A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime
aa r X i v : . [ m a t h . A P ] J a n A note on the nonexistence of global solutions to the semilinearwave equation with nonlinearity of derivative-type in thegeneralized Einstein – de Sitter spacetime
Makram Hamouda a , Mohamed Ali Hamza a , Alessandro Palmieri b a Basic Sciences Department, Deanship of Preparatory Year and Supporting Studies, P. O. Box 1982, ImamAbdulrahman Bin Faisal University, Dammam, KSA. b Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy
February 1, 2021
Abstract
In this paper, we establish blow-up results for the semilinear wave equation in generalizedEinstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on theintegral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian’s Integral Transform approach. Asupper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent whichdepends both on the space dimension and on the Lorentzian metric in the generalized Einstein-deSitter spacetime.
Keywords
Wave equation, Einstein-de Sitter spacetime, Integral transform, Blow-up, Glasseyexponent, Nonlinearity of derivative type
AMS Classification (2010)
Primary: 35B44, 35C15, 35L71; Secondary: 35A08, 35B33, 35L15
The aim of the present paper is to establish a blow-up result for local in time solutions to the Cauchyproblem with nonlinearity of derivative type | ∂ t u | p ∂ t u − t − k ∆ u + 2 t − ∂ t u = | ∂ t u | p , x ∈ R n , t > ,u (1 , x ) = εu ( x ) , x ∈ R n ,u t (1 , x ) = εu ( x ) , x ∈ R n , (1)where k ∈ (0 , p > ε is a positive constant describing the size of Cauchy data.In the related literature (see, for example, [5]), the differential operator with time-dependentcoefficients on the left-hand side of (1) is called the wave operator on the generalized Einstein-deSitter spacetime . This nomenclature is due to the fact that for k = and n = 3 the operator ∂ t − t − ∆ + 2 t − ∂ t coincides with the d’Alembertian operator in Einstein-de Sitter’s Lorentzianmetric.In recent years, many papers have been devoted to the study of blow-up results and lifespanestimates for the semilinear wave equation in the generalized Einstein - de Sitter (EdS) spacetime withpower nonlinearities [5, 23] and generalizations [31, 32, 24]. More specifically, it has been conjecturedthat the critical exponent for the semilinear Cauchy problem with power nonlinearity | u | p ∂ t u − t − k ∆ u + 2 t − ∂ t u = | u | p , x ∈ R n , t > ,u (1 , x ) = εu ( x ) , x ∈ R n ,u t (1 , x ) = εu ( x ) , x ∈ R n , (2)is given by the competition between two exponents, namely,max n p ( n, k ) , − k ) n o , (3)1here p ( n, k ) is the greatest root of the quadratic equation((1 − k ) n + 1)( p − + ((1 − k ) n − − k )( p − − . The results from the previously quoted papers have been obtained via four different approaches,namely, Kato’s type lemma, iteration argument (together with a slicing procedure in the critical case),comparison argument, and test function method. We can summarize these results by saying that forlocal in time solutions to (2) it has been shown the validity of a blow-up result (under suitable signand support assumptions for the Cauchy data) whenever the exponent of the power nonlinearity | u | p fulfills 1 < p max { p ( n, k ) , − k ) n } . Due to these nonexistence results, it has been conjecturethat the exponent in (3) is critical, even though the global existence of small data solutions is acompletely open problem to the best of the authors’ knowledge.The goal of the present work is to establish a blow-up result for (1) and to determine a candidateas critical exponent somehow related to the so-called Glassey exponent p Gla ( n ) . = ( n +1 n − if n > , ∞ if n = 1 , (4)which is the critical exponent for the semilinear wave equation with nonlinearity of derivative type[14, 17, 30, 29, 28, 1, 12, 33, 46, 13].Our approach consists in a modification of Zhou’s approach for the corresponding semilinear waveequation in [46]. Consequently, we will devote the first part of the paper to the proof of an integralrepresentation formula for the solution to the linear inhomogeneous Cauchy problem associated with(1) in the case n = 1 via Yagjian’s Integral Transform approach.We consider also a second semilinear model, that can be studied analogously with the tools of thepresent work. If we delete the linear damping term 2 t − ∂ t u in (1), we obtain somehow a semilinear Tricomi-type model with negative power in the speed of propagation and nonlinearity of derivative type.Hence, with minor modifications in our blow-up argument for (1) we will prove a blow-up result alsofor the following model ∂ t u − t − k ∆ u = | ∂ t u | p , x ∈ R n , t > ,u (1 , x ) = εu ( x ) , x ∈ R n ,u t (1 , x ) = εu ( x ) , x ∈ R n , (5)where k ∈ (0 , p > ε is a positive constant describing the size of Cauchy data.Recently, the semilinear generalized Tricomi equation with nonlinearity of derivative type, namely,the Cauchy problem ∂ t u − t ℓ ∆ u = | ∂ t u | p , x ∈ R n , t > ,u (0 , x ) = εu ( x ) , x ∈ R n ,u t (0 , x ) = εu ( x ) , x ∈ R n , (6)where ℓ > p > ε >
0, has been attracting a lot of attention and different approaches havebeen applied to establish several blow-up results for p in suitable ranges depending on n, ℓ . Amongthese approaches we find, on the one hand, comparison arguments based either on the fundamentalsolution for the operator ∂ t − t ℓ ∆ in [16] or on the employment of a special positive solution of thecorresponding homogeneous equation involving modified Bessel functions of second kind in [10], and,on the other hand, a modified test function method in [15]. In particular, the result that we are goingto show for (5) is consistent with those for (6) in [15, 10], although we shall use a different approachto prove this result. Moreover, we will see that the upper bound for the exponent p in the blow-upresult for (1) is a shift of the corresponding upper bound for (5). In this section, we state the main blow-up results for the semilinear models in (1) and (5), respectively.Since the speed of propagation in both semilinear models (1) and (5) is a k ( t ) . = t − k , in the followingit will be useful to employ the notation φ k ( t ) . = t − k − k , (7)2or the primitive of a k vanishing at t = 0 and the notation A k ( t ) . = φ k ( t ) − φ k (1) = ˆ t a k ( τ ) d τ, (8)for the distance function describing the amplitude of the curved light-cone. Theorem 1.1 (Blow-up result for the semilinear wave equation in EdS spacetime) . Let n > and k ∈ (0 , . We assume that ( u , u ) ∈ C ( R n ) × C ( R n ) are nonnegative functions and have supportscontained in B R . = { x ∈ R n : | x | < R } . Let us consider an exponent p in the nonlinearity of derivativetype such that < p p Gla (cid:0) (1 − k ) n + 2 k + 2 (cid:1) , (9) where the Glassey exponent p Gla is defined by (4) .Then, there exists ε = ε ( n, p, k, u , u , R ) > such that for any ε ∈ (0 , ε ] if u ∈ C ([1 , T ) × R n ) is a local in time solution to (1) such that supp u ( t, · ) ⊂ B R + A k ( t ) for any t ∈ [1 , T ) and with lifespan T = T ( ε ) , then, u blows up in finite time, that is, T < ∞ .Furthermore, the following upper bound estimate for the lifespan holds T ( ε ) ( Cε − (cid:0) p − − (1 − k ) n +2 k +12 (cid:1) − if < p < p Gla (cid:0) (1 − k ) n + 2 k + 2 (cid:1) , exp (cid:0) Cε − ( p − (cid:1) if p = p Gla (cid:0) (1 − k ) n + 2 k + 2) , (10) where the positive constant C is independent of ε . Theorem 1.2 (Blow-up result for the semilinear Tricomi-type equation) . Let n > and k ∈ (0 , .We assume that u = 0 and that u ∈ C ( R n ) is nonnegative function with support contained in B R for some R > . Let us consider an exponent p in the nonlinearity of derivative type such that < p p Gla (cid:0) (1 − k ) n + 2 k (cid:1) , (11) where the Glassey exponent p Gla is defined by (4) .Then, there exists ε = ε ( n, p, k, u , R ) > such that for any ε ∈ (0 , ε ] if u ∈ C ([1 , T ) × R n ) is a local in time solution to (5) such that supp u ( t, · ) ⊂ B R + A k ( t ) for any t ∈ [1 , T ) and with lifespan T = T ( ε ) , then, u blows up in finite time, that is, T < ∞ .Furthermore, the following upper bound estimate for the lifespan holds T ( ε ) ( Cε − (cid:0) p − − (1 − k ) n +2 k − (cid:1) − if < p < p Gla (cid:0) (1 − k ) n + 2 k (cid:1) , exp (cid:0) Cε − ( p − (cid:1) if p = p Gla (cid:0) (1 − k ) n + 2 k ) , (12) where the positive constant C is independent of ε .Remark . The upper bound for p in the blow-up range (9) is a shift of magnitude 2 of the Glasseyexponent that appears as upper bound in (11). This kind of phenomenon has already been observedin the semilinear model with power nonlinearity in [31, 24]. Remark . The upper bound p Gla (cid:0) (1 − k ) n + 2 k (cid:1) in Theorem 1.2 is consistent with the upper boundfor the semilinear generalized Tricomi with nonlinearity of derivative type (when the power in thespeed of propagation is positive and the Cauchy data are assumed at the initial time t = 0) see e.g.[15, 10]. Remark . In Theorem 1.2 we required a trivial first Cauchy data ( u = 0). This assumption is dueto the fact that, in general, the kernel function K (cid:0) t, x ; y ; 0 , , k (cid:1) , whose definition will be providedlater, see (18), is not a nonnegative function. In the series of papers [34, 35, 36, 44, 37, 45, 38, 39, 40, 41, 42, 22, 43], several integral represen-tation formulae for solutions to Cauchy problems associated with linear hyperbolic equations withvariable coefficients have been derived and applied both to study the necessity and the sufficiencypart concerning the problem of the global (in time) existence of solutions. The general scheme todetermine an integral representation in the above cited literature is the following: the desired formula3s obtained by considering the composition of two operators. On the one hand, the external operatoris an integral transformation, whose kernel is determined by the time-dependent coefficients and/or bythe lower-order terms in the associated partial differential operator. On the other hand, the internaloperator is a solution operator for a family of parameter dependent Cauchy problems (this step is oftencalled, in the above quoted literature, a revised Duhamel’s principle). In the special case in whichthe considered hyperbolic model is a wave equation with time-dependent speed of propagation andlower-order terms, the above mentioned solution operator associates with a given function the solutionto the Cauchy problem for the classical free wave equation with the given function as first initial dataand with vanishing second initial data.In the present section, we are going to use Yagjian’s Integral Transform approach in order todetermine an explicit integral representation formula for the linear Cauchy problem ∂ t u − t − k ∂ x u + µ t − ∂ t u + ν t − u = g ( t, x ) , x ∈ R , t > ,u (1 , x ) = u ( x ) , x ∈ R ,∂ t u (1 , x ) = u ( x ) , x ∈ R , (13)where µ, ν are nonnegative real parameters and k ∈ (0 , ν t − u simplifies the description of the symmetry of the second-order operator L k,µ,ν . = ∂ t − t − k ∂ x + µ t − ∂ t + ν t − with respect to the parameter µ . More in detail, the quantity δ = δ ( µ, ν ) . = ( µ − − ν , (14)has a crucial role in determining some properties of the fundamental solution of L k,µ,ν . In thespecial case k = 0 (the so-called wave operator with scale-invariant damping and mass), it is knownin the literature that the value of δ affects not only the fundamental solution of L ,µ,ν but alsothe critical exponents in the treatment of semilinear Cauchy problem associated with L ,µ,ν withpower nonlinearity [18, 25, 21, 19, 20, 26, 3, 2], nonlinearity of derivative type [27, 8], and combinednonlinearity [6, 7, 9].We shall see that even in the case k ∈ (0 ,
1) some properties of the fundamental solution of L k,µ,ν depend strongly on the value of δ . Of course, in the general case, we will find an interplay of δ and k in the description of the fundamental solution.We point out that, even though in this section we will focus on the case n = 1, analogously to whatis done in [34, 36, 44, 45, 22] it is possible to extend the integral representation even to the higherdimensional case by using the spherical means and the method of descent.Let us state now the representation formula for the solution to (13) in space dimension 1. Theorem 2.1.
Let n = 1 , k ∈ (0 , and let µ, ν be nonnegative constants. We assume u ∈ C ( R ) , u ∈ C ( R ) and g ∈ C , t,x ([1 , ∞ ) × R ) . Then, a representation formula for the solution u to (13) isgiven by u ( t, x ) = 12 t k − µ (cid:0) u ( x + A k ( t )) + u ( x − A k ( t )) (cid:1) + ˆ x + A k ( t ) x − A k ( t ) u ( y ) K (cid:0) t, x ; y ; µ, ν , k (cid:1) d y + ˆ x + A k ( t ) x − A k ( t ) u ( y ) K (cid:0) t, x ; y ; µ, ν , k (cid:1) d y + ˆ t ˆ x + φ k ( t ) − φ k ( b ) x − φ k ( t )+ φ k ( b ) g ( b, y ) E (cid:0) t, x ; b, y ; µ, ν , k (cid:1) d y d b. (15) Here the kernel function E is defined by E (cid:0) t, x ; b, y ; µ, ν , k (cid:1) . = c t − µ + −√ δ b µ + −√ δ (cid:0) ( φ k ( t ) + φ k ( b )) − ( y − x ) (cid:1) − γ × F (cid:18) γ, γ ; 1; ( φ k ( t ) − φ k ( b )) − ( y − x ) ( φ k ( t ) + φ k ( b )) − ( y − x ) (cid:19) , (16)4 here c = c ( µ, ν , k ) . = 2 − √ δ − k (1 − k ) − √ δ − k and γ = γ ( µ, ν , k ) . = 12 − √ δ − k ) , (17) and F ( α , α ; β ; z ) denotes the Gauss hypergeometric function, while the kernel functions K , K ap-pearing in the integral terms involving the Cauchy data are given by K (cid:0) t, x ; y ; µ, ν , k (cid:1) . = µE (cid:0) t, x ; 1 , y ; µ, ν , k (cid:1) − ∂∂b E (cid:0) t, x ; b, y ; µ, ν , k (cid:1)(cid:12)(cid:12)(cid:12) b =1 , (18) K (cid:0) t, x ; y ; µ, ν , k (cid:1) . = E (cid:0) t, x ; 1 , y ; µ, ν , k (cid:1) . (19) Proof.
We are going to prove the representation formula in (15) by means of a suitable change ofvariables that transforms (13) in a linear wave equation with scale-invariant damping and mass termsand allows us to employ a result from [22]. More specifically, we perform the transformation τ . = t − k − , z . = (1 − k ) x. (20)Setting v ( τ, z ) = u ( t, x ), by straightforward computations it follows that u solves (13) if and only if v is a solution to ∂ τ v − ∂ z v + µ − k − k (1 + τ ) − ∂ τ v + ν (1 − k ) (1 + τ ) − v = f ( τ, z ) , z ∈ R , τ > ,v ( τ = 0 , z ) = u (cid:0) z − k (cid:1) , z ∈ R ,∂ τ v ( τ = 0 , z ) = − k u (cid:0) z − k (cid:1) , z ∈ R , where f ( τ, z ) . = (1 − k ) − (1 + τ ) k − k g (cid:0) (1 + τ ) − k , z − k (cid:1) . According to [22, Theorem 1.1], we canrepresent v in the following way v = X j =0 v j , (21)where the addends { v j } j ∈{ , , , } are given by v ( τ, z ) . = (1 + τ ) − e µ (cid:0) u (cid:0) z + τ − k (cid:1) + u (cid:0) z − τ − k (cid:1)(cid:1) ,v ( τ, z ) . = 2 − √ e δ ˆ z + τz − τ u (cid:0) e y − k (cid:1) e K (cid:0) τ, z ; e y ; e µ, e ν (cid:1) d e y,v ( τ, z ) . = 2 − √ e δ ˆ z + τz − τ (cid:16) − k u (cid:0) e y − k (cid:1) + µ − k − k u (cid:0) e y − k (cid:1)(cid:17) e K (cid:0) τ, z ; e y ; e µ, e ν (cid:1) d e y,v ( τ, z ) . = 2 − √ e δ ˆ τ ˆ z + τ − e bz − τ + e b f ( e b, e y ) e E (cid:0) τ, z ; e b, e y ; e µ, e ν (cid:1) d e y d e b, and the kernel functions are given by e E (cid:0) τ, z ; e b, e y ; e µ, e ν (cid:1) . = (1 + τ ) − e µ − √ e δ (cid:0) e b (cid:1) e µ − √ e δ (cid:0) ( τ + e b + 2) − ( e y − z ) (cid:1) √ e δ − × F − √ e δ , − √ e δ ; 1; ( τ − e b ) − ( e y − z ) ( τ + e b + 2) − ( e y − z ) ! , e K (cid:0) τ, z ; e y ; e µ, e ν (cid:1) . = − ∂∂ e b e E (cid:0) τ, z ; e b, e y ; e µ, e ν (cid:1)(cid:12)(cid:12)(cid:12)e b =0 , e K (cid:0) τ, z ; e y ; e µ, e ν (cid:1) . = e E (cid:0) τ, z ; 0 , e y ; e µ, e ν (cid:1) , where F ( α , α ; β ; z ) is the Gauss hypergeometric function and e µ . = µ − k − k , e ν . = ν − k , e δ = e δ ( µ, ν , k ) . = ( e µ − − e ν = δ ( µ, ν )(1 − k ) . In order to show the validity of (15), we will transform back each term in (21) through (20).5et us begin with the function v . Recalling the definition of the function A k in (8), we can writeimmediately v ( τ, y ) = t k − µ (cid:0) u ( x + A k ( t )) + u ( x − A k ( t )) (cid:1) . (22)Let us deal with the term v . Using the explicit expression of f , we get v ( τ, z ) = 2 − √ e δ ˆ τ ˆ z + τ − e bz − τ + e b (1 − k ) − (1 + e b ) k − k g (cid:0) (1 + e b ) − k , e y − k (cid:1) e E (cid:0) τ, z ; e b, e y ; e µ, e ν (cid:1) d e y d e b. Carrying out the change of variables y = (1 − k ) − e y , b = (cid:0) e b (cid:1) − k and the transformation (20) inthe last integral, we arrive at v ( τ, z ) = 2 − √ δ − k ˆ t ˆ x + φ k ( t ) − φ k ( b ) x − φ k ( t )+ φ k ( b ) g ( b, y ) b k e E (cid:0) t − k − , (1 − k ) x ; b − k − , (1 − k ) y ; e µ, e ν (cid:1) d y d b = ˆ t ˆ x + φ k ( t ) − φ k ( b ) x − φ k ( t )+ φ k ( b ) g ( b, y ) E (cid:0) t, x ; b, y ; µ, ν , k (cid:1) d y d b, (23)where in the second step we used the identity e E (cid:0) t − k − , (1 − k ) x ; b − k − , (1 − k ) y ; e µ, e ν (cid:1) = 2 √ δ − k b − k E (cid:0) t, x ; b, y ; µ, ν , k (cid:1) , and the definition in (16).Finally, we deal with the functions v , v . Performing the change of variables y = (1 − k ) − e y andusing (20), we find v ( τ, z ) . = 2 − √ δ − k (1 − k ) ˆ x + A k ( t ) x − A k ( t ) u ( y ) e K (cid:0) t − k − , (1 − k ) x ; (1 − k ) y ; e µ, e ν (cid:1) d y,v ( τ, z ) . = 2 − √ δ − k ˆ x + A k ( t ) x − A k ( t ) (cid:0) u ( y ) + ( µ − k ) u ( y ) (cid:1) e K (cid:0) t − k − , (1 − k ) x ; (1 − k ) y ; e µ, e ν (cid:1) d y. By elementary computations we have e K (cid:0) t − k − , (1 − k ) x ; (1 − k ) y ; e µ, e ν (cid:1) = − √ δ − k ∂∂ e b (cid:16)(cid:0) e b (cid:1) − k − k E (cid:0) t, x ; (cid:0) e b (cid:1) − k , y ; µ, ν , k (cid:1)(cid:17)(cid:12)(cid:12)(cid:12)e b =0 , e K (cid:0) t − k − , (1 − k ) x ; (1 − k ) y ; e µ, e ν (cid:1) = 2 √ δ − k E (cid:0) t, x ; 1 , y ; µ, ν , k (cid:1) . Considering the transformation b = (cid:0) e b (cid:1) − k and using the relation (cid:18) ∂∂ e b (cid:19)e b =0 = 11 − k (cid:18) ∂∂b (cid:19) b =1 , we obtain(1 − k ) e K (cid:0) t − k − , (1 − k ) x ; (1 − k ) y ; e µ, e ν (cid:1) = − √ δ − k ∂∂b (cid:16) b − k E (cid:0) t, x ; b, y ; µ, ν , k (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) b =1 = 2 √ δ − k kE (cid:0) t, x ; 1 , y ; µ, ν , k (cid:1) − √ δ − k ∂∂b (cid:16) E (cid:0) t, x ; b, y ; µ, ν , k (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) b =1 . Combining the previous representations for e K , e K , we conclude v ( τ, z ) + v ( τ, z ) = ˆ x + A k ( t ) x − A k ( t ) u ( y ) (cid:18) µE (cid:0) t, x ; 1 , y ; µ, ν , k (cid:1) − ∂∂b (cid:16) E (cid:0) t, x ; b, y ; µ, ν , k (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) b =1 (cid:19) d y + ˆ x + A k ( t ) x − A k ( t ) u ( y ) E (cid:0) t, x ; 1 , y ; µ, ν , k (cid:1) d y. (24)Summarizing, from (21), (22), (23) and (24) it follows immediately (15). This completes the proof.6 orollary 2.2 (Representation formula in EdS spacetime) . Let n = 1 and k ∈ (0 , . We assume u ∈ C ( R ) , u ∈ C ( R ) and g ∈ C , t,x ([1 , ∞ ) × R ) . Then, a representation formula for the solution u to the Cauchy problem associated with the linear wave equation in Einstein-de Sitter spacetime ∂ t u − t − k ∂ x u + 2 t − ∂ t u = g ( t, x ) , x ∈ R , t > ,u (1 , x ) = u ( x ) , x ∈ R ,∂ t u (1 , x ) = u ( x ) , x ∈ R , (25) is given by u ( t, x ) = 12 t k − (cid:0) u ( x + A k ( t )) + u ( x − A k ( t )) (cid:1) + ˆ x + A k ( t ) x − A k ( t ) u ( y ) K (cid:0) t, x ; y ; 2 , , k (cid:1) d y + ˆ x + A k ( t ) x − A k ( t ) u ( y ) K (cid:0) t, x ; y ; 2 , , k (cid:1) d y + ˆ t ˆ x + φ k ( t ) − φ k ( b ) x − φ k ( t )+ φ k ( b ) g ( b, y ) E (cid:0) t, x ; b, y ; 2 , , k (cid:1) d y d b. (26) Here the kernel function E is defined by E (cid:0) t, x ; b, y ; 2 , , k (cid:1) . = c t − b (cid:0) ( φ k ( t ) + φ k ( b )) − ( y − x ) (cid:1) − γ F (cid:18) γ, γ ; 1; ( φ k ( t ) − φ k ( b )) − ( y − x ) ( φ k ( t ) + φ k ( b )) − ( y − x ) (cid:19) , (27) where c = c (2 , , k ) . = 2 − − k (1 − k ) k − k and γ = γ (2 , , k ) . = − k − k ) , (28) and F ( α , α ; β ; z ) denotes the Gauss hypergeometric function, while the kernel functions K , K ap-pearing in the integral terms involving the Cauchy data are given by (18) and (19) , respectively, forthe special values ( µ, ν ) = (2 , . Corollary 2.3 (Representation formula for the Tricomi-type equation) . Let n = 1 and k ∈ (0 , . Weassume u ∈ C ( R ) , u ∈ C ( R ) and g ∈ C , t,x ([1 , ∞ ) × R ) . Then, a representation formula for thesolution u to the Cauchy problem associated with the linear Tricomi-type equation ∂ t u − t − k ∂ x u = g ( t, x ) , x ∈ R , t > ,u (1 , x ) = u ( x ) , x ∈ R ,∂ t u (1 , x ) = u ( x ) , x ∈ R , (29) is given by u ( t, x ) = 12 t k (cid:0) u ( x + A k ( t )) + u ( x − A k ( t )) (cid:1) + ˆ x + A k ( t ) x − A k ( t ) u ( y ) K (cid:0) t, x ; y ; 0 , , k (cid:1) d y + ˆ x + A k ( t ) x − A k ( t ) u ( y ) K (cid:0) t, x ; y ; 0 , , k (cid:1) d y + ˆ t ˆ x + φ k ( t ) − φ k ( b ) x − φ k ( t )+ φ k ( b ) g ( b, y ) E (cid:0) t, x ; b, y ; 0 , , k (cid:1) d y d b. (30) Here the kernel function E is defined by E (cid:0) t, x ; b, y ; 0 , , k (cid:1) . = c (cid:0) ( φ k ( t ) + φ k ( b )) − ( y − x ) (cid:1) − γ F (cid:18) γ, γ ; 1; ( φ k ( t ) − φ k ( b )) − ( y − x ) ( φ k ( t ) + φ k ( b )) − ( y − x ) (cid:19) , (31) where c = c (0 , , k ) . = 2 − − k (1 − k ) k − k and γ = γ (0 , , k ) . = − k − k ) , (32) and F ( α , α ; β ; z ) denotes the Gauss hypergeometric function, while the kernel functions K , K ap-pearing in the integral terms involving the Cauchy data are given by (18) and (19) , respectively, forthe special values ( µ, ν ) = (0 , . emark . Note that the kernel functions in (27) and in (31) are strongly related, since the followingrelation holds E (cid:0) t, x ; b, y ; 2 , , k (cid:1) = bt E (cid:0) t, x ; b, y ; 0 , , k (cid:1) . Remark . Let us stress that representation formulae for the solutions to the wave equation in Einstein-de Sitter spacetime and to the generalized Tricomi equation (even in the case with speed of propagationwith negative power) have already been established in the literature (see for example [4] and [41],respectively) when the Cauchy data are prescribed at the initial time t = 0. However, since in thepresent work we prescribe the Cauchy data at the initial data t = 1 and the considered models are notinvariant by time translation (due to the presence of time-dependent coefficients), the representationformulae from Corollaries 2.2 and 2.3 are not redundant and will play a crucial role in the proof ofTheorems 1.1 and 1.2, respectively. Remark . In the next sections we shall estimate from below the kernel function E on suitable subsetsof the forward light-cone. According to this purpose, the following lower bound estimate for the Gausshypergeometric function is very helpful F ( α, α ; β ; z ) > , (33)for any z ∈ [0 ,
1) and for α ∈ R , β >
0. The previous estimate is a direct consequence of the seriesexpansion for F ( α, α ; β ; z ). Furthermore, for α ∈ R , β > β − α > F ( α, α ; β ; z ) can be estimated from above by a positive constant independent of z on[0 , In the present section, we prove the main blow-up results by using a generalization of Zhou’s blow-up argument on the characteristic line A k ( t ) − z = R . In place of the d’Alembert’s formula weshall employ the integral representation formulae from Theorems 1.1 and 1.2 obtained via Yagdjian’sIntegral Transform approach. The main steps in the proof are inspired by the computations in[46, 27, 16]. Let u be a local (in time) solution to the Cauchy problem (1). In order to reduce our problem to aone-dimensional problem in space, we will introduce an auxiliary function which depends on the timevariable and only on the first space variable. This step is achieved by integrating u with respect tothe remaining ( n −
1) spatial variables. Thus, if we denote x = ( z, w ) with z ∈ R and w ∈ R n − , then,we deal with the function U ( t, z ) . = ˆ R n − u ( t, z, w ) d w for any t > , z ∈ R . Hereafter, we consider just the case n > n = 1, simply by working with u instead of introducing U . In order to prescribe the initial values of U , we set U ( z ) . = ˆ R n − u ( z, w ) d w, U ( z ) . = ˆ R n − u ( z, w ) d w for any z ∈ R . According to the statement of Theorem 1.1 we require that u , u are compactly supported withsupport contained in B R . Hence, U , U are compactly supported too, with supports contained in( − R, R ). Analogously, since supp u ( t, · ) ⊂ B R + A k ( t ) for any t ∈ (1 , T ) we get the following supportcondition for U supp U ( t, · ) ⊂ ( − ( R + A k ( t )) , R + A k ( t )) for any t ∈ (1 , T ) . (34)Therefore, U solves the following Cauchy problem ∂ t U − t − k ∂ z U + 2 t − ∂ t U = ´ R n − | ∂ t u ( t, z, w ) | p d w, z ∈ R , t > , U (1 , z ) = ε U ( z ) , z ∈ R ,∂ t U (1 , z ) = ε U ( z ) , z ∈ R .
8y Corollary 2.2, we obtain the representation U ( t, z ) = 2 − εt k − (cid:0) U ( z + A k ( t )) + U ( z − A k ( t )) (cid:1) + ε ˆ z + A k ( t ) z − A k ( t ) U ( y ) K (cid:0) t, z ; y ; 2 , , k (cid:1) d y + ε ˆ z + A k ( t ) z − A k ( t ) U ( y ) K (cid:0) t, z ; y ; 2 , , k (cid:1) d y + ˆ t ˆ z + φ k ( t ) − φ k ( b ) z − φ k ( t )+ φ k ( b ) ˆ R n − | ∂ t u ( b, y, w ) | p d w E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d y d b, where the kernel function E is defined by (27), while K , K are defined in (18) and in (19), respectively,for ( µ, ν ) = (2 , u , u it results that U , U are nonnegative functions as well.Consequently, from the previous identity we get U ( t, z ) > ε ˆ z + A k ( t ) z − A k ( t ) (cid:0) U ( y ) K (cid:0) t, z ; y ; 2 , , k (cid:1) + U ( y ) K (cid:0) t, z ; y ; 2 , , k (cid:1)(cid:1) d y + ˆ t ˆ z + φ k ( t ) − φ k ( b ) z − φ k ( t )+ φ k ( b ) ˆ R n − | ∂ t u ( b, y, w ) | p d w E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d y d b . = εJ ( t, z ) + I ( t, z ) . We begin with the estimate of the integral J ( t, z ) that involves the Cauchy data.For any y ∈ [ z − A k ( t ) , z + A k ( t )], since γ (2 , , k ) = − k/ (2 − k ) < − γ (2 , , k ) =1 / (1 − k ) > K (cid:0) t, z ; y ; 2 , , k (cid:1) = E (cid:0) t, z ; 1 , y ; 2 , , k (cid:1) = ct − (cid:0) ( φ k ( t ) + φ k (1)) − ( y − z ) (cid:1) − γ F (cid:18) γ, γ ; 1; ( φ k ( t ) − φ k (1)) − ( y − z ) ( φ k ( t ) + φ k (1)) − ( y − z ) (cid:19) & t − (4 φ k (1) φ k ( t )) − γ & t k − . (35)Moreover, for any y ∈ [ z − A k ( t ) , z + A k ( t )] we may prove that K (cid:0) t, z ; y ; 2 , , k (cid:1) = 2 E (cid:0) t, z ; 1 , y ; 2 , , k (cid:1) − ∂∂b E (cid:0) t, z ; b, y ; 2 , , k (cid:1)(cid:12)(cid:12)(cid:12) b =1 & t k − . (36)Indeed, by straightforward computations we find ∂∂b E (cid:0) t, z ; b, y ; 2 , , k (cid:1) = ct − (cid:0) ( φ k ( t ) + φ k ( b )) − ( y − z ) (cid:1) − γ × (cid:26)(cid:18) kφ k ( b )( φ k ( t ) + φ k ( b ))( φ k ( t ) + φ k ( b )) − ( y − z ) (cid:19) F ( γ, γ ; 1; ζ ) + F ( γ + 1 , γ + 1; 2; ζ ) ∂ζ∂b (cid:27) , where ζ = ζ ( t, z ; b, y ; k ) . = ( φ k ( t ) − φ k ( b )) − ( y − z ) ( φ k ( t ) + φ k ( b )) − ( y − z ) . Due to ∂ζ∂b ( t, z ; b, y ; k ) = − b − k φ k ( t ) ( φ k ( t )) − ( φ k ( b )) − ( y − z ) [( φ k ( t ) + φ k ( b )) − ( y − z ) ] , it follows ∂ζ∂b ( t, z ; b, y ; k ) (cid:12)(cid:12)(cid:12) b =1 = − φ k ( t ) ( φ k ( t )) − ( φ k (1)) − ( y − z ) [( φ k ( t ) + φ k (1)) − ( y − z ) ] − φ k ( t ) φ k (1)( φ k ( t ) − φ k (1))[( φ k ( t ) + φ k (1)) − ( y − z ) ] , for any y ∈ [ z − A k ( t ) , z + A k ( t )]. Therefore, we may neglect the influence of the term F ( γ +1 , γ +1; 2; ζ )when we estimate the kernel K from below. Hence, K (cid:0) t, z ; y ; 2 , , k (cid:1) > ct − (cid:0) ( φ k ( t ) + φ k (1)) − ( y − z ) (cid:1) − γ (cid:18) − kφ k (1)( φ k ( t ) + φ k (1))( φ k ( t ) + φ k (1)) − ( y − z ) (cid:19) F ( γ, γ ; 1; ζ ) > ct − (cid:0) ( φ k ( t ) + φ k (1)) − ( y − z ) (cid:1) − γ (cid:0) − k (1 − ζ ) (cid:1) F ( γ, γ ; 1; ζ ) − ck (1 − k ) t − (cid:0) ( φ k ( t ) + φ k (1)) − ( y − z ) (cid:1) − γ − F ( γ, γ ; 1; ζ ) . (37)9sing the following estimate k (1 − k ) (cid:0) ( φ k ( t ) + φ k (1)) − ( y − z ) (cid:1) − k (1 − k ) (4 φ k ( t ) φ k (1)) − = k t − k k , for y ∈ [ z − A k ( t ) , z + A k ( t )] and t >
1, from (37) we derive K (cid:0) t, z ; y ; 2 , , k (cid:1) > c (cid:0) − k + k ζ (cid:1) t − (cid:0) ( φ k ( t ) + φ k (1)) − ( y − z ) (cid:1) − γ F ( γ, γ ; 1; ζ ) > c (cid:0) − k (cid:1) t − (cid:0) ( φ k ( t ) + φ k (1)) − A k ( t ) (cid:1) − γ F ( γ, γ ; 1; ζ )= c (cid:0) − k (cid:1) t − (4 φ k (1) φ k ( t )) − γ F ( γ, γ ; 1; ζ ) & t k − F ( γ, γ ; 1; ζ ) , which implies in turn (36) thanks to (33).Therefore, combining (35) and (36), we obtain J ( t, z ) & εt k − ˆ z + A k ( t ) z − A k ( t ) ( U ( y ) + U ( y )) d y. From now on, we will work on the characteristic line with equation A k ( t ) − z = R for z > R . For z > R such that A k ( t ) − z = R it holds[ − R, R ] ⊂ [ z − A k ( t ) , z + A k ( t )] . Consequently, using the support condition supp U , supp U ⊂ ( − R, R ), we conclude J ( t, z ) & εt k − ˆ R ( U ( y ) + U ( y )) d y = εt k − k u + u k L ( R n ) , (38)where in the last step we used Fubini’s theorem.Now we estimate the term I ( t, z ). Clearly, the following support conditionsupp ∂ t u ( t, · ) ⊂ B R + A k ( t ) , holds for any t ∈ (1 , T ) due the shape of the light-cone. With respect to the w ∈ R n − variable wecan express the previous support condition as follows:supp ∂ t u ( t, z, · ) ⊂ (cid:8) w ∈ R n − : | w | (cid:0) ( R + A k ( t )) − z (cid:1) / (cid:9) , for any t ∈ (1 , T ) and any z ∈ R . Then, from Hölder’s inequality we derive the estimate | ∂ t U ( b, y ) | = (cid:12)(cid:12)(cid:12) ˆ R n − ∂ t u ( b, y, w ) d w (cid:12)(cid:12)(cid:12) (cid:18) ˆ R n − | ∂ t u ( b, y, w ) | p d w (cid:19) p (cid:0) meas n − (cid:0) supp ∂ t u ( b, y, · ) (cid:1)(cid:1) − p . (cid:0) ( R + A k ( b )) − y (cid:1) n − ( − p ) (cid:18) ˆ R n − | ∂ t u ( b, y, w ) | p d w (cid:19) p . Hereafter, the unexpressed multiplicative constants will depend on n, k, R, p . From the previousinequality, we have ˆ R n − | ∂ t u ( b, y, w ) | p d w & (cid:0) ( R + A k ( b )) − y (cid:1) − n − ( p − | ∂ t U ( b, y ) | p , which provides in turn the following estimate I ( t, z ) & ˆ t ˆ z + φ k ( t ) − φ k ( b ) z − φ k ( t )+ φ k ( b ) (cid:0) ( R + A k ( b )) − y (cid:1) − n − ( p − | ∂ t U ( b, y ) | p E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d y d b = ˆ z + A k ( t ) z − A k ( t ) ˆ φ − k ( φ k ( t ) −| z − y | )1 (cid:0) ( R + A k ( b )) − y (cid:1) − n − ( p − | ∂ t U ( b, y ) | p E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d b d y, A k ( t ) − z = R , for z > R it holds [ z − A k ( t ) , z + A k ( t )] ⊃ [ R, z ]. Consequently, we can shrink thedomain of integration in the last lower bound estimate for I ( t, z ), obtaining I ( t, z ) & ˆ zR ˆ φ − k ( φ k ( t ) −| z − y | )1 (cid:0) ( R + A k ( b )) − y (cid:1) − n − ( p − | ∂ t U ( b, y ) | p E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d b d y. On the characteristic line A k ( t ) − z = R and for y ∈ [ R, z ], it holds φ − k ( φ k ( t ) − | z − y | ) = φ − k ( φ k ( t ) − z + y ) = φ − k ( φ k (1) + R + y ) = A − k ( R + y ) , where in the last step we used the relation A − k ( τ ) = φ − k ( φ k (1) + τ ) . Moreover, for y ∈ [ R, z ] it holds1 φ − k ( φ k (1) + y − R ) = A − k ( y − R ) , thanks to the monotonicity of φ − k . Thus, after a further shrinking of the domain of integration, weget I ( t, z ) & ˆ zR ˆ A − k ( y + R ) A − k ( y − R ) (cid:0) ( R + A k ( b )) − y (cid:1) − n − ( p − | ∂ t U ( b, y ) | p E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d b d y & ˆ zR (cid:0) ( R + A k ( A − k ( y + R ))) − y (cid:1) − n − ( p − ˆ A − k ( y + R ) A − k ( y − R ) | ∂ t U ( b, y ) | p E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d b d y & ˆ zR ( R + y ) − n − ( p − ˆ A − k ( y + R ) A − k ( y − R ) | ∂ t U ( b, y ) | p E (cid:0) t, z ; b, y ; 2 , , k (cid:1) d b d y, for z > R such that A k ( t ) − z = R . Now we estimate the kernel function E ( · ; 2 , , k ) from below on therestricted domain of integration. Due to 1 − γ (2 , , k ) > E involving the hypergeometricfunction can be controlled from above and from below with a positive constant. Consequently, for y ∈ [ R, z ] and b ∈ [ A − k ( y − R ) , A − k ( y + R )], on the characteristic A k ( t ) − z = R we have E (cid:0) t, z ; b, y ; 2 , , k (cid:1) = c bt (cid:0) ( φ k ( t ) + φ k ( b )) − ( y − z ) (cid:1) − γ F (cid:18) γ, γ ; 1; ( φ k ( t ) − φ k ( b )) − ( y − z ) ( φ k ( t ) + φ k ( b )) − ( y − z ) (cid:19) & bt (cid:0) ( φ k ( t ) + φ k ( b )) − ( y − z ) (cid:1) − γ = bt (cid:0) ( A k ( t ) + A k ( b ) + 2 φ k (1)) − ( y − z ) (cid:1) − γ & bt (cid:0) ( A k ( t ) − R + y + 2 φ k (1)) − ( y − z ) (cid:1) − γ = bt (cid:0) ( z + y + 2 φ k (1)) − ( y − z ) (cid:1) − γ = bt (cid:16) z + φ k (1))( y + φ k (1)) (cid:17) − γ & A − k ( y − R ) t ( z + R ) − γ ( y + R ) − γ , where we used that γ = γ (2 , , k ) is a negative parameter. We remark that for y ∈ [ R, z ], it holds A − k ( y − R ) A − k ( y + R ) = φ − k ( φ k (1) + y − R ) φ − k ( φ k (1) + y + R ) = (cid:18) φ k (1) + y − Rφ k (1) + y + R (cid:19) − k = (cid:18) − Rφ k (1) + y + R (cid:19) − k > (cid:18) − Rφ k (1) + 2 R (cid:19) − k = C R,k . Combining the lower bound estimate for E on the domain of integration and the last inequality, wearrive at I ( t, z ) & ( z + R ) − γ t − ˆ zR ( R + y ) − n − ( p − − γ A − k ( y + R ) ˆ A − k ( y + R ) A − k ( y − R ) | ∂ t U ( b, y ) | p d b d y & ( z + R ) − γ t − ˆ zR ( R + y ) − n − ( p − − γ + − k ˆ A − k ( y + R ) A − k ( y − R ) | ∂ t U ( b, y ) | p d b d y, (39)11or z > R such that A k ( t ) − z = R .Applying Jensen’s inequality, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ A − k ( y + R ) A − k ( y − R ) ∂ t U ( b, y ) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:0) A − k ( y + R ) − A − k ( y − R ) (cid:1) p − ˆ A − k ( y + R ) A − k ( y − R ) | U t ( b, y ) | p d b (2 R ) p − (cid:18) max τ ∈ [ y − R,y + R ] dd τ A − k ( τ ) (cid:19) p − ˆ A − k ( y + R ) A − k ( y − R ) | ∂ t U ( b, y ) | p d b . ( φ k (1) + y + R ) k − k ( p − ˆ A − k ( y + R ) A − k ( y − R ) | ∂ t U ( b, y ) | p d b, (40)where we used dd τ A − k ( τ ) = (cid:0) (1 − k )( φ k (1) + τ ) (cid:1) k − k .Combining the fundamental theorem of calculus, (39) and (40), on the characteristic A k ( t ) − z = R we find I ( A − k ( z + R ) , z ) & ( z + R ) − γ − − k ˆ zR ( R + y ) − n − ( p − − γ + − k − k − k ( p − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ A − k ( y + R ) A − k ( y − R ) ∂ t U ( b, y ) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p d y = ( z + R ) − γ − − k ˆ zR ( R + y ) − n − ( p − − γ + − k − k − k ( p − | U ( A − k ( y + R ) , y ) | p d y, (41)where in the second step we can apply the relation U ( A − k ( y − R ) , y ) = 0 due to (34).Using together (38) and (41), on the characteristic line A k ( t ) − z = R for z > R , we obtain( R + z ) γ + − k U ( A − k ( z + R ) , z ) > Cε k u + u k L ( R n ) + C ˆ zR ( R + y ) − n − ( p − − γ + − k − k − k ( p − | U ( A − k ( y + R ) , y ) | p d y, (42)where C = C ( n, k, R, p ) > U ( z ) . = ( R + z ) γ + − k U ( A − k ( z + R ) , z ) , z > R, whose dynamic will be employed to prove the blow-up result.Rewriting (42) through U , we find U ( z ) > Cε k u + u k L ( R n ) + C ˆ zR ( R + y )( − n − − k − k − γ ) ( p − − γ | U ( y ) | p d y for z > R. (43)Finally, we apply a comparison argument to U . We define the auxiliary function G in the followingway G ( z ) . = M ε + C ˆ zR ( R + y )( − n − − k − k − γ ) ( p − − γ | U ( y ) | p d y for z > R, where M . = C k u + u k L ( R n ) . From (43) we get immediately U > G . Furthermore, G fulfills theordinary differential inequality G ′ ( z ) = C ( R + z )( − n − − k − k − γ ) ( p − − γ | U ( z ) | p > C ( R + z )( − n − − k − k − γ ) ( p − − γ ( G ( z )) p and satisfies the initial condition G ( R ) = M ε . If p satisfies (cid:18) − n − − k − k − γ (cid:19) ( p − − γ = − , (44)then, ( M ε ) − p − G ( z ) − p > C ( p −
1) log (cid:18) R + z R (cid:19) . (45)12e underline that (44) is equivalent to p = p Gla (cid:0) (1 − k ) n + 2 k + 2 (cid:1) .Otherwise, since G ( z ) > z > R , it follows( M ε ) − p − G ( z ) − p > C (1 − k ) p − − (1 − k ) n +2 k +12 (cid:16) ( R + z ) − γ − (1 − k ) n +2 k +12(1 − k ) ( p − − (2 R ) − γ − (1 − k ) n +2 k +12(1 − k ) ( p − (cid:17) . (46)Thus, if p ∈ (cid:0) , p Gla (cid:0) (1 − k ) n + 2 k + 2 (cid:1)(cid:1) , then, the multiplicative factor on the right-hand side of(46) is positive. So, we let ε = ε ( n, p, k, u , u , R ) sufficiently small such that for any ε ∈ (0 , ε ] itresults G ( z ) > " M ε ) − p − C (1 − k ) p − − (1 − k ) n +2 k +12 ( R + z ) − γ − (1 − k ) n +2 k +12(1 − k ) ( p − − p − . (47)In the limit case p = p Gla (cid:0) (1 − k ) n + 2 k + 2 (cid:1) , from (45) we get the estimate U ( z ) > G ( z ) > (cid:2) ( M ε ) − p − C ( p −
1) log (cid:0) R + z R (cid:1)(cid:3) − p − , that provides the blow-up in finite time of U ( z ) and the lifespan estimatelog T ( ε ) . ε − ( p − . Otherwise, in the case p ∈ (cid:0) , p Gla (cid:0) (1 − k ) n + 2 k + 2 (cid:1)(cid:1) , the right-hand side of the inequality in (47)blows up for A k ( t ) = R + z ≈ ε − (1 − k ) (cid:0) p − − (1 − k ) n +2 k +12 (cid:1) − . Therefore, G (and U , in turn) blows up and the following upper bound estimates holds T ( ε ) . ε − (cid:0) p − − (1 − k ) n +2 k +12 (cid:1) − . This completes the proof of Theorem 1.1.
The proof of Theorem 1.2 is analogous to the one of Theorem 1.1. Let us just sketch the key pointsin the blow-up argument and emphasize the modifications that we have to carry out in comparisonto the previous case. Given a local solution u to (5), we may introduce also in this case the function U = U ( t, z ) for t > z ∈ R by integrating with respect to the last ( n − u = 0 and Corollary 2.3, the following representation holds U ( t, z ) = ε e J ( t, z ) + e I ( t, z ) , where e J ( t, z ) . = ˆ z + A k ( t ) z − A k ( t ) U ( y ) K (cid:0) t, z ; y ; 0 , , k (cid:1) d y, e I ( t, z ) . = ˆ t ˆ z + φ k ( t ) − φ k ( b ) z − φ k ( t )+ φ k ( b ) ˆ R n − | ∂ t u ( b, y, w ) | p d w E (cid:0) t, z ; b, y ; 0 , , k (cid:1) d y d b. In order to estimate from below the terms e J, e I , we can proceed very similarly as in the previous proof,keeping in mind the relation pointed out in Remark 4 on the fundamental solution E . More precisely,in place of (38), we get e J ( t, z ) & ε t k k u k L ( R n ) , while instead of (41), we find e I ( A − k ( z + R ) , z ) & ( z + R ) − γ ˆ zR ( R + y ) − n − ( p − − γ − k − k ( p − | U ( A − k ( y + R ) , y ) | p d y,
13n the characteristic A k ( t ) − z = R , for z > R . Therefore, the functional that we have to consider inorder to study the blow-up dynamic is e U ( z ) . = ( z + R ) γ U ( A − k ( z + R ) , z ) . Consequently, the ordinary integral inequality for e U is given by e U ( z ) > e Cε k u k L ( R n ) + e C ˆ zR ( R + y )( − n − − k − k − γ ) ( p − − γ | e U ( y ) | p d y for z > R, where e C > y + R ) in the lastintegral is equal to − p = p Gla ((1 − k ) n + 2 k ). Repeating the same kind of computationsas in Section 3.1, we obtain the blow-up in finite time for e U provided that p ∈ (1 , p Gla ((1 − k ) n + 2 k )]and the corresponding upper bound estimates for the lifespan. We concluded the proof of Theorem1.2. In the present paper, we proved blow-up results for the semilinear model ∂ t u − t − k ∆ u + µt − ∂ t u = | ∂ t u | p , x ∈ R n , t > ,u (1 , x ) = εu ( x ) , x ∈ R n ,u t (1 , x ) = εu ( x ) , x ∈ R n , (48)when µ ∈ { , } , provided that the Cauchy data fulfill suitable sign and support assumptions and thatthe exponent of the nonlinear term belongs to the following range1 < p p Gla ((1 − k ) n + 2 k + µ ) . Due to the consistency of this result with other results known in the literature for special values of theparameters k and µ (namely, for k = 0 and/or µ = 0), we conjecture that the previous upper bound for p could be the critical exponent for (48). Nevertheless, in order to prove the validity of this conjecturethe proof of the global (in time) existence of small data solutions in the case p > p Gla ((1 − k ) n +2 k + µ )is necessary.We point out that in the general case µ >
0, the blow-up argument from Theorems 1.1 and 1.2does not work sharply, especially for µ in the interval ( k, − k ). In the forthcoming paper [11], wewill study systematically the blow-up dynamic for (48) via a completely different approach. Acknowledgments
A. Palmieri is supported by the GNAMPA project ’Problemi stazionari e di evoluzione nelle equazionidi campo nonlineari dispersive’.
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