On the strauss index of semilinear tricomi equation

In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution \begin{document}$ u $\end{document} of the multi-dimensional semilinear Tricomi equation \begin{document}$ \begin{equation*} \partial_t^2 u-t\, \Delta u = |u|^p, \quad \big(u(0, \cdot), \partial_t u(0, \cdot)\big) = (u_0, u_1), \end{equation*} $\end{document} where \begin{document}$ t>0 $\end{document} , \begin{document}$ x\in \mathbb R^n $\end{document} , \begin{document}$n\geq2$\end{document} , \begin{document}$ p>1 $\end{document} , and \begin{document}$ u_i\in C_0^{\infty}( \mathbb R^n) $\end{document} ( \begin{document}$ i = 0, 1 $\end{document} ). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of \begin{document}$\|u(t, \cdot)\|_{L^\infty_x(\mathbb R)}$\end{document} is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of \begin{document}$u$\end{document} in \begin{document}$t$\end{document} , then the crucial weighted Strichartz estimates are established and the global existence of solution \begin{document}$ u $\end{document} is proved when \begin{document}$ p>5 $\end{document} .


1.
Introduction. In the article, we consider the 1-D semilinear Tricomi equation where R 1+1 + = {t : t ≥ 0} × {x : x ∈ R}, p > 1, u i ∈ C ∞ 0 (R) (i = 0, 1). Our chief goal in this paper is to establish p = p crit = 5 as the critical exponent such that small-data weak solution u to (1.1) exists globally when p > p crit . Otherwise, the weak solution u to (1.1) generally blows up in finite time when 1 < p ≤ p crit .
Before we describe the content of this paper in detail, let us give a bit of historical background. Firstly, we replace the Tricomi operator with the classical wave operator, and consider (1.2) where p > 1, n ≥ 2, and v i ∈ C ∞ 0 (R n ) (i = 0, 1). Let p c = p c (n) denote the positive root of the quadratic equation (n − 1) p 2 − (n + 1) p − 2 = 0.
Strauss [30] made the following conjecture: Strauss Conjecture. If p > p c (n), then small data solutions of problem (1.2) exist globally. If 1 < p < p c (n), then small data solutions of problem (1.2) blow up in finite time.
This conjecture was almost solved in last three decades. In 1979, John [16] showed that when n = 3, the global solution v always exists if p > p c = 1 + √ 2 and ε > 0 is small, meanwhile, v will blow up in finite time if p < p c . In addition, this conjecture was shortly verified when n = 2 by Glassey [9]. On the other hand, John's blowup result was then extended by Sideris [28], showing that, for all n, there can be blowup for arbitrarily small data if p < p c . Along the other direction, Zhou [37] showed that when n = 4, in which case p c = 2, there is always global existence for small data if p > p c . Finally, Georgiev, Lindblad and Sogge in [18] and [10] prove the global existence for n ≥ 4 and p > p c (except some exceptional values of p). On the other hand, for the critical case of p = p c , the solution v generally blows up (for n = 2, 3 with p = p c (2), p c (3), see [27]; for n ≥ 4 and p = p c (n), see [36]).
(2) By the first inequality of (1.7), we have r ∈ (1, 3 2 ), and the global existence interval of p has a lower bound Now let us turn back to the 1-D Cauchy problem for Tricomi equation (1.1). Since the local existence of weak solution u to (1.1) under minimal regularity assumptions has been established in [26] and [34], without loss of generality, as in [13,14], we focus on the global small-data weak solution problem to (1.1) starting at a fixed positive time. Moreover, in order to establish the global existence result, we need to apply the Strichartz estimates with a characteristic weight ((φ(t) + M ) 2 − |x| 2 ) γ , however, the characteristic cone for Tricomi operator is φ 2 (t) = |x| 2 , which admits a cusp singularity at t = 0, due to this difficulty, we can only establish inhomogeneous Strichartz estimates for the case t is away from zero. Thus, without loss of generality, we make some reduction of the problem (1.1). More specifically, by the local existence theory in [26] and [34], for givenε > 0, there exists a fixed smallT > 0, such that if ε <ε, then the lifespan T of the local solution of (1.1) satisfies T ≥T . To this end, it is reasonable that one replaces nonlinearity |u| p in (1.1) with the nonlinear function F p (t, u) = (1 − χ(t))F p (u) + χ(t)|u| p , where F p is a C ∞ function satisfying F p (0) = 0 and |F p (u)| ≤ C(1 + |u|) p−2 |u| 2 (this assumption is reasonable since p > p crit > 2) and χ ∈ C ∞ (R) fulfills (1.9) In the following, in place of (1.1) we shall also study the problem From now on we denote φ(t) = 2 3 t 3 2 . The main result in this paper is the following theorem: Theorem 1.1 (Global existence for p > p crit ). Assume that p > p crit = 5. Then there exists a constant ε 0 ∈ (0,ε) such that, for all 0 < ε ≤ ε 0 , problem (1.10) admits a global weak solution u satisfying Note that for each r ∈ (1, 3 2 ), 3+2r 3−2r > 5, thus we have especially improved the lower bound of p in [7] to the sharp value p > 5. Remark 1.3. For 1 < p < 5, the blowup of weak solution u to (1.1) has been shown in [7]. For the critical case of p = 5, we also establish the finite time blowup in [15]. Remark 1.4. By the proof procedure of Theorem 1.1, the nonlinear term |u| p in (1.1) can be any C 1 function F (p) satisfying |∂ j u F (p) (u)| ≤ C j |u| p−j , j = 0, 1. Thus Theorem 1.1 still holds for the nonlinear terms like −|u| p and ±|u| p−1 u. Remark 1.5. By the proof of Theorem 1.1 we know that the approximate solution sequences u k → u ∈ L p+1 ( , these results together with the local existence result give that u is a weak solution of (1.10) in the sense of distribution. Remark 1.6. For the sake of brevity, we restrict ourselves in this paper to the study of the semilinear Tricomi equation instead of the generalized semilinear Tricomi equation In fact, by the arguments analogous to those in the proofs of Theorems 1.1 and the procedures in [12,13], we can establish the same result to Theorem 1.1 for the generalized semilinear Tricomi equation with the critical power p crit (m) = 1 + 4 m . Remark 1.7. For the 1-D linear wave equation for any q satisfying 1 ≤ q < ∞. More precisely, there is no global Strichartz-type inequality for solutions v to the 1-D linear wave equation. Thus, for the 1-D semilinear wave equation ∂ 2 t w − ∂ 2 x w = |w| p (p > 1), direct computation shows that the local weak solution w will generally blow up in finite time, see for example [8]. This, however, is not the case for the 1-D linear Tricomi equation: Since we can establish some weighted Strichartz estimates (see Theorem 2.1 and Theorem 3.2), we see that the global small data weak solution u exists for p > 5. . It is easily verified that p 1 < p crit = 5 in Theorems 1.1. This is due to that we need an extra condition (2.2) for the Strichartz estimate of linear homogeneous equation (see Theorem 2.1).
The linear equation ∂ 2 t u − t∂ 2 x u = 0 is the well-known Tricomi equation which arises from transonic gas dynamics (see [3,23]). There are extensive results for both linear and semilinear Tricomi equations in n space dimensions (n ∈ N). For instances, the authors of [1,32,35] have computed the forward fundamental solution of the linear Tricomi equation ∂ 2 t u − t∆u = 0 explicitly. The authors of [11,19,20,21,22] have obtained a series of interesting results on the existence and uniqueness of solutions u to the semilinear Tricomi equation x, u) in bounded domains, under certain restrictions on the nonlinearity f (t, x, u). The authors of [2,24,25,26] have established the local existence as well as the singularity structure of low regularity solutions to the Cauchy problem for semilinear Tricomi equations in the degenerate hyperbolic region and the elliptic-hyperbolic mixed region, respectively. We have additionally given a complete study of the global existence versus blowup problem for small-data solutions u to the semilinear Tricomi equation ∂ 2 t u − t∆u = |u| p for n ≥ 2 (see [12,13,14]). In addition, Lin and Tu in [17] gave the upper bound of lifespan when 1 < p ≤ p c . However, since the stationary phase method is not valid in the 1-D case, we need to utilize some different ideas to get the necessary estimates for the solution of (1.10). In the present paper, we shall systematically study the 1-D semilinear Tricomi equation We now comment on the proof of Theorem 1.1. To prove the global existence in Theorem 1.1, we are required to establish weighted Strichartz estimates for the

DAOYIN HE, INGO WITT AND HUICHENG YIN
Tricomi operator ∂ 2 t − t∂ 2 x as in [13]. We first consider the linear homogeneous equation . In this process, a series of inequalities are derived by applying an explicit formula for the solution v and by utilizing a basic observation from [10] together with some further delicate analysis. Here we point out that since the stationary phase method used to treat the M-D problem in [13,18] is not applicable in the 1-D case, one cannot obtain a suitable L 1 -L ∞ estimate for v and then use interpolation between the L 1 -L ∞ estimate and the L 2 -L 2 energy estimate to establish the Strichartztype estimate for v as in [13]. To overcome this difficulty, we write v in an explicit formula (2.9), then use the decay rate of the amplitude function in (2.9), based on the resulting inequalities (2.20), we are able to establish weighted Strichartz estimates in Theorem 2.1. For the inhomogeneous estimate Theorem 3.2, the method for symmetric wave equation from [10] motivates us to apply a dual argument, then Theorem 3.2 follows by precise computation. Combining Theorem 2.1 and Theorem 3.2, together with the contraction mapping principle, we complete the proof of Theorem 1.1.
This paper is organized as follows: In Section 2, certain weighted Strichartz estimates for the linear homogeneous Tricomi equation are established. In Section 3, related weighted Strichartz estimates are derived for the linear inhomogeneous Tricomi equation. Applying the results of Section 2 and Setion 3, Theorem 1.1 is finally proved in Section 4.

2.
Mixed-norm estimate for homogeneous equation. In order to establish the global existence of weak solution u to problem (1.1), we shall derive some mixed space-time norm estimates for the corresponding linear problem.
In this section, we consider the homogeneous problem where f, g ∈ C ∞ 0 (R) and supp(f, g) ⊆ {x : |x| ≤ R − 1} for some fixed constant R > 1. We derive a weighted space-time estimate of Strichartz-type for the solution v.
where C is a positive constant depending on q, γ, and δ.
Proof. It follows from [34] that the solution v of (2.1) can be expressed as where the symbols V j (t, ξ) (j = 1, 2) of the Fourier integral operators V j (t, D x ) are Here, z = 2iφ(t)|ξ|, ξ ∈ R, i = √ −1, and H ± are smooth functions of the variable z. By [31], one has that, for β ∈ N, Then where C > 0 is a generic constant, and for β ∈ N, Next we analyze v 2 (t, x). It follows from [5] or [34] that where Φ is the confluent hypergeometric function which is analytic with respect to the variable z = 2iφ(t)|ξ|. Thus for |z| ≤ 1, Similarly, one has Thus we arrive at

DAOYIN HE, INGO WITT AND HUICHENG YIN
where, for β ∈ N 0 , Substituting (2.9) and (2.10) into (2.8) yields where a l (l = 1, 2) satisfies The decay 1 + φ(t)|ξ| − 1 6 is the key observation in the proof of Theorem 2.1, since we can not apply stationary phase method in 1-D case. To To estimate (Af )(t, x), we now study its corresponding dyadic operators where j ∈ Z. Note that the kernel of operator A j is where |y| ≤ R because of supp f ⊆ {x : |x| ≤ R}. By (3.29) of [18], we have that for any N ∈ R + , where λ j = 2 j . Since the solution v of (2.1) is smooth and has compact support on the variable x for any fixed time, one easily knows that (2.3) holds in domain [0,T ]×R for any fixed T > 0. Therefore, in order to prove (2.3), it suffices to consider the case of φ(t) R. At this time, the following two cases will be studied separately.

Mixed-norm estimate for the inhomogeneous equation.
In this section, we turn to the inhomogeneous Tricomi equation Since the stationary phase method is not applicable to the solution w in case n = 1, one cannot obtain a suitable L 1 -L ∞ estimate and then use interpolation to obtain the Strichartz-type estimate as in [13]. To overcome this difficulty, we shall cite a conclusion from [10] and subsequently use the representation formula for w to establish the space-time mixed norm estimate by a sophisticated analysis. |u−ξ| δ |ξ| β |u| α dξ. Then f L q ((0,∞)) ≤ C g L r ((0,∞)) , provided that
Proof. To prove (3.10), we first consider the special case that F (t, x) = 0 for |x| > φ(t) − φ( T0 4 ). By finite propagation speed for the hyperbolic equation (3.1), we have that the integration domain in (3.10) is just Q = {(t, x) : t ≥ T0 2 , |x| ≤ φ(t)+R −1}. In fact, for the Tricomi equation, the speed of the propagation is a(t) = √ t, then by Example 2a in [33] (Page 308), we have where A(t) = t 0 max τ ≤s a(τ ) ds = φ(t). Then observe that Q can be covered by a finite number of angular domains {Q j } N0 j=1 , N 0 ≈ RT where χ Qj is the characteristic function of Q j , and Then supp w j ⊆ Q j . Since the Tricomi equation is invariant under the translation with respect to the variable x, it follows from Theorem 3.2 that , (3.11) where ν j ∈ R n corresponds to the coordinate shift of the space variable x from Q 1 to Q j , and Next we derive (3.10) by utilizing (3.11) and the condition of t ≥ T0 4 . At first, we illustrate that there exists a constant δ > 0 such that for (t, x) ∈ Q j , (3.12) To prove (3.12) for 1 ≤ j ≤ N 0 , it only suffices to consider the two extreme cases: ν j = 0 (corresponding to j = 1) and |ν j0 | = R − 1 + φ 3T0 8 (choosing j 0 such that |ν j0 | = max 1≤j≤N0 |ν j | = R − 1 + φ 3T0 8 . Note that |ν j0 | > R − 1 holds so that the domain Q can be covered by ∪ N0 j=1 Q j ). For ν j = 0, (3.12) is equivalent to We now illustrate that (3.13) is correct. By |x| ≤ φ(t) − φ T0 4 for (t, x) ∈ Q 1 , then in order to show (3.13) it suffices to prove This is equivalent to Obviously, this is easily achieved by t ≥ T0 4 and the smallness of δ. For ν j0 = R − 1 + φ 3T0 8 , the argument on (3.12) is a little involved. First, note that for fixed t > 0, the domain Q is symmetric with respect to the variable x, thus we can assume ν j0 = ν = R − 1 + φ 3T0 8 . In this case, (3.12) is equivalent to (3.14) If 1 − |ν j | < 0, then by |ν j | ≤ R − 1 + φ( 3T0 8 ) and the smallness of T 0 , the last line in (3.21) is bounded from below by while in the case of 1 − |ν j | ≥ 0, it follows from (3.21) that On the other hand, if 2φ(t) < R 2 − 1, then Therefore, , which derives (3.10).
4. Proof of Theorem 1.1. To establish the global existence, we set two cases.
4.1. The case p crit < p < p 0 = 9. By the local existence and regularity of weak solution u to (1.10) (see, e.g., [26] and the references therein), one has that u ∈ C ∞ ([0,T 2 ] × R) exists for the sameT as in (1.9), and u has compact support in the variable x. Furthermore, for any N ∈ N, Then we can take u T 2 , x , ∂ t u T Now we use standard Picard iteration to prove Theorem 1.1. Let u −1 ≡ 0, and for k ∈ N, let u k be the solution to the equation By (2.2), (3.8) and (3.9), for p > p crit , we have to pick a number γ satisfying γ < min 1 p (p + 1) , .

DAOYIN HE, INGO WITT AND HUICHENG YIN
Let j = −1, assuming that M k ≤ 2M 0 ≤ 2C 0 ε, then in view of M −1 = 0, we conclude from (4.2) that This yields that, if we choose ε small enough such that Thus we have obtained the boundedness of the sequence {u k } in the space L q (R 1+1 + ) when the constantsT and ε > 0 are chosen to be sufficiently small. Similarly, we have which shows the existence of a function u ∈ L q (T 2 , ∞) × R with u k → u ∈ L q (T 2 , ∞) × R . Moreover, from the boundedness of the sequence {M k } and the computations above, one easily obtains Hence, u is a weak solution to (1.10).

4.2.
The case p ≥ p 0 = 9. In this case, Theorem 1.1 can be proven in a manner that is analogous to the proof of [12, Theorem 1.2]. We just give the sketch of the proof here.
The first step is to establish the Strichartz estimates for linear equations. To this end, we study the linear Cauchy problem Note that the solution u of (4.3) can be written as where v solves the homogeneous problem and w solves the inhomogeneous problem with zero initial data x) ∈ R 1+1 + , w(0, ·) = 0, ∂ t w(0, ·) = 0. If g ≡ 0 in (4.4), we intend to establish the Strichartz-type inequality v L q t L r x ≤ C f Ḣs (R n ) , where q ≥ 1 and r ≥ 1 are suitable constants related to s. One obtains by a scaling argument that those indices should satisfy Setting r = q and s = 1 3 in (2.5), we find that q = q 0 ≡ 10 > 2.
Next we treat the inhomogeneous problem (4.5). Based on Lemmas 4.1, we establish the following estimate: where γ = 1 2 − 5 3q , q 0 ≤ q < ∞, and the constant C > 0 only depends on q. Proof. As in the proof of Lemma 3.4 in [12], we can write where a(t, τ, ξ) satisfies To treat (AF )(t, x) conveniently, we introduce the more general operator where 0 ≤ α < 1 2 is a parameter. As in the proof of Lemma 4.1, we shall use the Littlewood-Paley argument with a bump function β. Define the operator Note that our aim is to establish the following inequality for γ = 1 2 − 5 3q and q 0 ≤ q < ∞, which is equivalent to prove In terms of the definition of operator A α in (4.9) with α = γ − 1 3 , in order to complete the proof of (4.6), it suffices to establish (4.10) Note that p 0 < 2 < q < ∞. To derive (4.10), it follows from the proof of Lemma 3.1 in [12] that we only need to prove (4.11) By interpolation, it suffices to prove that (4.11) holds for the special cases q = q 0 and q = ∞. Denote the corresponding indices α by α 0 and α 1 . A direct computation yields α 0 = 1 2 − 5 3q0 − 1 3 = 0 and α 1 = 1 6 . The treatment for q = q 0 is exactly the same as in [12], we only need to handle the case q = ∞.
Relying on Lemmas 4.1 and 4.2, we have: Lemma 4.3. Let w solve (4.5). Then , where γ = 1 2 − 5 3q , q 0 ≤ q < ∞, and the constant C > 0 only depends on m, n, and q. Based on these estimates above, we are able to establish the global existence result for p ≥ p conf , the procedure is similar to that of Part 1 in [12,Section 4], and we obtain Proposition 4.4 (Global existence for p > p conf = 9). Assume that 9 ≤ p < ∞. Then there exists a constant ε 0 > 0 such that problem (1.1) admits a global weak solution u ∈ L r (R 1+1 + ) whenever u 0 H s + u 1 H s− 2 3 ≤ ε 0 , where s = 1 2 − 4 3(p−1) and r = 5 4 (p − 1). Combining both cases in Subsection 4.1 and Subsection 4.2, we have completed the proof of Theorem 1.1.