On semilinear Tricomi equations with critical exponents or in two space dimensions

TL;DR

This paper investigates semilinear Tricomi equations, focusing on critical exponents and two-dimensional cases, extending previous research to deepen understanding of solution behaviors in these settings.

Contribution

It provides new insights into the behavior of solutions to semilinear Tricomi equations with critical exponents, especially in two-dimensional space.

Findings

Analysis of solution existence and blow-up phenomena

Identification of critical exponent thresholds

Extension of previous results to two-dimensional cases

Abstract

This paper is a complement of our recent works on the semilinear Tricomi equations in [8] and[9].

Equations

$$\{\begin{array}{rl}& {\partial }_t^{2}u-t\Delta u=|u{|}^p,\\ & u(0,\cdot )=f(x),{\partial }_{t}u(0,\cdot )=g(x),\end{array}$$

$$(3n-2)p^2-3np-6=0,$$

$$u\in C^{1}\big{(}[0,\infty),H^{1}(\mathbb{R}^{n})\big{)}\cap C\big{(}[0,\infty),L^{2}(\mathbb{R}^{n})\big{)}.$$

$$\sum _{|\alpha |\le 1}(\Vert Z^{\alpha }f{\Vert }_{{\dot{H}}^s({\mathbb{R}}^2)}+\Vert Z^{\alpha }g{\Vert }_{{\dot{H}}^{s-\frac{2}{3}}({\mathbb{R}}^2)})<\epsilon ,$$

$$u\in L^{q}_{t}L^{p}_{r}L_{\theta}^{2}(\mathbb{R}_{+}^{1+2}),\quad\text{i.e., $\biggl{(}\int_{0}^{\infty}\bigl{(}\int_{0}^{\infty}\bigl{(}\int_{0}^{2\pi}|u(t,rcos\theta,rsin\theta)|^{2}d\theta\bigr{)}^{\frac{p}{2}}dr\bigr{)}^{\frac{q}{p}}dt\biggr{)}^{\frac{1}{q}}<\infty$},$$

$$G(t)\ge$$

$$G^{\prime \prime }(t)\ge$$

$$G^{\prime \prime }(t)={\int }_{{\mathbb{R}}^n}|u(t,x){|}^p\mathrm{d}x\ge C(M+t{)}^{-\frac{3}{2}n(p-1)}|G(t){|}^p.$$

$$\bar{u}=\frac{1}{{\omega }_n}{\int }_{{\mathbb{S}}^{n-1}}u(t,r\theta )\mathrm{d}\theta$$

$${\partial }_t^2\bar{u}-t\overline{\Delta u}=\overline{|u{|}^p}.$$

$$|\bar{u}|=\bigg{|}\frac{1}{\omega_{n}}\int_{\mathbb{S}^{n-1}}u(t,r\theta)\mathrm{d}\theta\bigg{|}\leq\bigg{(}\frac{1}{\omega_{n}}\int_{\mathbb{S}^{n-1}}|u|^{p}\mathrm{d}\theta\bigg{)}^{\frac{1}{p}}\bigg{(}\int_{\mathbb{S}^{n-1}}\frac{\mathrm{d}\theta}{\omega_{n}}\bigg{)}^{\frac{1}{p^{\prime}}}\leq\big{(}\overline{|u|^{p}}\big{)}^{\frac{1}{p}}.$$

$${\partial }_t^2\bar{u}-t\Delta \bar{u}\ge |\bar{u}{|}^p.$$

$$\mathbf{R}(u)(t,\rho )={\int }_{x\cdot \omega =\rho }u(t,x)\mathrm{d}{S}_x,$$

$$\begin{array}{rl}\mathbf{R}(u)(t,\rho )& ={\int }_{\{x^{\prime }:x^{\prime }\cdot \omega =0\}}u(t,\rho \omega +x^{\prime })\mathrm{d}{S}_x\\ & =c_n{\int }_{|\rho |}^{\infty }u(t,r)(r^2-{\rho }^2{)}^{\frac{n-3}{2}}r\mathrm{d}r.\end{array}$$

$$\mathbf{R}(\Delta u)(t,\rho )={\partial }_{\rho }^2\mathbf{R}(u)(t,\rho ).$$

$$\{\begin{array}{rl}& {\partial }_t^2\mathbf{R}(u)-t{\partial }_{\rho }^2\mathbf{R}(u)=\mathbf{R}(|u{|}^p),(t,\rho )\in {\mathbb{R}}_{+}^{1+1},\\ & \mathbf{R}(u)(0,\rho )=\mathbf{R}(f),{\partial }_t\mathbf{R}(u)(0,\rho )=\mathbf{R}(g).\end{array}$$

$$\begin{split}\mathbf{R}(u)&(t,\rho)=C\int_{0}^{1}v_{\mathbf{R}(f)}\big{(}\phi(t)s,\rho\big{)}(1-s^{2})^{-\frac{5}{6}}\mathrm{d}s+Ct\int_{0}^{1}v_{\mathbf{R}(g)}\big{(}\phi(t)s,\rho\big{)}(1-s^{2})^{-\frac{1}{6}}\mathrm{d}s\\ &+C\int_{0}^{t}\int_{\rho-\phi(t)+\phi(s)}^{\rho+\phi(t)-\phi(s)}\big{(}\phi(t)+\phi(s)+\rho-\rho_{1}\big{)}^{-\gamma}\big{(}\phi(t)+\phi(s)-\rho+\rho_{1}\big{)}^{-\gamma}F\big{(}\frac{1}{6},\frac{1}{6},1,z\big{)}\\ &\qquad\qquad\qquad\qquad\quad\times\mathbf{R}(|u|^{p})(s,\rho_{1})\mathrm{d}\rho_{1}\mathrm{d}s,\end{split}$$

$$\{\begin{array}{rl}& {\partial }_t^{2}v-{\partial }_x^{2}v=0,(t,x)\in {\mathbb{R}}_{+}^{1+1},\\ & v(0,x)=\phi ,{\partial }_{t}v(0,x)=0.\end{array}$$

$$\mathbf{R}(u)(t,\rho )$$

$$\displaystyle\qquad\qquad\qquad\qquad\quad\times F\big{(}\frac{1}{6},\frac{1}{6},1,z\big{)}\mathbf{R}(|u|^{p})(s,\rho_{1})\mathrm{d}\rho_{1}\mathrm{d}s.$$

$$z=\frac{(\varphi (t)-\varphi (s){)}^2-(\rho -{\rho }_1{)}^2}{(\varphi (t)+\varphi (s){)}^2-(\rho -{\rho }_1{)}^2}\in [0,1].$$

$$\begin{array}{rl}F(\frac{1}{6},\frac{1}{6},1,z)=& \frac{1}{\Gamma (\frac{1}{6})\Gamma (\frac{5}{6})}{\int }_0^1{t}^{-\frac{5}{6}}(1-t{)}^{-\frac{1}{6}}(1-zt{)}^{-\frac{1}{6}}dt\\ \ge & \frac{1}{\Gamma (\frac{1}{6})\Gamma (\frac{5}{6})}{\int }_0^1{t}^{-\frac{5}{6}}(1-t{)}^{-\frac{1}{6}}dt\\ =& \frac{1}{\Gamma (\frac{1}{6})\Gamma (\frac{5}{6})}B(\frac{1}{6},\frac{5}{6})\\ =& \frac{1}{\Gamma (1)}=1.\end{array}$$

$$\mathbf{R}(u)(t,\rho)\geq C\int_{0}^{t}\int_{\rho-\phi(t)+\phi(s)}^{\rho+\phi(t)-\phi(s)}\big{(}(\phi(t)+\phi(s))^{2}-(\rho-\rho_{1})^{2}\big{)}^{-\frac{1}{6}}\mathbf{R}(|u|^{p})(s,\rho_{1})\mathrm{d}\rho_{1}\mathrm{d}s.$$

$$|{\rho }_1\omega +y|=\sqrt{|{\rho }_1{|}^2+y^2}\ge |{\rho }_1|>\varphi (s)+M.$$

$$\mathbf{R}(|u{|}^p)(s,{\rho }_1)={\int }_{\{y:y\cdot \omega =0\}}u(s,{\rho }_1\omega +y)\mathrm{d}{S}_y=0.$$

$$0\le \varphi (s)\le \varphi (s_1)=:\frac{\varphi (t)-\rho -M}{2},$$

$$\rho+\phi(t)-\phi(s)\geq\phi(s)+M,\quad\rho-\phi(t)+\phi(s)\leq-\big{(}\phi(s)+M\big{)}.$$

$$\begin{split}\mathbf{R}(u)(t,\rho)&\geq C\int_{0}^{s_{1}}\int_{\rho-\phi(t)+\phi(s)}^{\rho+\phi(t)-\phi(s)}\big{(}(\phi(t)+\phi(s))^{2}-(\rho-\rho_{1})^{2}\big{)}^{-\frac{1}{6}}\mathbf{R}(|u|^{p})(s,\rho_{1})\mathrm{d}\rho_{1}\mathrm{d}s\\ &=C\int_{0}^{s_{1}}\int_{-\infty}^{\infty}\big{(}(\phi(t)+\phi(s))^{2}-(\rho-\rho_{1})^{2}\big{)}^{-\frac{1}{6}}\mathbf{R}(|u|^{p})(s,\rho_{1})\mathrm{d}\rho_{1}\mathrm{d}s.\end{split}$$

$$\varphi (t)+\varphi (s)+\rho -{\rho }_1$$

$$\varphi (t)+\varphi (s)-\rho +{\rho }_1$$

$$\mathbf{R}(u)(t,\rho)\geq C\int_{0}^{s_{1}}\int_{-\infty}^{\infty}\big{(}\phi(t)-\rho\big{)}^{-\frac{1}{6}}\phi(t)^{-\frac{1}{6}}\mathbf{R}(|u|^{p})(s,\rho_{1})\mathrm{d}\rho_{1}\mathrm{d}s\\ \begin{aligned} &=\big{(}\phi(t)-\rho\big{)}^{-\frac{1}{6}}\phi(t)^{-\frac{1}{6}}\int_{0}^{s_{1}}\int_{-\infty}^{\infty}\int_{\{y^{\prime}:y^{\prime}\cdot\omega=0\}}u(s,\rho_{1}\omega+y^{\prime})\mathrm{d}S_{y^{\prime}}\mathrm{d}\rho_{1}\mathrm{d}s\\ &=\big{(}\phi(t)-\rho\big{)}^{-\frac{1}{6}}\phi(t)^{-\frac{1}{6}}\int_{0}^{s_{1}}\int_{\mathbb{R}^{n}}|u(s,y)|^{p}\mathrm{d}y\mathrm{d}s.\end{aligned}$$

$${\int }_{{\mathbb{R}}^n}|u(s,y){|}^p\mathrm{d}y\ge C{s}^{\frac{p}{2}}{\left(M+\varphi (s)\right)}^{n-1-\frac{n}{2}p}.$$

Full text

Keywords: Tricomi equation, critical exponent, blowup, global existence, Strichartz estimate.

Mathematical Subject Classification 2000: 35L70, 35L65, 35L67

1 Introduction

In this paper, we continue to be concerned with the global existence or blowup of solutions $u$ to the semilinear Tricomi equation

[TABLE]

where $t\geq 0$, $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ ($n\geq 2$), $p>1$, and $u_{i}\in C_{0}^{\infty}(B(0,M))$ $(i=0,1)$ with $B(0,M)=\{x:|x|=\sqrt{x_{1}^{2}+...+x_{n}^{2}}<M\}$ and $M>1$. For the local well-posedness and optimal regularities of solution $u$ to problem (1.1), the readers may consult [19, 18, 20, 21, 29] and the references therein. In [8]-[9], we have determined a critical exponent $p_{crit}(n)$ and a conformal exponent $p_{conf}(n)$ (>p_{crit}(n)\big{)} for (1.1) as follows (corresponding to the case of $m=1$ in the generalized equation Tricomi equation $\partial_{t}^{2}u-t^{m}\Delta u=|u|^{p}$): $p_{crit}(n)$ is the positive root of the algebraic equation

[TABLE]

and $p_{conf}(n)=\frac{3n+6}{3n-2}$. It is shown in [8] that for all $n\geq 2$, the solution $u$ of (1.1) generally blows up in finite time when $1<p<p_{crit}(n)$, and meanwhile $u$ exists globally when $p\geq p_{conf}(n)$ for small initial data and $n\geq 2$. In [9], we prove that the small data solution $u$ of (1.1) exists globally when $n\geq 3$ and $p_{crit}(n)<p<p_{conf}(n)$. Therefore, collecting the results in [8]-[9], we have given a detailed study on the blowup or global existence of small data solution $u$ to problem (1.1) for $n\geq 3$ except $p=p_{crit}(n)$, and for $n=2$ with $p\geq p_{conf}(n)$ except $p_{crit}(n)<p<p_{conf}(n)$. In this paper, firstly, we establish the finite time blowup result for problem (1.1) when $n\geq 2$ and $p=p_{crit}(n)$, secondly, we prove the global existence of small data solution $u$ to problem (1.1) when $n=2$ and $p_{crit}(n)<p<p_{conf}(n)$.

Theorem 1.1**.**

Let $n\geq 2$ and $p=p_{crit}(n)$. Suppose that the initial data $f,g\in C^{\infty}_{0}(\mathbb{R}^{n})$ are non-negative and positive somewhere, then problem (1.1) admits no global solution $u$ with

[TABLE]

Theorem 1.2**.**

Let $n=2$. For $p_{crit}(n)<p\leq p_{conf}(n)$, suppose that the initial data $(f,g)$ satisfy

[TABLE]

where $\varepsilon>0$ is a sufficiently small constant, $s=1-\frac{4}{3(p-1)}$, and $Z=\{\partial_{1},\partial_{2},x_{1}\partial_{2}-x_{2}\partial_{1}\}$. Then problem (1.1) admits a global solution $u$ such that

[TABLE]

where $x_{1}=rcos\theta$ and $x_{2}=rsin\theta$ with $r\geq 0$ and $\theta\in[0,2\pi]$, $q=\frac{p(p-1)}{3-p}>2$ for $p_{crit}(2)<p\leq\frac{7}{3}$; $q=\frac{(3p+1)(p-11)}{11-3p}>2$ for $\frac{7}{3}<p\leq\frac{4+\sqrt{17}}{3}$; $q=\frac{p^{2}-1}{5-p}>2$ for $\frac{4+\sqrt{17}}{3}<p\leq p_{conf}(2)=3$.

Remark 1.1*.*

For the semilinear wave equation $\partial_{t}^{2}u-\Delta u=|u|^{p}$ $(p>1)$, the critical exponent $p_{0}(n)$ in Strauss’ conjecture (see [26]) is determined by the algebraic equation $(n-1)p_{0}^{2}(n)-(n+1)p_{0}(n)-2=0$ (so far the global existence of small data solution $u$ for $p>p_{0}(n)$ or the blowup of solution $u$ for $1<p<p_{0}(n)$ have been proved in [4]-[6], [12]-[13], [23] and the references therein). The finite time blowup for the critical wave equations $\partial_{t}^{2}u-\Delta u=|u|^{p_{0}(n)}$ has been established in [4],[12], [22], and [31]-[32], respectively. Motivated by the techniques in [31] and [8], we prove the blowup result for the critical semilinear Tricomi equation in (1.1).

Remark 1.2*.*

For brevity, we only study the semilinear Tricomi equation instead of the generalized semilinear Tricomi equation $\partial_{t}^{2}u-t^{m}\Delta u=|u|^{p}$ $(m\in\mathbb{N})$ in problem (1.1). In fact, by the methods in Theorem 1.1 and Theorem 1.2, one can establish the analogous results to Theorem 1.1-Theorem 1.2 for the generalized semilinear Tricomi equation.

Remark 1.3*.*

It follows from a direct computation that $p_{crit}(2)=\frac{3+\sqrt{33}}{4}$ and $p_{conf}(2)=3$ in Theorem 1.2.

For $n=1$, the linear equation $\partial_{t}^{2}u-t\partial_{x}^{2}u=0$ is the well-known Tricomi equation which arises in transonic gas dynamics. There are extensive results for both linear and semilinear Tricomi equations in $n$ space dimensions $(n\in\mathbb{N})$. For instances, with respect to the linear Tricomi equation, the authors in [1], [28] and [30] have computed its fundamental solution explicitly; with respect to the semilinear Tricomi equation $\partial_{t}^{2}u-t\Delta u=f(t,x,u)$, under some certain assumptions on the function $f(t,x,u)$, the authors in [7] and [14]-[17] have obtained a series of interesting results on the existence and uniqueness of solution $u$ in bounded domains under Tricomi, Goursat or Dirichlet boundary conditions respectively in the mixed type case, in the degenerate hyperbolic setting or in the degenerate elliptic setting; with respect to the Cauchy problem of semilinear Tricomi equations, the authors in [19, 18, 20, 21] established the local existence as well as the singularity structure of low regularity solutions in the degenerate hyperbolic region and the elliptic-hyperbolic mixed region, respectively. In addition, by establishing some classes of $L^{p}$-$L^{q}$ estimates for the solution $v$ of linear equation $\partial_{t}^{2}v-t\triangle v=F(t,x)$, the author in [29] obtained some results about the global existence or the blowup of solutions to problem (1.1) when the exponent $p$ belongs to a certain range, however, there was a gap between the global existence interval and the blowup interval. By establishing the Strichartz inequality and the weighted Strichartz inequality for the linear Tricomi equation, respectively, we have shown the global existence of small data solution $u$ to problem (1.1) for $p>p_{crit}(n)$ ($n\geq 3$) in [8]- [9].

We now comment on the proof of Theorem 1.1 and Theorem 1.2. To prove Theorem 1.1, we define the function $G(t)=\int_{\mathbb{R}^{n}}u(t,x)\ \mathrm{d}x$. By applying some crucial techniques for the modified Bessel function as in [11, 20], and motivated by [31] and [8], we can derive a Riccati-type ordinary differential inequality for $G(t)$ through a delicate analysis of (1.1), which is stronger than the ordinary differential inequality in [8] (see (2.1) of [8]). From this and Lemma 2.1 in [31], the blowup result for $p=p_{crit}(n)$ in Theorem 1.1 is established under the positivity assumptions of $f$ and $g$. To prove the global existence result in Theorem 1.2, we require to establish angular Strichartz estimates for the Tricomi operator $\partial_{t}^{2}-t\Delta$ as in the treatment on the 2-D linear wave operator in [24]. In this process, a series of inequalities are derived by applying an explicit formula for the solutions of linear Tricomi equations and by utilizing some basic properties of related Fourier integral operators and some classical results in harmonic analysis. Based on the resulting Strichartz inequalities and the contractible mapping principle, we eventually complete the proof of Theorem 1.2. Here we point out that compared with the techniques of [24] for deriving the Strichartz inequality with angular mixed-norm of 2-D linear wave equation, due to the influences of degeneracy and variable coefficients in the linear equation, it is more involved and complicated to give the related analysis on the resulting Fourier integral operator from linear Tricomi equation.

This paper is organized as follows: In Section 2, we complete the proof of Theorem 1.1. In Section 3, some Strichartz estimates with angular mixed norms for the linear Tricomi equation are established. In Section 4, by applying the results in Section 3, Theorem 1.2 is proved.

2 Proof of Theorem 1.1

Before starting the proof of Theorem 1.1, we cite a blowup lemma from [31].

Lemma 2.1**.**

Let $p>1$, $a\geq 1$, and $(p-1)a=q-2$. Suppose $G\in C^{2}[0,T)$ satisfies that for $t\geq T_{0}>0$,

[TABLE]

where $K_{0}$, $K_{1}$, $T_{0}$ and $M$ are some positive constants. Fixing $K_{1}$, there exists a positive constant $c_{0}$, independent of $M$ and $T_{0}$ such that if $K_{0}\geq c_{0}$, then $T<\infty$.

With this lemma and $G(t)=\int_{\mathbb{R}^{n}}u(t,x)\ \mathrm{d}x$, our subsequent tasks are to derive (2.1) and (2.2) for the solution $u$ of problem (1.1). It follows from Section 2 of [8] that

[TABLE]

This means that (2.2) holds for $q=\frac{3}{2}\,n(p-1)$. Next, strongly motivated by the techniques in [31] and [8], we focus on the derivation of (2.1), which is divided into the following three steps:

Step 1. Some reductions

Let

[TABLE]

be the spherical average of $u$. Then applying the spherical average on both sides of (1.1) yields

[TABLE]

By Daboux’s identity, one has $\overline{\Delta u}=\Delta\bar{u}$. On the other hand, it follows from Hölder’s inequality that

[TABLE]

This, together with (2.4), yields

[TABLE]

Thus we can assume that $u$ is radial since the blowup of $\bar{u}$ obviously yields the blowup of $u$. Let $\omega\in\mathbb{R}^{n}$ be a unit vector. The Radon transform of $u$ with respect to the variable $x$ is defined as

[TABLE]

where $\rho\in\mathbb{R}$, $\mathrm{d}S_{x}$ is the Lebesque measure on the hyper-plane $\{x:x\cdot\omega=\rho\}$. From (2.6) and the radial assumption of $u(t,\cdot)$, it is easy to see

[TABLE]

Obviously, $\mathbf{R}(u)(t,\rho)$ is independent of $\omega$.

Step 2. The lower bound of $\mathbf{R}(u)$

From Page 3 of [10], we have

[TABLE]

Since $u$ is a solution of (1.1), it follows from (2.8) that $\mathbf{R}(u)$ solves

[TABLE]

By Lemma 2.1 in [30] and Theorem 3.1 in [28], we have

[TABLE]

where $C>0$ is a constant, $z=\frac{(\rho-\rho_{1}+\phi(t)-\phi(s))(\rho-\rho_{1}-\phi(t)+\phi(s))}{(\rho-\rho_{1}+\phi(t)+\phi(s))(\rho-\rho_{1}-\phi(t)-\phi(s))}$ with $\phi(t)=\frac{2}{3}t^{\frac{3}{2}}$, F\big{(}\frac{1}{6},\frac{1}{6},1,z\big{)} is the hypergeometric function, and the function $v_{\varphi}$ solves the 1-D wave equation

[TABLE]

Note that (2.7) together with the non-negativity of $f$ and $g$ shows $\mathbf{R}(f)\geq 0$ and $\mathbf{R}(g)\geq 0$. In addition, by D’Alembert’s formula, we obtain $v_{\mathbf{R}(f)}\geq 0$ and $v_{\mathbf{R}(g)}\geq 0$. Hence,

[TABLE]

Note that

[TABLE]

Then by page 59 of [3], we arrive at

[TABLE]

Therefore,

[TABLE]

Notice that the support of $u(s,\cdot)$ is contained in the ball B\big{(}0,M+\phi(s)\big{)}=:\{x\in\mathbb{R}^{n}:|x|\leq M+\phi(s)\}. On the other hand, if $|\rho_{1}|>M+\phi(s)$, then for any vector $y\in\mathbb{R}^{n}$ which is perpendicular to $\omega$, one has

[TABLE]

This yields that for $|\rho_{1}|>M+\phi(s)$,

[TABLE]

Thus, \operatorname{supp}\mathbf{R}(|u|^{p})(s,\cdot)\subseteq B\big{(}0,M+\phi(s)\big{)} holds. From now on, we can assume $\rho\geq 0$. If

[TABLE]

then

[TABLE]

From this, we arrive at

[TABLE]

By (2.11), one has

[TABLE]

Together with this and (2.12), we deduce

[TABLE]

On the other hand, by (2.17) of [8], one has

[TABLE]

Substituting (2.14) into (2.13) yields

[TABLE]

To guarantee that the integral in (2.15) is convergent, we shall need

[TABLE]

This is achieved by $p=p_{\textup{crit}}(n)<p_{\textup{conf}}(n)=\frac{3n+6}{3n-2}$ and direct computation. Thus we conclude that for $n\geq 2$,

[TABLE]

Step 3. The lower bound of $\int_{\mathbb{R}^{n}}|u(t,x)|^{p}\mathrm{d}x$

Following (2.16) of [31], one can introduce the transformation

[TABLE]

and derive

[TABLE]

In fact, if $n\geq 3$, then it is easy to see that

[TABLE]

where $M(|f|)$ is the maximal function of $f$. Hence there exists a constant $C>0$ such that (2.17) holds.

For $n=2$, at first we prove that $\mathbf{T}$ maps $L^{\infty}$ to $L^{\infty}$ and $L^{1}$ to $L^{1,\infty}$ (weak $L^{1}$ space), respectively. If so, by the Marcinkiewicz interpolation theorem, then (2.17) holds for $n=2$.

In fact, it follows from a direct computation that for $\rho>0$,

[TABLE]

which yields the $L^{\infty}-L^{\infty}$ estimate of operator $\mathbf{T}$. Next we derive the $L^{1}-L^{1,\infty}$ estimate of $\mathbf{T}$. Suppose $f\in L^{1}([0,\phi(t)+M])$. Let

[TABLE]

Denote d_{\varphi}(\alpha)=\big{|}\{0\leq\rho\leq\phi(t)+M:\varphi(\rho)>\alpha\}\big{|} as the distribution function of $\varphi$. It is known that for $0<\alpha<\infty$ and measurable functions $f_{1}$, $f_{2}$

[TABLE]

Note that

[TABLE]

In addition,

[TABLE]

Since $\frac{1}{|r|^{\frac{1}{2}}}\in L^{2,\infty}([0,\phi(t)+M])$ and $f\in L^{1}([0,\phi(t)+M])$, by Young’s inequality, we have $h\in L^{2,\infty}([0,\phi(t)+M])$. Therefore,

[TABLE]

which means $\mathbf{T}(f)(\rho)=g(\rho)h(\rho)\in L^{1,\infty}([0,\phi(t)+M])$. Then an application of Marcinkiewicz interpolation theorem yields

[TABLE]

where $C_{0}>0$ is a uniform constant independent of $t$. Due to $\operatorname{supp}u(t,\cdot)\subseteq[0,\phi(t)+M]$, the inequality (2.18) is enough for the application in the proof of Theorem 1.1.

Applying (2.17) or (2.18) to the function

[TABLE]

we have

[TABLE]

When $r\geq\rho$, we arrive at

[TABLE]

Next we only treat the case of $1<p\leq 2$ since the treatment for $p>2$ is completely similar. When $1<p\leq 2$, it follows from (2.19) that

[TABLE]

On the other hand,

[TABLE]

Substituting (2.22) into (2.21) yields

[TABLE]

By the bound of $\mathbf{R}(u)$ in (2.16), we deduce

[TABLE]

If $\rho\in(0,\phi(t)-M-1)$, then there exists a constant $C_{M}>0$ such that for all $\phi(t)>2(M+1)$,

[TABLE]

This observation together with (2.23) yields

[TABLE]

Note that for $p=p_{crit}(n)$,

[TABLE]

Thus we have from (2.24) that for $p=p_{crit}(n)$,

[TABLE]

Note that the term \ln\big{(}\phi(t)-M+1\big{)} can be sufficiently large when $t$ is large, and if the power of $t$ in the right hand side of (2) satisfies

[TABLE]

then there is a large constant $K_{0}>0$ such that for large $t>0$ and $p=p_{crit}(n)$,

[TABLE]

and

[TABLE]

Next we turn to verify (2.26). By the condition

[TABLE]

direct computation yields

[TABLE]

If $n\geq 3$, then

[TABLE]

If $n=2$, then

[TABLE]

Hence (2.26) is valid for all $n\geq 2$. By (2.27) and (2.3), choosing a=\frac{p}{2}+\frac{3}{2}\big{(}n-1-\frac{np}{2}\big{)}+2 and $q=\frac{3n}{2}(p-1)$ with $p=p_{crit}(n)$ in Lemma 2.1, then all the assumptions of Lemma 2.1 hold. Therefore, Theorem 1.1 is shown by Lemma 2.1.

3 Strichartz estimates in angular mixed norm spaces

Before establishing Strichartz estimates for the linear Tricomi operator, we recall two important results. The first one is a minor variant of [13, Lemma 3.8], and the second one comes from [2, Theorem 1.2].

Lemma 3.1**.**

Let $\displaystyle\beta\in C_{0}^{\infty}\left((1/2,2)\right)$ and $\displaystyle\sum\limits_{j=-\infty}^{\infty}\beta\left(2^{-j}\tau\right)\equiv 1$ for $\tau>0$. Define the Littlewood-Paley operators as

[TABLE]

Then

[TABLE]

Lemma 3.2**.**

Suppose that $1\leq p<q\leq\infty$. Let $T:L^{p}(\mathbb{R})\rightarrow L^{q}(\mathbb{R})$ be a bounded linear operator which is defined by

[TABLE]

where $K(x,y)$ is locally integrable. Define

[TABLE]

Then

[TABLE]

To prove Theorem 1.2, we shall require to get certain Strichartz estimates in $\mathbb{R}_{+}^{1+2}$ for 2-D linear Tricomi operator. For this purpose, we study the following linear Cauchy problem

[TABLE]

Note that the solution $u$ of (3.1) can be written as

[TABLE]

where $v$ solves the homogeneous problem

[TABLE]

and $w$ solves the inhomogeneous problem with zero initial data

[TABLE]

Let $\dot{H}^{s}(\mathbb{R}^{2})$ denote the homogeneous Sobolev space with norm

[TABLE]

where

[TABLE]

It follows from [29] that the solution $v$ of (3.2) can be expressed as

[TABLE]

where the symbols $V_{j}(t,\xi)$ ($j=1,2$) of the Fourier integral operators $V_{j}(t,D_{x})$ are

[TABLE]

and

[TABLE]

here $z=2i\phi(t)|\xi|$, $i=\sqrt{-1}$, and $H_{\pm}$ are smooth functions of the variable $z$. By [27], one knows that for $\beta\in\mathbb{N}_{0}^{n}$,

[TABLE]

We only estimate $V_{1}(t,D_{x})f(x)$ since the estimation on $V_{2}(t,D_{x})g(x)$ is similar. Indeed, up to a factor of $t\,\phi(t)^{-\frac{5}{6}}=C\phi(t)^{-\frac{1}{6}}$, the powers of $t$ appearing in $V_{1}(t,D_{x})f(x)$ or $V_{2}(t,D_{x})g(x)$ are the same.

Choose a cut-off function $\chi(s)\in C^{\infty}(\mathbb{R})$ with $\chi(s)=\left\{\enspace\begin{aligned} 1,\quad&s\geq 2\\ 0,\quad&s\leq 1\end{aligned}\right.$. Then

[TABLE]

By (3.4), (3.6) and (3.7), we derive that

[TABLE]

where $C>0$ is a generic constant, and for $\beta\in\mathbb{N}_{0}^{n}$,

[TABLE]

Next we analyze $v_{2}(t,x)$. It follows from [3] or [29] that

[TABLE]

where $\Phi$ is the confluent hypergeometric function which is analytic with respect to the variable $z=2i\phi(t)|\xi|$. Then

[TABLE]

Similarly, one has

[TABLE]

Thus we arrive at

[TABLE]

where, for $\beta\in\mathbb{N}_{0}^{n}$,

[TABLE]

Substituting (3.9) and (3.10) into (3.8) yields

[TABLE]

where $a_{l}$ $(l=1,2)$ satisfies

[TABLE]

Next we only treat the integral $\int_{\mathbb{R}^{n}}e^{i\left(x\cdot\xi-\phi(t)|\xi|\right)}a_{2}(t,\xi)\hat{f}(\xi)\,d\xi$ since the treatment of the integral $\int_{\mathbb{R}^{n}}e^{i\left(x\cdot\xi+\phi(t)|\xi|\right)}a_{1}(t,\xi)\hat{f}(\xi)\,d\xi$ is similar. Denote

[TABLE]

We will show that

[TABLE]

where $q\geq 2$ and $r\geq 2$ are some suitable constants related to $s$. One obtains by a scaling argument that those indices in (3.13) should satisfy

[TABLE]

On the other hand, by another scaling argument similar to Knapp’s counter example, we get the second restriction on the indices in (3.13)

[TABLE]

In fact, for small $\delta>0$, set $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2}\backslash\{0\}$ and denote

[TABLE]

Let $\hat{f}(\xi)=\chi_{D}(\xi)$ be the characteristic function of domain $D$. Note that on domain $D$ it holds

[TABLE]

By (3.12), one has

[TABLE]

Choose a domain $R$ in $\mathbb{R}_{+}\times\mathbb{R}^{2}$ as

[TABLE]

For $(t,x)\in R$ and $\xi\in D$, then the phase function in (3.16) is essentially equivalent to a constant and we have

[TABLE]

Therefore if we take $s=0$ in (3.13), then a direct computation yields

[TABLE]

Since $\delta>0$ is small, in order to get (3.13), we shall need

[TABLE]

which gives restriction (3.15). Now our task is to prove

Lemma 3.3**.**

Let operator $A$ be defined by (3.12). Assume that $(q,r)\neq(\infty,\infty)$,

[TABLE]

Then

[TABLE]

where $s=2(\frac{1}{2}-\frac{1}{r})-\frac{2}{3}\cdot\frac{1}{q}.$

Proof.

The main step in the proof of (3.17) is to show that

[TABLE]

Indeed, once (3.18) is proved, then by the support condition of $f$, we know that

[TABLE]

This together with Lemma 3.1 yields (3.17).

To prove (3.18), we follow some ideas of [24] and use the interpolation method. The first case is $q=\infty$ and $s=1-\frac{2}{r}$. Since Hardy-Littlewood-Sobolev estimate gives $\dot{H}^{1-\frac{2}{r}}(\mathbb{R}^{2})\subseteq L^{r}(\mathbb{R}^{2})$ for $2\leq r<\infty$, we clearly have

[TABLE]

It follows from (3.11) and (3.12) that if $\hat{f}(\xi)=0$ for $|\xi|\notin[\frac{1}{2},1]$,

[TABLE]

By interpolation, if we can conclude that for $2\leq q<\infty$,

[TABLE]

then (3.18) is immediately proved. Next we turn to the proof of (3.20). By the support condition for $\hat{f}$, we arrive at

[TABLE]

here $\xi=(\rho\cos\omega,\rho\sin\omega)$. Expanding the angular part of $\hat{f}$ by Fourier series yields that there are coefficients $c_{k}(\rho)$ $(k\in\mathbb{Z})$ vanishing for $\rho\notin[\frac{1}{2},1]$ such that

[TABLE]

This means

[TABLE]

By Plancherel’s theorem for $\mathbb{S}^{1}$ and $\mathbb{R}$, we have

[TABLE]

where $\hat{c}_{k}(s)$ is the one-dimensional Fourier transform of $c_{k}(\rho)$. Recall that (see [25], p.137)

[TABLE]

where $k\in\mathbb{Z}$, and $J_{k}$ is the $k$-th Bessel function defined by

[TABLE]

Choose a cut-off function $\beta\in C_{0}^{\infty}(\mathbb{R})$ such that

[TABLE]

Let $\alpha(t,\rho)=\rho\beta(\rho)a_{2}(t,\rho)$. Then by (3.24) and the support condition of $c_{k}$, we have

[TABLE]

where $\hat{\alpha}_{t}(\xi)$ stands for the Fourier transformation of $\alpha(t,\rho)$ with respect to the variable $\rho$. Direct computation yields that for any $r\geq 0$,

[TABLE]

To proceed further, we shall need a control of the integral in (3) with respect to the variable $\theta$, which is similar to Lemma 2.1 in [24].

Lemma 3.4**.**

Let $\hat{\alpha}(t,\xi)$ be defined as above and a number $N\in\mathbb{N}$ be fixed. Then there is a uniform constant $C>0$, which is independent of the variables $b\in\mathbb{R}$ and $r\geq 0$, so that the following inequalities hold:

[TABLE]

[TABLE]

Proof of Lemma 3.4..

By the definition of function $\hat{\alpha}_{t}$, we only need to study the integral

[TABLE]

Case I. $\mathbf{|b|\geq 2r}$ In this case, we have

[TABLE]

Since $\rho\beta(\rho)\in\mathcal{S}(\mathbb{R})$ and $a_{2}$ satisfies (3.11), direct computation yields

[TABLE]

which just corresponds to (3.26).

Case II. $\mathbf{0\leq r\leq 1}$

For $|b|>2$, it is reduced to Case I. For $|b|\leq 2$, by a direct computation, we have

[TABLE]

Case III. $\mathbf{r>1}$ and $\mathbf{|b|\leq 2r}$ In this case, we intend to prove that

[TABLE]

and

[TABLE]

To show (3.28), it only suffices to estimate the first integral in (3.28). Let $u=1-\cos\theta$, we then have

[TABLE]

We further set $\bar{u}=ru$, then the last integral in (3) can be controlled by

[TABLE]

If \big{|}r-|b|\big{|}\geq 2, then

[TABLE]

For $II_{1}$, we can repeat the analysis in Case I and integrate by parts to get

[TABLE]

For $II_{2}$, integrating by parts yields

[TABLE]

If \big{|}r-|b|\big{|}\leq 2, then by similar computation,

[TABLE]

Thus it follows from (3)-(3) that the proof of (3.28) is finished.

To show (3.29), we set $u=r\cos\theta$. Then

[TABLE]

Collecting all the analysis above in Case I-Case III yields the proof of Lemma 3.4. ∎

By Lemma 3.4, we have

Claim**.**

For $\delta>0$, there is a constant $C_{\delta}>0$, which is independent of $t\in\mathbb{R}_{+}$ and $r\geq 0$, such that

[TABLE]

proof of claim.

If $0\leq r\leq 1$, or $|\phi(t)-s|\geq 2r$, then for $\delta>0$, it is easy to see that (3.26) yields the expected estimate (3.36).

If $|\phi(t)-s|\leq 2r$ and $r>1$, then by (3.27), one has

[TABLE]

Next we treat the integral in (3). By $r>1$, we have

[TABLE]

For the second part of the integral in (3), if $|\phi(t)-s|\leq 1$ and $r>1$, then

[TABLE]

If $|\phi(t)-s|>1$ and $r>1$, we then set $\eta=|\phi(t)-s|$ and derive that for $0<\delta<1/2$, the integral in (3) can be controlled by

[TABLE]

Note that

[TABLE]

and

[TABLE]

Then collecting (3), (3.39) and (3) yields

[TABLE]

Hence the claim is proved. ∎

It follows from Claim, (3) and Hölder’s inequality that

[TABLE]

For any $q\geq 2$, we can choose a constant $\delta>0$ such that

[TABLE]

Then by Minkowski’s inequality, we have that for $q\geq 2$,

[TABLE]

To handle (3.42), we require to compute

[TABLE]

If $s\leq 0$, then by (3.41), we arrive at

[TABLE]

If $s>0$, we then write $s=\phi(\bar{s})$ and conclude

[TABLE]

By (3.41), in order to estimate (3), we only need to compute the following integral for $\bar{s}>0$,

[TABLE]

A direct computation yields

[TABLE]

Combining (3.42) with (3.43)-(3), we conclude that for $\hat{f}(\xi)=0$ when $|\xi|\notin[\frac{1}{2},1]$,

[TABLE]

Thus we have proved (3.20) and futher (3.17). Namely, Lemma 3.3 is proved. ∎

Next we turn to estimate the solution $w$ of problem (3.3). Note that $w$ can be written as

[TABLE]

To estimate $w(t,x)$, it suffices to treat the term $\int_{0}^{t}V_{2}(t,D_{x})V_{1}(\tau,D_{x})F(\tau,x)d\tau$ since the treatment on the term $\int_{0}^{t}V_{1}(t,D_{x})V_{2}(\tau,D_{x})F(\tau,x)d\tau$ is completely analogous.

If we repeat the reduction of (3.23)-(3.24) in [8], we then have

[TABLE]

where the amplitude function $a$ satisfies

[TABLE]

By a dual argument similar to the proof of Lemma 3.4 in [8], we can prove that if $\hat{F}(\tau,\xi)=0$ when $|\xi|\notin[\frac{1}{2},1]$, then

[TABLE]

where

[TABLE]

and

[TABLE]

Then an application of Lemma 3.2 yields that for $\hat{F}(\tau,\xi)=0$ when $|\xi|\notin[\frac{1}{2},1]$,

[TABLE]

Utilizing Lemma 3.1 to remove the restriction on the support of $\hat{F}$ in (3.49), we then get the following estimate for problem (3.3).

Lemma 3.5**.**

Let $w$ be the solution of (3.3). If $q,r,\tilde{q},\tilde{r}\geq 2$ and satisfy (3.47)-(3.48), then

[TABLE]

4 Proof of Theorem 1.2

First we consider the linear problem (3.1). Recall the definition of vector fields $\{Z\}$ in Theorem 1.2, then by Lemma 3.3 and energy estimates, we have

[TABLE]

where $q,r,\tilde{q},\tilde{r}\geq 2$ satisfy (3.48) and

[TABLE]

Note that the nonlinear term in (1.1) is $|u|^{p}$, we then have $q=p\tilde{q}^{\prime}$ and $r=p\tilde{r}^{\prime}$. This together with condition (4.2) yields

[TABLE]

Meanwhile the conditions on $r$ and $q$ become

[TABLE]

Case I. Choosing $r=p$ in (4.3) and (4.4)

In this case, it follows from (4.3) that $q=\frac{p(p-1)}{3-p}$. Then

[TABLE]

From Lemma 3.1 we require $q\geq 2$, which leads to

[TABLE]

This condition is fulfilled by $p>p_{\textup{crit}}(2)$ and Remark 1.3. Furthermore, we have

[TABLE]

On the other hand, by Lemma 3.1 we also require $1\leq\tilde{q}^{\prime}\leq 2$ and $1\leq\tilde{r}^{\prime}\leq 2$, which is equivalent to $2\leq p\leq 7/3$. In addition, one needs

[TABLE]

which holds by $\tilde{q}^{\prime}\geq 1$. Collecting all these observations above, we intend to prove the global existence of $u$ to problem (1.1) by an iteration argument in the range

[TABLE]

provided the initial data are small. More specifically, let $u_{0}$ solve the Cauchy problem (3.2), we then define $u_{k}$ ($k\geq 1$) by solving

[TABLE]

The first step is to show that if

[TABLE]

and $\varepsilon>0$ is small enough, then

[TABLE]

is uniformly small, where $q=\frac{p(p-1)}{3-p}$ and $s=1-\frac{4}{3(p-1)}$.

For $k=0$, it follows from Lemma 3.3 and the energy estimate that

[TABLE]

For $k\geq 1$, (3.1) yields

[TABLE]

To control the right hand side of (4.7), we note that for a function $v(x)=v(|x|,\theta)$ $(x\in\mathbb{R}^{2})$ with $\sum_{|\alpha|\leq 1}|Z^{\alpha}v|\in L^{2}_{\theta}$,

[TABLE]

Since $\partial_{\theta}=x_{1}\partial_{2}-x_{2}\partial_{1}\in\{Z\}$, we have

[TABLE]

which derives

[TABLE]

Thus we have

[TABLE]

If $M_{k-1}\leq 2C_{0}\varepsilon$, then for small $\varepsilon>0$,

[TABLE]

Define

[TABLE]

Then by (4.9) and direct computation similar to (4), we get that for small $\varepsilon>0$,

[TABLE]

This means that there exists a function $u\in L^{q}_{t}L^{r}_{|x|}L_{\theta}^{2}$ such that $u_{k}\to u$ in $L^{q}_{t}L^{r}_{|x|}L_{\theta}^{2}$. In addition,

[TABLE]

which means $|u_{k}|^{p}\to|u|^{p}$ in $L^{1}_{\textup{loc}}(\mathbb{R}_{+}\times\mathbb{R}^{2})$ and hence in the sense of distribution. Therefore $u$ is a global weak solution of (1.1) and the proof of Theorem 1.2 is completed for $p_{\textup{crit}}(2)<p\leq\frac{7}{3}$.

Case II. Choosing $r=p+\frac{1}{3}$ in (4.3) and (4.4)

In this case, it follows from (4.3) that $q=\frac{(3p+1)(p-11)}{11-3p}$. Then $q\geq 2$ is equivalent to

[TABLE]

which leads to $p\geq\frac{\sqrt{73}-2}{3}$. Note that

[TABLE]

In addition,

[TABLE]

and

[TABLE]

On the other hand,

[TABLE]

which derives

[TABLE]

Note that the following condition in Lemma 3.1 is also required

[TABLE]

Substituting

[TABLE]

and

[TABLE]

into (4.10) yields

[TABLE]

Since $\frac{13+\sqrt{193}}{12}\approx 2.141$ and $\frac{4+\sqrt{17}}{3}\approx 2.708$, we obtain that the admissible range for $p$ in Case II is

[TABLE]

Hence, we can use (4.1) and repeat the computation from (4) to (4.9) to get a global weak solution $u\in L^{q}_{t}L^{p+1/3}_{r}L_{\theta}^{2}(\mathbb{R}_{+}^{1+2})$ of problem (1.1), where $q=\frac{(3p+1)(p-11)}{11-3p}$ and $\frac{7+\sqrt{409}}{12}\leq p\leq\frac{4+\sqrt{17}}{3}$.

Case III. Choosing $r=p+1$ and $q=\frac{p^{2}-1}{5-p}$ in (4.3) and (4.4)

In this case, by (4.3) we get $q=\frac{p^{2}-1}{5-p}$. Then $q\geq 2$ is equivalent to

[TABLE]

which derives

[TABLE]

In addition,

[TABLE]

and

[TABLE]

On the other hand,

[TABLE]

which leads to

[TABLE]

Note that the following condition in Lemma 3.1 is also required

[TABLE]

This means

[TABLE]

Since $2\sqrt{3}-1\approx 2.464$, $\frac{5+\sqrt{33}}{4}\approx 2.686$ and $\frac{5+2\sqrt{7}}{3}\approx 3.43>3$, the admissible range for $p$ in Case III is

[TABLE]

Then we can use (4.1) and repeat the computation from (4.5) to (4.9) to get a global weak solution $u\in L^{q}_{t}L^{p+1}_{r}L_{\theta}^{2}(\mathbb{R}_{+}^{1+2})$, where $q=\frac{p^{2}-1}{5-p}>2$ and $\frac{5+\sqrt{33}}{4}\leq p\leq 3$.

Note that $\frac{7+\sqrt{409}}{12}<\frac{7}{3}<\frac{4+\sqrt{17}}{3}$ and $\frac{5+\sqrt{33}}{4}<\frac{4+\sqrt{17}}{3}<3$. Then

[TABLE]

Therefore collecting the proofs in Case I-Case III, we obtain Theorem 1.2.