A blow-up result for semi-linear structurally damped σ -evolution equations
aa r X i v : . [ m a t h . A P ] S e p A BLOW-UP RESULT FOR SEMI-LINEAR STRUCTURALLY DAMPED σ -EVOLUTION EQUATIONS TUAN ANH DAO AND MICHAEL REISSIG
Abstract.
We would like to prove a blow-up result for semi-linear structurally damped σ -evolutionequations, where σ ě δ P r , σ q are assumed to be any fractional numbers. To deal with thefractional Laplacian operators p´ ∆ q σ and p´ ∆ q δ as well-known non-local operators, in general, itseems difficult to apply the standard test function method directly. For this reason, in this paperwe shall construct new test functions to overcome this difficulty. Introduction
The main goal of this paper is to discuss the critical exponent to the following Cauchy problemfor semi-linear structurally damped σ -evolution models: u tt ` p´ ∆ q σ u ` p´ ∆ q δ u t “ | u | p ,u p , x q “ u p x q , u t p , x q “ u p x q , (1)with some σ ě δ P r , σ q and a given real number p ą
1. Here, critical exponent p crit “ p crit p n q means that for some range of admissible p ą p crit there exists a global (in time) Sobolev solutionfor small initial data from a suitable function space. Moreover, one can find suitable small datasuch that there exists no global (in time) Sobolev solution if 1 ă p ď p crit . In other words, we have,in general, only local (in time) Sobolev solutions under this assumption for the exponent p .For the local existence of Sobolev solutions to (1), we address the interested readers to Proposition9 . ‚ For given nonnegative f and g we write f À g if there exists a constant C ą f ď Cg .We write f « g if g À f À g . ‚ As usual, H a with a ě L . ‚ We denote by r b s the integer part of b P R . ‚ Moreover, we introduce the following two parameters: k ´ : “ min t σ ; 2 δ u and k ` : “ max t σ ; 2 δ u if δ P r , σ q . In order to state our main result, we recall the global (in time) existence result of small dataenergy solutions to (1) in the following theorem.
Mathematics Subject Classification.
Key words and phrases. σ -evolution equations; structural damping; critical exponent; blow-up; test functions.The PhD study of MSc. T.A. Dao is supported by Vietnamese Government’s Scholarship (Grant number:2015/911). Theorem 1.1 ( Global existence ) . Let m P r , q and n ą m k ´ with m “ m ´ . We assumethe conditions m ď p ă 8 if n ď k ` , m ď p ď nn ´ k ` if n P ´ k ` , k ` ´ m ı . Moreover, we suppose the following condition: p ą ` m p k ` ` σ q n ´ m k ´ . (2) Then, there exists a constant ε ą such that for any small data p u , u q P ` L m X H k ` ˘ ˆ ` L m X L ˘ satisfying the assumption } u } L m X H k ` ` } u } L m X L ď ε , we have a uniquely determined global (in time) small data energy solution u P C pr , , H k ` q X C pr , , L q to (1). Moreover, the following estimates hold: } u p t, ¨q} L À p ` t q ´ n p k `´ δ q p m ´ q` k ´ p k `´ δ q ` } u } L m X H k ` ` } u } L m X L ˘ , ›› | D | k ` u p t, ¨q ›› L À p ` t q ´ n p k `´ δ q p m ´ q´ k `´ k ´ p k `´ δ q ` } u } L m X H k ` ` } u } L m X L ˘ , } u t p t, ¨q} L À p ` t q ´ n p k `´ δ q p m ´ q´ σ ´ k ´ k `´ δ ` } u } L m X H k ` ` } u } L m X L ˘ . We are going to prove the following main result.
Theorem 1.2 ( Blow-up ) . Let σ ě , δ P r , σ q and n ą k ´ . We assume that we choose theinitial data u “ and u P L satisfying the following relation: ż R n u p x q dx ą . (3) Moreover, we suppose the condition p P ´ , ` σn ´ k ´ ¯ . (4) Then, there is no global (in time) Sobolev solution u P C ` r , , H σ ˘ to (1). Remark 1.1.
We want to underline that the lifespan T ε of Sobolev solutions to given data p , εu q for any small positive constant ε in Theorem 1.2 can be estimated as follows: T ε ď Cε ´ p σ ´ k ´qp p ´ q σ ´p n ´ k ´qp p ´ q with C ą . (5) Remark 1.2.
If we choose m “ p crit is given by p crit p n q “ ` σn ´ δ if δ P ” , σ ı and 4 δ ă n ď σ. However, in the case δ P p σ , σ q there appears a gap between the exponents given by 1 ` δ ` σn ´ σ fromTheorem 1.1 and 1 ` σn ´ σ from Theorem 1.2 for 2 σ ă n ď δ . BLOW-UP RESULT FOR SEMI-LINEAR STRUCTURALLY DAMPED σ -EVOLUTION EQUATIONS 3 Preliminaries
In this section, we collect some preliminary knowledge needed in our proofs.
Definition 2.1 ([8, 11]) . Let s P p , q . Let X be a suitable set of functions defined on R n . Then,the fractional Laplacian p´ ∆ q s in R n is a non-local operator given by p´ ∆ q s : v P X Ñ p´ ∆ q s v p x q : “ C n,s p.v. ż R n v p x q ´ v p y q| x ´ y | n ` s dy as long as the right-hand side exists, where p.v. stands for Cauchy’s principal value, C n,s : “ s Γ p n ` s q π n Γ p´ s q is a normalization constant and Γ denotes the Gamma function. Lemma 2.1.
Let (cid:10) x (cid:11) “ p ` | x | q and q ą . Then, the following estimate holds for any multi-index α satisfying | α | ě : ˇˇ B αx (cid:10) x (cid:11) ´ q ˇˇ À (cid:10) x (cid:11) ´ q ´| α | . Proof.
First, we recall the following formula of derivatives of composed functions for | α | ě B αx h ` f p x q ˘ “ | α | ÿ k “ h p k q ` f p x q ˘ ¨˚˚˝ ÿ γ `¨¨¨` γ k ď α | γ |`¨¨¨`| γ k |“| α | , | γ i |ě ` B γ x f p x q ˘ ¨ ¨ ¨ ` B γ k x f p x q ˘˛‹‹‚ , where h “ h p z q and h p k q p z q “ d k h p z q dz k . Applying this formula with h p z q “ z ´ q and f p x q “ ` | x | we obtain ˇˇ B αx (cid:10) x (cid:11) ´ q ˇˇ ď | α | ÿ k “ p ` | x | q ´ q ´ k ¨˚˚˝ ÿ γ `¨¨¨` γ k ď α | γ |`¨¨¨`| γ k |“| α | , | γ i |ě ˇˇ B γ x p ` | x | q ˇˇ ¨ ¨ ¨ ˇˇ B γ k x p ` | x | q ˇˇ˛‹‹‚ ď C | α | ÿ k “ p ` | x | q ´ q ´ k $’’’’&’’’’% ď | x | ď , ¨˚˚˝ ÿ γ `¨¨¨` γ k ď α | γ |`¨¨¨`| γ k |“| α | , | γ i |ě | x | ´| γ | ¨ ¨ ¨ | x | ´| γ k | ˛‹‹‚ if | x | ě , ď C | α | ÿ k “ p ` | x | q ´ q ´ k ď | x | ď , | x | k ´| α | if | x | ě , ď C | α | (cid:10) x (cid:11) ´ q ´ if 0 ď | x | ď ,C | α | (cid:10) x (cid:11) ´ q | x | ´| α | if | x | ě , where C and C are some suitable constants. This completes the proof. (cid:3) Lemma 2.2.
Let m P Z , s P p , q and γ : “ m ` s . If v P H γ p R n q , then it holds p´ ∆ q γ v p x q “ p´ ∆ q m ` p´ ∆ q s v p x q ˘ “ p´ ∆ q s ` p´ ∆ q m v p x q ˘ . One can find the proof of Lemma 2.2 in Remark 3 . Lemma 2.3.
Let m P Z , s P p , q and γ : “ m ` s . Let (cid:10) x (cid:11) “ p ` | x | q and q ą . Then, thefollowing estimates hold for all x P R n : ˇˇ p´ ∆ q γ (cid:10) x (cid:11) ´ q ˇˇ À $’&’% (cid:10) x (cid:11) ´ q ´ γ if ă q ` m ă n, (cid:10) x (cid:11) ´ n ´ s log p e ` | x |q if q ` m “ n, (cid:10) x (cid:11) ´ n ´ s if q ` m ą n. (6) TUAN ANH DAO AND MICHAEL REISSIG
Proof.
We follow ideas from the proof of Lemma 1 . m “ s “ ,that is, the case γ “ is generalized to any fractional number γ ą
0. To do this, for any s P p , q we shall divide the proof into two cases: m “ m ě m “
0. Denoting by ψ “ ψ p x q : “ (cid:10) x (cid:11) ´ q we write p´ ∆ q s (cid:10) x (cid:11) ´ q “p´ ∆ q s p ψ qp x q . According to Definition 2.1 of fractional Laplacian as a singular integral operator,we have p´ ∆ q s p ψ qp x q : “ C n,δ p.v. ż R n ψ p x q ´ ψ p y q| x ´ y | n ` s dy. A standard change of variables leads to p´ ∆ q s p ψ qp x q “ ´ C n,s p.v. ż R n ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy “ ´ C n,s ε Ñ ` ż ε ď| y |ď ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy ´ C n,s ż | y |ě ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy. To deal with the first integral, after using a second order Taylor expansion for ψ we arrive at | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q|| y | n ` s À }B x ψ } L | y | n ` s ´ . Thanks to the above estimate and s P p , q , we may remove the principal value of the integral atthe origin to conclude p´ ∆ q s p ψ qp x q “ ´ C n,s ż R n ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy. To prove the desired estimates, we shall divide our considerations into two cases. In the first subcase t x : | x | ď u , we can proceed as follows: ˇˇ p´ ∆ q s p ψ qp x q ˇˇ À ż | y |ď | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q|| y | n ` s dy ` ż | y |ě | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q|| y | n ` s dy À }B x ψ } L ż | y |ď | y | n ` s ´ dy ` } ψ } L ż | y |ě | y | n ` s dy. Due to the boundedness of the above two integrals, it follows immediately ˇˇ p´ ∆ q s p ψ qp x q ˇˇ À | x | ď . (7)In order to deal with the second subcase t x : | x | ě u , we can re-write p´ ∆ q s p ψ qp x q “ ´ C n,s ż | y |ě | x | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy ´ C n,s ż | x |ď| y |ď | x | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy ´ C n,s ż | y |ď | x | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy. (8) BLOW-UP RESULT FOR SEMI-LINEAR STRUCTURALLY DAMPED σ -EVOLUTION EQUATIONS 5 For the first integral, we notice that the relations | x ` y | ě | y | ´ | x | ě | x | and | x ´ y | ě | y | ´ | x | ě | x | hold for | y | ě | x | . Since ψ is a decreasing function, we obtain the following estimate: ˇˇˇ ż | y |ě | x | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy ˇˇˇ ď | ψ p x q| ż | y |ě | x | | y | n ` s dy À (cid:10) x (cid:11) ´ q ż | y |ě | x | | y | ` s d | y |À (cid:10) x (cid:11) ´ q | x | ´ s À (cid:10) x (cid:11) ´ q ´ s ` due to | x | « (cid:10) x (cid:11) for | x | ě ˘ . (9)It is clear that | y | « | x | in the second integral domain. Moreover, it follows ! y : 12 | x | ď | y | ď | x | ) Ă y : | x ` y | ď | x | ( , (10) ! y : 12 | x | ď | y | ď | x | ) Ă y : | x ´ y | ď | x | ( . (11)For this reason, we arrive at ˇˇˇ ż | x |ď| y |ď | x | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy ˇˇˇ À | x | ´ n ´ s ´ ż | x ` y |ď | x | ψ p x ` y q dy ` ż | x ´ y |ď | x | ψ p x ´ y q dy ` ψ p x q ż | x |ď| y |ď | x | dy ¯ À | x | ´ n ´ s ´ ż | x ` y |ď | x | ψ p x ` y q dy ` (cid:10) x (cid:11) ´ q | x | n ¯ , (12)where we used the relation ż | x ` y |ď | x | ψ p x ` y q dy “ ż | x ´ y |ď | x | ψ p x ´ y q dy. By the change of variables r “ | x ` y | , we apply the inequality 1 ` r ě p ` r q to get ż | x ` y |ď | x | ψ p x ` y q dy À ż r ď | x | p ` r q ´ q r n ´ dr À ż r ď | x | p ` r q n ´ q ´ dr À $’&’% p ` | x |q n ´ q if 0 ă q ă n, log p e ` | x |q if q “ n, q ą n. (13)By | x | « (cid:10) x (cid:11) for | x | ě
1, combining (12) and (13) leads to ˇˇˇ ż | x |ď| y |ď | x | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy ˇˇˇ À $’&’% (cid:10) x (cid:11) ´ q ´ s if 0 ă q ă n, (cid:10) x (cid:11) ´ n ´ s log p e ` | x |q if q “ n, (cid:10) x (cid:11) ´ n ´ s if q ą n. (14)For the third integral in (8), using again the second order Taylor expansion for ψ we obtain ˇˇˇ ż | y |ď | x | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q| y | n ` s dy ˇˇˇ ď ż | y |ď | x | | ψ p x ` y q ` ψ p x ´ y q ´ ψ p x q|| y | n ` s dy À ż | y |ď | x | max θ Pr , s ˇˇ B x ψ p x ˘ θy q ˇˇ | y | n ` s ´ dy À ż | y |ď | x | max θ Pr , s (cid:10) x ˘ θy (cid:11) ´ q ´ | y | n ` s ´ dy À (cid:10) x (cid:11) ´ q ´ ż | y |ď | x | | y | ´ s d | y | À (cid:10) x (cid:11) ´ q ´ s . (15) TUAN ANH DAO AND MICHAEL REISSIG
Here we used the relation | x ˘ θy | ě | x | ´ θ | y | ě | x | ´ | x | “ | x | . From (8), (9), (14) and (15) wearrive at the following estimates for | x | ě ˇˇ p´ ∆ q s p ψ qp x q ˇˇ À $’&’% (cid:10) x (cid:11) ´ q ´ s if 0 ă q ă n, (cid:10) x (cid:11) ´ n ´ s log p e ` | x |q if q “ n, (cid:10) x (cid:11) ´ n ´ s if q ą n. (16)Finally, combining (7) and (16) we may conclude all desired estimates for m “ m ě
1. First, a straight-forward calculation gives the followingrelation: ´ ∆ (cid:10) x (cid:11) ´ r “ r ´ p n ´ r ´ q (cid:10) x (cid:11) ´ r ´ ` p r ` q (cid:10) x (cid:11) ´ r ´ ¯ for any r ą . (17)By induction argument, carrying out m steps of (17) we obtain the following formula for any m ě p´ ∆ q m (cid:10) x (cid:11) ´ q “ p´ q m m ´ ź j “ p q ` j q ´ m ź j “ p´ n ` q ` j q (cid:10) x (cid:11) ´ q ´ m ´ C m m ź j “ p´ n ` q ` j qp q ` m q (cid:10) x (cid:11) ´ q ´ m ´ ` C m m ź j “ p´ n ` q ` j qp q ` m qp q ` m ` q (cid:10) x (cid:11) ´ q ´ m ´ ` ¨ ¨ ¨ ` p´ q m m ´ ź j “ p q ` m ` j q (cid:10) x (cid:11) ´ q ´ m ¯ . (18)Then, thanks to Lemma 2.2, we derive p´ ∆ q γ (cid:10) x (cid:11) ´ q “ p´ ∆ q s ` p´ ∆ q m (cid:10) x (cid:11) ´ q ˘ “ p´ q m m ´ ź j “ p q ` j q ´ m ź j “ p´ n ` q ` j q p´ ∆ q s (cid:10) x (cid:11) ´ q ´ m ´ C m m ź j “ p´ n ` q ` j qp q ` m q p´ ∆ q s (cid:10) x (cid:11) ´ q ´ m ´ ` C m m ź j “ p´ n ` q ` j qp q ` m qp q ` m ` q p´ ∆ q s (cid:10) x (cid:11) ´ q ´ m ´ ` ¨ ¨ ¨ ` p´ q m m ´ ź j “ p q ` m ` j q p´ ∆ q s (cid:10) x (cid:11) ´ q ´ m ¯ . (19)For this reason, in order to conclude the desired estimates, we only indicate the following estimatesfor k “ , ¨ ¨ ¨ , m : ˇˇ p´ ∆ q s (cid:10) x (cid:11) ´ q ´ p m ` k q ˇˇ À $’&’% (cid:10) x (cid:11) ´ q ´ γ if 0 ă q ` m ă n, (cid:10) x (cid:11) ´ n ´ s log p e ` | x |q if q ` m “ n, (cid:10) x (cid:11) ´ n ´ s if q ` m ą n. (20)Indeed, substituting q by q ` p m ` k q with k “ , ¨ ¨ ¨ , m and γ “ s into (6) leads to ˇˇ p´ ∆ q s (cid:10) x (cid:11) ´ q ´ p m ` k q ˇˇ À $’&’% (cid:10) x (cid:11) ´ q ´ γ if 0 ă q ` p m ` k q ă n, (cid:10) x (cid:11) ´ n ´ s log p e ` | x |q if q ` p m ` k q “ n, (cid:10) x (cid:11) ´ n ´ s if q ` p m ` k q ą n. BLOW-UP RESULT FOR SEMI-LINEAR STRUCTURALLY DAMPED σ -EVOLUTION EQUATIONS 7 From these estimates, it follows immediately (20) to conclude (6) for any m ě
1. Summarizing,the proof of Lemma 2.3 is completed. (cid:3)
Lemma 2.4.
Let s P p , q . Let ψ be a smooth function satisfying B x ψ P L . For any R ą , let ψ R be a function defined by ψ R p x q : “ ψ ` R ´ x ˘ for all x P R n . Then, p´ ∆ q s p ψ R q satisfies the following scaling properties for all x P R n : p´ ∆ q s p ψ R qp x q “ R ´ s ` p´ ∆ q s ψ ˘` R ´ x ˘ . Proof.
Thanks to the assumption B x ψ P L , following the proof of Lemma 2.3 we may remove theprincipal value of the integral at the origin to conclude p´ ∆ q s p ψ R qp x q “ ´ C n,s ż R n ψ R p x ` y q ` ψ R p x ´ y q ´ ψ R p x q| y | n ` s dy “ ´ C n,s R ´ s ż R n ψ ` R ´ x ` R ´ y ˘ ` ψ ` R ´ x ´ R ´ y ˘ ´ ψ ` R ´ x ˘ | R ´ y | n ` s d p R ´ y q“ R ´ s ` p´ ∆ q s ψ ˘` R ´ x ˘ . This completes the proof. (cid:3) Proof of the blow-up result
We divide the proof of Theorem 1.2 into several cases.3.1.
The case that both parameters σ and δ are integers. The proof of this case can befound in the paper [4].3.2.
The case that the parameter σ is integer and the parameter δ is fractional from p , q . Proof.
First, we introduce the function ϕ “ ϕ p| x |q : “ (cid:10) x (cid:11) ´ n ´ δ and the function η “ η p t q havingthe following properties:1 . η P C pr , and η p t q “ $’&’% ď t ď , decreasing for ď t ď , t ě , . η ´ p p p t q ` | η p t q| p ` | η p t q| p ˘ ď C for any t P ” , ı , (21)where p is the conjugate of p ą
1. Let R be a large parameter in r , . We define the followingtest function: φ R p t, x q : “ η R p t q ϕ R p x q , where η R p t q : “ η ` R ´ α t ˘ and ϕ R p x q : “ ϕ ` R ´ x ˘ with a fixed parameter α : “ σ ´ k ´ . We definethe functionals I R : “ ż ż R n | u p t, x q| p φ R p t, x q dxdt “ ż R α ż R n | u p t, x q| p φ R p t, x q dxdt and I R,t : “ ż R αRα ż R n | u p t, x q| p φ R p t, x q dxdt. TUAN ANH DAO AND MICHAEL REISSIG
Let us assume that u “ u p t, x q is a global (in time) Sobolev solution from C ` r , , H σ ˘ to (1).After multiplying the equation (1) by φ R “ φ R p t, x q , we carry out partial integration to derive0 ď I R “ ´ ż R n u p x q ϕ R p x q dx ` ż R αRα ż R n u p t, x qB t η R p t q ϕ R p x q dxdt ` ż ż R n η R p t q ϕ R p x q p´ ∆ q σ u p t, x q dxdt ´ ż R αRα ż R n B t η R p t q ϕ R p x q p´ ∆ q δ u p t, x q dxdt “ : ´ ż R n u p x q ϕ R p x q dx ` J ` J ´ J . (22)Applying H¨older’s inequality with p ` p “ | J | ď ż R αRα ż R n | u p t, x q| ˇˇ B t η R p t q ˇˇ ϕ R p x q dxdt À ´ ż R αRα ż R n ˇˇˇ u p t, x q φ p R p t, x q ˇˇˇ p dxdt ¯ p ´ ż R αRα ż R n ˇˇˇ φ ´ p R p t, x qB t η R p t q ϕ R p x q ˇˇˇ p dxdt ¯ p À I p R,t ´ ż R αRα ż R n η ´ p p R p t q ˇˇ B t η R p t q ˇˇ p ϕ R p x q dxdt ¯ p . By the change of variables ˜ t : “ R ´ α t and ˜ x : “ R ´ x , a straight-forward calculation gives | J | À I p R,t R ´ α ` n ` αp ´ ż R n (cid:10) ˜ x (cid:11) ´ n ´ δ d ˜ x ¯ p . (23)Here we used B t η R p t q “ R ´ α η p ˜ t q and the assumption (21). Now let us turn to estimate J and J . First, we notice that by Parseval-Plancherel formula it holds: ż R n v p x q p´ ∆ q γ v p x q dx “ ż R n | ξ | γ p v p ξ q p v p ξ q dξ “ ż R n v p x q p´ ∆ q γ v p x q dx, for any γ ą v , v P H γ . Here p v j “ p v j p ξ q stands for the Fourier transform with respectto the spatial variables of v j “ v j p x q , j “ ,
2. Using ϕ R P H σ and u P C ` r , , H σ ˘ we mayconclude ż R n ϕ R p x q p´ ∆ q σ u p t, x q dx “ ż R n u p t, x q p´ ∆ q σ ϕ R p x q dx, ż R n ϕ R p x q p´ ∆ q δ u p t, x q dx “ ż R n u p t, x q p´ ∆ q δ ϕ R p x q dx, that is, ` ϕ R , p´ ∆ q σ u p t, ¨q ˘ L “ ` p´ ∆ q σ ϕ R , u p t, ¨q ˘ L , ` ϕ R , p´ ∆ q δ u p t, ¨q ˘ L “ ` p´ ∆ q δ ϕ R , u p t, ¨q ˘ L . We can see that under the assumptions both scalar products are well defined. Hence, we obtain J “ ż ż R n η R p t q ϕ R p x q p´ ∆ q σ u p t, x q dxdt “ ż ż R n η R p t q u p t, x q p´ ∆ q σ ϕ R p x q dxdt, and J “ ż R αRα ż R n B t η R p t q ϕ R p x q p´ ∆ q δ u p t, x q dxdt “ ż R αRα ż R n B t η R p t q u p t, x q p´ ∆ q δ ϕ R p x q dxdt. BLOW-UP RESULT FOR SEMI-LINEAR STRUCTURALLY DAMPED σ -EVOLUTION EQUATIONS 9 Applying H¨older’s inequality again as we estimated J leads to | J | ď I p R ´ ż R α ż R n η R p t q ϕ ´ p p R p x q ˇˇ p´ ∆ q σ ϕ R p x q ˇˇ p dxdt ¯ p , and | J | ď I p R,t ´ ż R αRα ż R n η ´ p p R p t q ˇˇ B t η R p t q ˇˇ p ϕ ´ p p R p x q ˇˇ p´ ∆ q δ ϕ R p x q ˇˇ p dxdt ¯ p . In order to control the above two integrals, the key tools rely on results from Lemmas 2.1, 2.3 and2.4. Namely, at first carrying out the change of variables ˜ t : “ R ´ α t and ˜ x : “ R ´ x we arrive at | J | À I p R R ´ σ ` n ` αp ´ ż ż R n η p ˜ t q ϕ ´ p p p ˜ x q ˇˇ p´ ∆ q σ p ϕ qp ˜ x q ˇˇ p d ˜ xd ˜ t ¯ p À I p R R ´ σ ` n ` αp ´ ż R n ϕ ´ p p p ˜ x q ˇˇ p´ ∆ q σ p ϕ qp ˜ x q ˇˇ p d ˜ x ¯ p , where we note ( σ is an integer) that p´ ∆ q σ ϕ R p x q “ R ´ σ p´ ∆ q σ ϕ p ˜ x q . Using Lemma 2.1 impliesthe following estimate: | J | À I p R R ´ σ ` n ` αp ´ ż R n (cid:10) ˜ x (cid:11) ´ n ´ δ ´ σp d ˜ x ¯ p . (24)Next carrying out again the change of variables ˜ t : “ R ´ α t and ˜ x : “ R ´ x and employing Lemma2.4 we can proceed J as follows: | J | À I p R,t R ´ δ ´ α ` n ` αp ´ ż ż R n η ´ p p p ˜ t q ˇˇ η p ˜ t q ˇˇ p ϕ ´ p p p ˜ x q ˇˇ p´ ∆ q δ p ϕ qp ˜ x q ˇˇ p d ˜ xd ˜ t ¯ p À I p R,t R ´ δ ´ α ` n ` αp ´ ż R n ϕ ´ p p p ˜ x q ˇˇ p´ ∆ q δ p ϕ qp ˜ x q ˇˇ p d ˜ x ¯ p . Here we used B t η R p t q “ R ´ α η p ˜ t q and the assumption (21). To deal with the last integral, we applyLemma 2.3 with q “ n ` δ and γ “ δ , that is, m “ s “ δ to get | J | À I p R,t R ´ δ ´ α ` n ` αp ´ ż R n (cid:10) ˜ x (cid:11) ´ n ´ δ d ˜ x ¯ p . (25)Because of assumption (3), there exists a sufficiently large constant R ą ż R n u p x q ϕ R p x q dx ą R ą R . Combining the estimates from (22) to (26) we may arrive at0 ă ż R n u p x q ϕ R p x q dx À I p R,t ` R ´ α ` n ` αp ` R ´ α ´ δ ` n ` αp ˘ ` I p R R ´ σ ` n ` αp ´ I R (27) À I p R R ´ σ ` n ` αp ´ I R (28)for all R ą R . Moreover, applying the inequality A y γ ´ y ď A ´ γ for any A ą , y ě ă γ ă ă ż R n u p x q ϕ R p x q dx À R ´ σp ` n ` α (29)for all R ą R . It is clear that the assumption (4) is equivalent to ´ σp ` n ` α ă
0. For thisreason, letting R Ñ 8 in (29) we obtain ż R n u p x q dx “ . This is a contradiction to the assumption (3). Summarizing, the proof is completed. (cid:3)
Let us now consider the case of subcritical exponent to explain the estimate for lifespan T ε ofsolutions in Remark 1.1. We assume that u “ u p t, x q is a local (in time) Sobolev solution to (1) in r , T q ˆ R n . In order to prove the lifespan estimate, we replace the initial data p , u q by p , εf q with a small constant ε ą
0, where f P L satisfies the assumption (3). Hence, there exists asufficiently large constant R ą ż R n f p x q ϕ R p x q dx ě c ą R ą R . Repeating the steps in the above proofs we arrive at the following estimate: ε ď C R ´ σp ` n ` α ď C T ´ σp n ´ αα with R “ T α . Finally, letting T Ñ T ´ ε we may conclude (5). Remark 3.1.
We want to underline that in the special case σ “ δ “ the authors in [5] haveinvestigated the critical exponent p crit “ p crit p n q “ ` n ´ . If we plug σ “ δ “ into thestatements of Theorem 1.2, then the obtained results for the critical exponent p crit coincide.3.3. The case that the parameter σ is integer and the parameter δ is fractional from p , σ q . Proof.
We follow ideas from the proof of Section 3.2. At first, we denote s δ : “ δ ´ r δ s . Let usintroduce test functions η “ η p t q as in Section 3.2 and ϕ “ ϕ p x q : “ (cid:10) x (cid:11) ´ n ´ s δ . We can repeatexactly, the estimates for J and J as we did in the proof of Section 3.2 to conclude | J | À I p R,t R ´ α ` n ` αp , (30) | J | À I p R R ´ σ ` n ` αp . (31)Let us turn to estimate J , where δ is any fractional number in p , σ q . In the first step, applyingParseval-Plancherel formula and H¨older’s inequality lead to | J | ď I p R,t ´ ż R αRα ż R n η ´ p p R p t q ˇˇ B t η R p t q ˇˇ p ϕ ´ p p R p x q ˇˇ p´ ∆ q δ ϕ R p x q ˇˇ p dxdt ¯ p . Now we can re-write δ “ m δ ` s δ , where m δ : “ r δ s ě s δ is a fractional number in p , q . Employing Lemma 2.2 we derive p´ ∆ q δ ϕ R p x q “ p´ ∆ q s δ ` p´ ∆ q m δ ϕ R p x q ˘ . By the change of variables ˜ x : “ R ´ x we also notice that p´ ∆ q m δ ϕ R p x q “ R ´ m δ p´ ∆ q m δ p ϕ qp ˜ x q since m δ is an integer. Using the formula (18) we re-write p´ ∆ q m δ ϕ R p x q “ p´ q m δ R ´ m δ m δ ´ ź j “ p q ` j q ´ m δ ź j “ p´ n ` q ` j q (cid:10) ˜ x (cid:11) ´ q ´ m δ ´ C m δ m δ ź j “ p´ n ` q ` j qp q ` m δ q (cid:10) ˜ x (cid:11) ´ q ´ m δ ´ ` C m δ m δ ź j “ p´ n ` q ` j qp q ` m δ qp q ` m δ ` q (cid:10) ˜ x (cid:11) ´ q ´ m δ ´ ` ¨ ¨ ¨ ` p´ q m δ m δ ´ ź j “ p q ` m δ ` j q (cid:10) ˜ x (cid:11) ´ q ´ m δ ¯ , BLOW-UP RESULT FOR SEMI-LINEAR STRUCTURALLY DAMPED σ -EVOLUTION EQUATIONS 11 where q : “ n ` s δ . For simplicity, we introduce the following functions: ϕ k p x q : “ (cid:10) x (cid:11) ´ q ´ m δ ´ k and ϕ k,R p x q : “ ϕ k p R ´ x q “ (cid:10) ˜ x (cid:11) ´ q ´ m δ ´ k with k “ , ¨ ¨ ¨ , m δ . As a result, by Lemma 2.4 we arrive at p´ ∆ q δ ϕ R p x q “ p´ q m δ R ´ m δ m δ ´ ź j “ p q ` j q ´ m δ ź j “ p´ n ` q ` j q p´ ∆ q s δ p ϕ ,R qp x q´ C m δ m δ ź j “ p´ n ` q ` j qp q ` m δ q p´ ∆ q s δ p ϕ ,R qp x q` C m δ m δ ź j “ p´ n ` q ` j qp q ` m δ qp q ` m δ ` q p´ ∆ q s δ p ϕ ,R qp x q` ¨ ¨ ¨ ` p´ q m δ m δ ´ ź j “ p q ` m δ ` j q p´ ∆ q s δ p ϕ m δ ,R qp x q ¯ “ p´ q m δ R ´ m δ ´ s δ m δ ´ ź j “ p q ` j q ´ m δ ź j “ p´ n ` q ` j q p´ ∆ q s δ p ϕ qp ˜ x q´ C m δ m δ ź j “ p´ n ` q ` j qp q ` m δ q p´ ∆ q s δ p ϕ qp ˜ x q` C m δ m δ ź j “ p´ n ` q ` j qp q ` m δ qp q ` m δ ` q p´ ∆ q s δ p ϕ qp ˜ x q` ¨ ¨ ¨ ` p´ q m δ m δ ´ ź j “ p q ` m δ ` j q p´ ∆ q s δ p ϕ m δ qp ˜ x q ¯ “ R ´ δ p´ ∆ q δ p ϕ qp ˜ x q . For this reason, performing the change of variables ˜ t : “ R ´ α t we obtain | J | À I p R,t R ´ δ ´ α ` n ` αp ´ ż ż R n η ´ p p p ˜ t q ˇˇ η p ˜ t q ˇˇ p ϕ ´ p p p ˜ x q ˇˇ p´ ∆ q δ p ϕ qp ˜ x q ˇˇ p d ˜ xd ˜ t ¯ p À I p R,t R ´ δ ´ α ` n ` αp ´ ż R n ϕ ´ p p p ˜ x q ˇˇ p´ ∆ q δ p ϕ qp ˜ x q ˇˇ p d ˜ x ¯ p . Here we used B t η R p t q “ R ´ α η p ˜ t q and the assumption (21). After applying Lemma 2.3 with q “ n ` s δ and γ “ δ , i.e. m “ m δ and s “ s δ , we may conclude | J | À I p R,t R ´ δ ´ α ` n ` αp ´ ż R n (cid:10) ˜ x (cid:11) ´ n ´ s δ d ˜ x ¯ p À I p R,t R ´ δ ´ α ` n ` αp . (32)Finally, combining (30) to (32) and repeating arguments as in Section 3.2 we may complete theproof of Theorem 1.2. (cid:3) The case that the parameter σ is fractional from p , and the parameter δ isinteger. Proof.
We follow ideas from the proofs of Sections 3.2 and 3.3. At first, we denote s σ : “ σ ´ r σ s .Let us introduce test functions η “ η p t q as in Section 3.2 and ϕ “ ϕ p x q : “ (cid:10) x (cid:11) ´ n ´ s σ . Then,repeating the proof of Sections 3.2 and 3.3 we may conclude what we wanted to prove. (cid:3) The case that the parameter σ is fractional from p , and the parameter δ isfractional from p , q . Proof.
We follow ideas from the proofs of Sections 3.2 and 3.4. At first, we denote s σ : “ σ ´ r σ s .Next, we put s ˚ : “ min t s σ , δ u . It is obvious that s ˚ is fractional from p , q . Let us introducetest functions η “ η p t q as in Section 3.2 and ϕ “ ϕ p x q : “ (cid:10) x (cid:11) ´ n ´ s ˚ . Then, repeating the proof ofSections 3.2 and 3.4 we may conclude what we wanted to prove. (cid:3) The case that the parameter σ is fractional from p , and the parameter δ isfractional from p , σ q . Proof.
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Tuan Anh DaoSchool of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1Dai Co Viet road, Hanoi, VietnamFaculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Pr¨uferstr. 9, 09596,Freiberg, Germany
E-mail address : [email protected] Michael ReissigFaculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Pr¨uferstr. 9, 09596,Freiberg, Germany
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