | q } dx ds. Using the definition of F ( t ), the equation (4.2) yields(4.3) F ′ ( t ) = F ′ (0) + Z t Z R N {| u t ( x, s ) | p + | u ( x, s ) | q } dx ds. Differentiating the above equation in time, we infer that(4.4) F ′′ ( t ) = Z R N {| u t ( x, t ) | p + | u ( x, t ) | q } dx. Integrating (4.4) twice in time over (0 , t ) and using the positivity of F (0) and F ′ (0), wededuce that F ( t ) ≥ Z t Z s Z R N {| u t ( x, τ ) | p + | u ( x, τ ) | q } dx dτ ds. (4.5)Thanks to the H¨older’s inequality and the estimates (3.10) and (3.24), we have(4.6) Z R N | u t ( x, t ) | p dx ≥ G p ( t ) (cid:18)Z | x |≤ ξ ( t )+ R (cid:16) ψ ( x, t ) (cid:17) pp − dx (cid:19) − ( p − ≥ Cρ − p ( t ) e − pξ ( t ) ε p ( ξ ( t )) − ( N − p − , ∀ t ≥ T . From the expression of ξ ( t ), given by (1.9), (3.3) and (3.22), we deduce that(4.7) ρ ( t ) e ξ ( t ) ≤ Ct − m , ∀ t ≥ T / . Combining (4.6) and (4.7), we get(4.8) Z R N | u t ( x, t ) | p dx ≥ Cε p t − ( m +1)( N − p − − mp , ∀ t ≥ T . Plugging the above inequality into (4.5), we obtain(4.9) F ( t ) ≥ Cε p t − ( m +1)( N − p − − mp , ∀ t ≥ T . Otherwise, we have (cid:16) Z R N u ( x, t ) dx (cid:17) q ≤ Z | x |≤ ξ ( t )+ R | u ( x, t ) | q dx (cid:16) Z | x |≤ ξ ( t )+ R dx (cid:17) q − , (4.10)which implies that F q ( t ) ≤ C (1 + t ) N ( q − m +1) Z | x |≤ ξ ( t )+ R | u ( x, t ) | q dx. (4.11) This choice is possible thanks to the fact that the energy solution u verifies supp( u ) ⊂ { ( x, t ) ∈ R N × [0 , ∞ ) : | x | ≤ ξ ( t ) + R } . athering (4.4) and (4.11), we deduce that(4.12) F ′′ ( t ) ≥ C F q ( t )(1 + t ) N ( q − m +1) , ∀ t > . From (4.3) we have F ′ ( t ) >
0. Then, multiplying (4.12) by F ′ ( t ) and integrating theobtained inequality, we get(4.13) (cid:16) F ′ ( t ) (cid:17) ≥ C F q +1 ( t )(1 + t ) N ( q − m +1) + (cid:0) ( F ′ (0)) − CF q +1 (0) (cid:1) , ∀ t > . Since we consider here small initial data, we can easily see that the last term in theright-hand side of (4.13) is positive, and more precisely this holds for ε small enough.Therefore (4.13) gives(4.14) F ′ ( t ) F δ ( t ) ≥ C F q − − δ ( t )(1 + t ) N ( q − m +1)2 , ∀ t > , for δ > t , t ], for all t > t ≥ T , and using (4.9), weobtain(4.15)1 δ (cid:16) F δ ( t ) − F δ ( t ) (cid:17) ≥ C ( ε p ) q − − δ Z t t (1 + s ) (2 − ( m +1)( N − p − − mp )( q − − δ ) (1 + s ) N ( q − m +1)2 ds, ∀ t > t ≥ T . Neglecting the second term in the left-hand side of (4.15) and using the definition ofΛ( p, q, N, m ) (given by (2.3)) yield(4.16) 1 F δ ( t ) ≥ Cδε p ( q − − pδ Z t t (1 + s ) − Λ( p,q,N,m )4 − δ ( − ( m +1)( N − p − − mp ) ds. Thanks to the hypothesis (2.2), we can choose δ = δ small enough such that γ := − Λ( p,q,N,m )4 − δ (cid:16) − ( m +1)( N − p − − mp (cid:17) > −
1. Then, the estimate (4.16) implies that(4.17) 1 F δ ( t ) ≥ Cε p ( q − − pδ (cid:0) (1 + t ) γ +1 − (1 + t ) γ +1 (cid:1) , ∀ t > t ≥ T . From (4.9), we infer that(4.18) ε p ( q − (cid:0) (1 + t ) γ +1 − (1 + t ) γ +1 (cid:1) ≤ C (1 + t ) − δ ( − ( m +1)( N − p − − mp ) , ∀ t > t ≥ T . Consequently, we have ε p ( q − (1 + t ) γ +1 ≤ C (1 + t ) − δ ( − ( m +1)( N − p − − mp )(4.19) + ε p ( q − (1 + t ) γ +1 , ∀ t > t ≥ T . ince − Λ( p,q,N,m )4 + 1 >
0, then for all ε >
0, we choose ˜ T such that(4.20) ˜ T = C − − Λ( p,q,N,m ) ε − p ( q − − Λ( p,q,N,m ) . Finally, we set t = max( T , ˜ T ) and we plug (4.20) in (4.19) to obtain that(4.21) t ≤ γ +1 (1 + t ) ≤ Cε − p ( q − − Λ( p,q,N,m ) . The proof of Theorem 2.2 is now complete. (cid:3) Proof of Theorem 2.3.
The purpose of this section is to give the details on the proof of Theorem 2.3 which isrelated to the solution of (1.7). For that aim, we use the computations already obtainedin Section 3 . First, we note that Lemmas 3.2 and 3.3 remain true for the solution of(1.7) instead of (1.1) (see Remark 3.1) since we only used the non negativity of thenonlinear terms and not their types.In fact, the result in this section is somehow an application (and does not constitutein any case the main objective of this work) of the result obtained for problem (1.1) withmixed nonlinearities in Theorem 2.2. Indeed, we note here that the blow-up result forthe system (1.7) is also obtained in [18]; the result there is the same as in Theorem 2.3.However, our approach here is totally different. Moreover, we believe that our methodcan be used to study the blow-up of the system with generalized Tricomi term. Thiswill be the subject of a forthcoming work.In order to prove the blow-up result for (1.7) we will use the estimate (3.35) (initiallyproved for (1.1)) with omitting the nonlinear term | u ( x, t ) | q . Hence, the analogous ofthe estimate (3.35) reads as follows:(5.1) G ′ ( t ) + 3Γ( t )4 G ( t ) ≥ ε C m ( f, g ) t m Z R N | u t ( x, t ) | p ψ ( x, t ) dx + t m Z t Z R N | u t ( x, s ) | p ψ ( x, s ) dxds, ∀ t ≥ ˜ T . Now, we introduce the following functional: H ( t ) := 18 Z t ˜ T Z R N | u t ( x, s ) | p ψ ( x, s ) dxds + C ε , where C = min( C m ( f, g ) / , C G ) ( C G is defined in Lemma 3.3) and ˜ T > T is chosensuch that t m − t )32 > t ≥ ˜ T (this is possible thanks to (3.19) and (3.9)).Let F ( t ) := G ( t ) − H ( t ) , hich verifies(5.2) F ′ ( t ) + 3Γ( t )4 F ( t ) ≥ (cid:18) t m − t )32 (cid:19) Z t ˜ T Z R N | u t ( x, s ) | p ψ ( x, s ) dxds + 78 Z R N | u t ( x, t ) | p ψ ( x, t ) dx + C (cid:18) t m − t )32 (cid:19) ε ≥ , ∀ t ≥ ˜ T . Multiplying (5.2) by ρ / ( t ) and integrating over ( ˜ T , t ), we obtain F ( t ) ≥ F ( ˜ T ) ρ / ( t ) ρ / ( ˜ T ) , ∀ t ≥ ˜ T , (5.3)where ρ ( t ) is defined by (3.3).Hence, we have F ( ˜ T ) = G ( ˜ T ) − C ε ≥ G ( ˜ T ) − C G ε ≥ C = min( C m ( f, g ) / , C G ) ≤ C G .Therefore we deduce that(5.4) G ( t ) ≥ H ( t ) , ∀ t ≥ ˜ T . On the other hand, using H¨older’s inequality and the estimates (3.10) and (3.24), weinfer that(5.5) Z R N | u t ( x, t ) | p ψ ( x, t ) dx ≥ G p ( t ) (cid:18)Z | x |≤ ξ ( t )+ R ψ ( x, t ) dx (cid:19) − ( p − ≥ CG p ( t ) ρ − ( p − ( t ) e − ( p − ξ ( t ) ( ξ ( t )) − ( N − p − . Thanks to (4.7), we obtain(5.6) Z R N | u t ( x, t ) | p ψ ( x, t ) dx ≥ CG p ( t ) t − [( N − m +1) − m ]( p − , ∀ t ≥ ˜ T . Using the above estimate and (5.4), we deduce that(5.7) H ′ ( t ) ≥ CH p ( t ) t − [( N − m +1) − m ]( p − , ∀ t ≥ ˜ T . Recall that H ( ˜ T ) = C ε >
0, we easily obtain the upper bound of the lifespan estimateas in Theorem 2.3 which concludes the proof.6.
Appendix
The appendix is aimed to give some light on the property (iii) in Section 3, namely(3.9), of the function ρ ( t ), the solution of (3.2), by showing some elementary results onthe behavior of ρ ( t ) for t large enough. Hence, we first rewrite the expression of ρ ( t ) asfollows:(6.1) ρ ( t ) = ( α m t K m +2 ( ξ ( t )) , ∀ t > , , at t = 0 , here K ν ( z ) = Z ∞ exp( − z cosh ζ ) cosh( νζ ) dζ , ν ∈ R . Using the fact that(6.2) ddz K ν ( z ) = − K ν +1 ( z ) + νz K ν ( z ) , we infer that(6.3) ρ ′ ( t ) ρ ( t ) = 1 t − K m +32 m +2 ( ξ ( t )) K m +2 ( ξ ( t )) t m , ∀ t > . From [3], we have the following property of the function K µ ( t ):(6.4) K ν ( t ) = r π t e − t (1 + O ( t − ) , as t → ∞ . Combining (6.3) and (6.4), we deduce the property (3.9).Finally, we refer the reader to [2, 3] for more details about the properties of thefunction K ν ( t ). References [1] W. Dai, Wei, D. Fang and C. Wang,
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Email address : [email protected] (M. Hamouda) Email address : [email protected] (M.A. Hamza)[email protected] (M.A. Hamza)