Blow-up of solutions to the coupled Tricomi equations with derivative type nonlinearities

1.   Introduction and main results

In this work, we consider the following Cauchy problem of coupled Tricomi equations with derivative type nonlinearities

where $m > 0$, $n\geq1$. The initial values possess compact supports

where $R\geq2$ is a constant.

Firstly, we recall some known results related to the single Tricomi equation

where $f(u, u_{t}) = |u|^{p},$ $|u_{t}|^{p},$ $|u_{t}|^{p}+|u|^{q}.$ Lin and Tu ^[27] study blow-up dynamics of problem (1.3) with $f(u, u_{t}) = |u|^{p}$ by using the test function method and iteration argument in the sub-critical case. An iteration procedure together with slicing method is employed to prove formation of singularity of solution in the critical case. He et al. ^[11] investigate blow-up result of solution by constructing the Riccati type differential inequality. Lucentea and Palmieri ^[29] establish blow-up dynamics for the Tricomi equation with derivative type nonlinearity $f(u, u_{t}) = |u_{t}|^{p}$ in the sub-critical and critical cases by utilizing the integral representation formula. Upper bound lifespan estimates of solution are obtained by using the test function method (see ^[22]). For the Tricomi equation with combined nonlinearities $f(u, u_{t}) = |u_{t}|^{p}+|u|^{q}$, blow-up results and lifespan estimates of solution are established by employing the iteration argument (see ^[1]). Hamouda and Hamza ^[6] study properties of solution to the liner Cauchy problem corresponding to problem (1.3). Formation of singularities of solution are obtained by constructing ordinary differential inequality. More detailed illustration related to the study of the Tricomi equation can be found in ^{[10,12,13,33,40]}.

Equation (1.3) reduces to the well known semilinear wave equation when $m = 0,$ namely

Problem (1.4) is concerned with the Strauss conjecture when $f(u, u_{t}) = |u|^{p}$ (see ^[41]). We note the critical exponent $p_{S}(1) = \infty$. For $n\geq2,$ $p_{S}(n)$ is the positive root of quadratic equation

The critical exponent $p_{S}(n)$ for $n\geq2$ divides the interval $(1, \infty)$ into two parts. For $p\in(1, p_{S}(n)],$ the solution blows up in finite time (see ^{[7,9,16,17,20,21,25,30,31,42,43,44]}). While the solution exists globally (in time) when $p\in(p_{S}(n), \infty)$ (see ^{[2,4,23,24,26,28,45]}). For $f(u, u_{t}) = |u_{t}|^{p}$, Eq (1.4) is related to the Glassey conjecture ^[3], where the critical exponent is characterized by $p_{G}(n) = \frac{n+1}{n-1}.$ Concretely, Lai and Tu ^[19] investigate problem (1.4) with space dependent damping $\frac{\mu}{(1+|x|)^{\beta}}u_{t}\, (\beta > 2)$ and $f(u, u_t) = |u|^{p}, \, |u_{t}|^{p}$, respectively. Blow-up results and lifespan estimates of solutions are obtained by using the test function method. For problem (1.4) involving mixed nonlinearities $f(u, u_{t}) = |u_{t}|^{p}+|u|^{q},$ Han and Zhou ^[8] obtain upper bound lifespan estimates of solution to the Cauchy problem by constructing proper test function and the ordinary differential inequalities. Lai and Takamura ^[18] illustrate blow-up results and upper bound lifespan estimates of solution to the problem with time dependent damping term $\frac{\mu}{(1+t)^{\beta}}u_{t}\, (\beta > 1)$ by making use of a multiplier and iteration argument. Formation of singularities for solution to problem (1.4) with scale invariant damping $\frac{\mu}{ 1+t }u_{t}$ are investigated by applying the test function approach (see ^[5]).

Recently, blow-up dynamics of the Cauchy problem for coupled system of semilinear wave equations

attracts extensive attention (see ^{[15,32,35,36,37,38,39]}). Using the test function method, Ikeda et al. ^[15] obtain blow-up results and lifespan estimates of solutions to problem (1.5) with power nonlinear terms $f_{1}(v, v_{t}) = |v|^{p}$, $f_{2}(u, u_{t}) = |u|^{q}$, derivative nonlinear terms $f_{1}(v, v_{t}) = |v_{t}|^{p}$, $f_{2}(u, u_{t}) = |u_{t}|^{q}$ and mixed nonlinear terms $f_{1}(v, v_{t}) = |v|^{p}$, $f_{2}(u, u_{t}) = |u_{t}|^{q}$, respectively. Palmieri and Takamura ^[38] discuss the coupled system of semilinear wave equations (1.5) with time dependent weak damping terms and power nonlinearities. Upper bound lifespan estimates of solutions are derived by utlizing the iteration argument. Palmieri and Takamura ^[35] consider the coupled system of semilinear time dependent damped wave equations of derivative type nonlinearities in the scattering case. Upper bound lifespan estimates of solutions in the sub-critical case are obtained by taking advantage of the Kato lemma. While in the critical case, an iteration procedure based on the slicing method is employed. Formation of singularities of solutions to problem (1.5) with time dependent damping and mixed nonlinear terms in the scattering case are established through the iteration argument (see ^[37]). Ikeda et al. ^[14] study the Cauchy problem for coupled system of semilinear Tricomi equations

where the nonlinear terms are power type nonlinearities $f_{1}(v, v_{t}) = |v|^{p}$ and $f_{2}(u, u_{t}) = |u|^{q}$. Blow-up dynamics and lifespan estimates of solutions are obtained by constructing test functions which are related to the Gauss hypergeometric functions.

Inspired by the results in ^{[14,15,22,29]}, we consider blow-up dynamics of the Cauchy problem for coupled Tricomi equations (1.1). For the Cauchy problem of single Tricomi equation with derivative nonlinear term, upper bound lifespan estimates of solution are established by using the test function method (see ^[22]) and integral representation formula (see ^[29]), respectively. Concerning the Cauchy problem for coupled semilinear wave equations and coupled Tricomi equations with power type nonlinearities, formation of singularities are established by employing the test function method (see ^[14,15]). We obtain blow-up results and upper bound lifespan estimates of solutions to problem (1.1) by using test function method and iteration argument, respectively. It is worth to mention that the test function utilized in the proof of Theorem 1.1 is different from the test functions employed in ^[14,15], which are related to the Gauss hypergeometric functions. Lucente and Palmieri ^[29] establish upper bound lifespan estimate of solution to the Cauchy problem for single Tricomi equation by using integral representation formula and constructing ordinary differential inequality. While in this paper, we combine the integral representation formula with iteration method to present the proof of Theorem 1.2. The results obtained in this paper can be regarded as an extended work in ^{[14,15,22,29]}. To our best knowledge, the results in Theorems 1.1 and 1.2 are new. In addition, we present a comparison for lifespan estimates in Theorems 1.1 and 1.2 in a special case (see Remark 1.1).

Throughout this paper, we use the following expressions

$C$ denotes the positive constant independent of $\varepsilon$, which may vary from line to line. $A\lesssim B$ stands for $A\leq CB$, where $C$ is a positive constant.

From the local existence result of solution to the Cauchy problem of single Tricomi equation with derivative type nonlinearity $|u_{t}|^{p}$ in Theorem 2 in ^[22], we can obtain the existence and uniqueness of solutions to the coupled system (1.1) by using the Banach fixed point theorem. We omit the details for simplicity.

The main results in this paper are described as follows.

Theorem 1.1. Assume that the initial data $u_{0}, v_{0}\in H^{1}(\mathbb{R}^{n})$, $u_{1}, v_{1}\in H^{1-\frac{1}{m+1}}(\mathbb{R}^{n})$ satisfy

where $\Gamma(s) = \int_{0}^{+\infty}z^{s-1}e^{-z}dz$ is the Gamma function for $s > 0$. Suppose that $(u, v)$ are a pair of solutions to problem $(1.1)$ which satisfy

Then, there exists a small positive constant$\varepsilon_{0} = \varepsilon_{0}(n, m, p, q, R, u_{0}, u_{1}, v_{0}, v_{1})$ such that the lifespan estimates satisfy

where $\widetilde{F}_{GG} = \max\{F_{GG}(n, m, p, q), F_{GG}(n, m, q, p)\}$ and $\varepsilon\in(0, \varepsilon_{0}]$.

Theorem 1.2. Assume that the initial data $u_{0}, v_{0}\in C_{0}^{2}(\mathbb{R}^{n}),$ $u_{1}, v_{1}\in C_{0}^{1}(\mathbb{R}^{n})$ are non-negative functions. Suppose that $(u, v)$ are a pair of solutions to problem $(1.1)$ which satisfy

Then, the lifespan estimates satisfy

where $\widetilde{\Lambda}_{GG} = \max\{(m+1)\Lambda_{GG}(n, m, p, q), (m+1)\Lambda_{GG}(n, m, q, p\}$.

Remark 1.1. Direct calculation shows $\widetilde{F}_{GG}(n, m, p, q) = m+\widetilde{\Lambda}_{GG}(n, m, p, q).$ Therefore, the first lifespan estimate in $(1.7)$ is better than the the first lifespan estimate in $(1.8)$ in the sub-critical case. We conjecture that the curve in the $p-q$ plane which satisfies $\widetilde{F}_{GG}(n, m, p, q) = 0$ is the critical curve. We will verify this conjecture in our future work. We observe that Ikeda et al. ^[15] obtain the following upper bound lifespan estimates of solutions to problem $(1.1)$ with $m = 0$, namely

It is worth to mention that the lifespan estimates in $(1.7)$ and the lifespan estimates in $(1.8)$ are coincide with the lifespan estimates in $(1.9)$ when $m = 0$.

Remark 1.2. Problem $(1.1)$ is equivalent to the Cauchy problem for single Tricomi equation when $p = q,$ which has been studied in ^[22,29]. Lifespan estimates of solutions in $(1.7)$ and $(1.8)$ in the case $p = q$ are coincide with the lifespan estimates of solutions in ^[22,29].

2.   Proof of Theorem 1.1

2.1.   Basic definition and related lemmas

Firstly, we illustrate the definition of weak solutions.

Definition 2.1. Assume that the pair of functions $(u, v)$ satisfy

Then, $(u, v)$ are called weak solutions of problem $(1.1)$ on $[0, T)$ if

where $\Psi(t, x)\in C_{0}^{\infty}([0, T)\times \mathbb{R}^{n})$ and $T\in(1, T(\varepsilon))$. Here, $T(\varepsilon)$ represents the upper bound lifespan estimate of solutions to problem $(1.1)$, which satisfies

Lemma 2.2. The cut off function $\eta(t)\in C^{\infty}([0, \infty))$ is defined by

which satisfies$|\eta'(t)|, |\eta''(t)| < C.$Let $\eta_{T}(t) = \eta(t/T)$ and $k > 1.$ It holds that

Suppose that the function $\theta(t)\in C^{\infty}([0, \infty))$ satisfies

The proof of Lemma 2.2 is easy, we omit the details.

Lemma 2.3. (Lemma 4 in ^[22]) Let $m > -\frac{1}{2}$ and $\gamma = \frac{m}{2(m+1)}.$ Assume that

where $K_{\upsilon}(\cdot)$ stands for modified Bessel function.It holds that $y(t)\in C^{1}([0, \infty))$ $\cap C^{\infty}(0, \infty)$ which satisfies

Moreover, $y(t)$ possesses the following properties:

(1) $y(t) > 0,$ $y'(t) < 0,$

(2) $\lim\limits_{t\rightarrow0^{+}}y(t) = 2^{\gamma-\frac{1}{2}}(m+1)^{\gamma+\frac{1}{2}}\Gamma(\gamma+\frac{1}{2}) = c_{0}(\gamma) > 0,$

(3) $\lim\limits_{t\rightarrow0^{+}}\dfrac{y'(t)}{t^{2m}} = -c_{0}(-\gamma) < 0,$

(4) $y(t) = \sqrt{\dfrac{(m+1)\pi}{2}}t^{\frac{m}{2}}\exp(-\dfrac{t^{m+1}}{m+1})\times(1+O(t^{-(m+1)})),$ for large $t > 0$,

(5) $y'(t) = -\sqrt{\dfrac{(m+1)\pi}{2}}t^{\frac{3m}{2}}\exp(-\dfrac{t^{m+1}}{m+1})\times(1+O(t^{-(m+1)})),$ for large $t > 0$.

2.2.   Proof of Theorem 1.1

We introduce the following test function

where

It holds that $\Delta\phi(x) = \phi(x)$ and

We note $\Psi(t, x)\in C_{0}^{\infty}([0, \infty)\times\mathbb{R}^{n})$ and

Applying the first equality in (2.1) and (2.3), we obtain

Using (2.2), (2.5) and the fact $\partial_{t}\eta_{M}(t) < 0$, we conclude

where $c_{1} = c_{0}(\gamma) \int_{\mathbb{R}^{n}}u_{0}(x)\phi dx+c_{0}(-\gamma)\int_{\mathbb{R}^{n}}u_{1}(x)\phi dx > 0.$

Taking $k\geq q' = \frac{q}{q-1},$ we obtain

where we have used the estimates in Lemma 2.3 and (2.4).

From (2.6)–(2.9), we arrive at

Similarly, making use of the second equality in (2.1) and (2.3) with $k\geq p'$ yields

where $c_{2} = c_{0}(\gamma) \int_{\mathbb{R}^{n}}v_{0}(x)\phi dx+c_{0}(-\gamma)\int_{\mathbb{R}^{n}}v_{1}(x)\phi dx > 0.$

Combining (2.10) and (2.11), we have

We set

It holds that

Let $w(t, x) = |v_{t}|^{p}t^{-2m}|\partial_{t}y|\phi.$ We define $Y(M) = Y[|v_{t}|^{p}t^{-2m}|\partial_{t}y|\phi](M).$ From (2.12), we have

Solving the ordinary differential inequality (2.14), we have

For $F_{GG}(n, m, p, q) = 0$ and $p = q,$ we obtain

Therefore, we derive

On the other hand, we define $Y_{1}(M) = {Y}[|u_{t}|^{q}t^{-2m}|\partial_{t}y|\phi](M)$. From (2.13), we deduce

Direct computation gives rise to

Combining (2.15)–(2.17), we conclude the lifespan estimates in (1.7). This completes the proof of Theorem 1.1.

3.   Proof of Theorem 1.2

In this section, we utilize the iterative method to characterize blow-up results of problem (1.1).

3.1.   Proof for the sub-critical case

Firstly, for Cauchy problem of the Tricomi equation

we have the following integral representation formula.

Lemma 3.1. (Proposition 2.1 in ^[29]) Let $n = 1$ and $m > 0$. Suppose that $u_{0}\in C_{0}^{2}(\mathbb{R}),$ $u_{1}\in C_{0}^{1}(\mathbb{R})$ and$h(t, x)\in C([0, \infty), C^{1}(\mathbb{R})).$ Then, the solution $u(t, x)$ to problem $(3.1)$ can be represented by

where

The kernel function $E(t, x; b, y; m)$ is defined by

where $F(a, b; c;z)$ stands for the Gauss hypergeometric function $F(a, b; c;z) = \sum\limits_{n = 0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}$ with the Pochhammer symbol $(d)_{0} = 1$ and $(d)_{n} = \prod_{k = 1}^{n}(d+k-1)$ for $n\in\mathbb{N}$.

Let $(u, v)$ be a pair of solutions to problem (1.1). We set the space variable $x = (z, w)$, where $z\in\mathbb{R}$ and $w\in\mathbb{R}^{n-1}$. We define $U(t, z) = \int_{\mathbb{R}^{n-1}}u(t, z, w)dw,$ $V(t, z) = \int_{\mathbb{R}^{n-1}}v(t, z, w)dw.$ Then, $(U(t, z), V(t, z))$ satisfy

where

From (3.2), we deduce that $U(t, z)$ and $V(t, z)$ can be represented as

Here, we only present the proof of lower bound estimate for $U(t, z)$. The lower bound estimate for $V(t, z)$ can be obtained in an analogous way.

From the fact $\varphi_{m}(t)-y+z\leq2\varphi_{m}(t)$ for $y\in [z-\varphi_{m}(t), z+\varphi_{m}(t)]$, we have

We derive lower bound estimates for $I_{4}$ and $I_{5}$ on the characteristic line $\varphi_{m}(t)-z = R$ for $z\geq R$. We note $[-R, R]\subset[z-\varphi_{m}(t), z+\varphi_{m}(t)]$. It holds that

where $K = \min\{2^{2(\gamma-1)}a_{m}R^{\gamma-1}, 2^{-2\gamma}b_{m}R^{-\gamma}\}.$

Using the Hölder inequality, we achieve

which implies

It follows that

By shrinking the domain of integration, we obtain

Since

we have

Employing the fact $F(a, a; c;z)\geq1$ for $a\in \mathbb{R},$ $c > 0$ and $z\in [0, 1)$ (more details can be found in ^[34]), we derive

Therefore, we have

It follows that

Making use of $V(\varphi_{m}^{-1}(y-R), y) = 0$ and the Jensen inequality, we deduce

where $C_{1} = 4^{-\gamma}c_{m}(4R)^{-\frac{n-1}{2}(p-1)}(2R(m+1))^{-\frac{p-1}{m+1}}.$

Combining (3.6)–(3.8), we have

Setting $\widetilde{U}(z) = (z+R)^{\gamma}U(t, z)$ and $\widetilde{V}(z) = (z+R)^{\gamma}V(t, z)$, we deduce

where $M = K\|u_{0}+u_{1}\|_{L^{1}(\mathbb{R}^{n})}.$ Similarly, we obtain

where $N = K\|v_{0}+v_{1}\|_{L^{1}(\mathbb{R}^{n})}$ and $C_{2} = 4^{-\gamma}c_{m}(4R)^{-\frac{n-1}{2}(q-1)}(2R(m+1))^{-\frac{q-1}{m+1}}.$

We are in the position to apply the iteration argument. In the sub-critical case, we assume

where $\{\alpha_{j}\}_{j\in \mathbb{N}}$, $\{\beta_{j}\}_{j\in \mathbb{N}}$ and $\{\theta_{j}\}_{j\in \mathbb{N}}$ are sequences of non-negative real numbers. We set $\alpha_{0} = 0,$ $\beta_{0} = 0$, $\theta_{0} = M\varepsilon$. From (3.9), we deduce that (3.11) holds with $j = 0$. Plugging (3.11) into (3.10) yields

Substituting (3.12) into (3.9), we have

Let

It is deduced from (3.14) that

where $A = \frac{n-1}{2}(pq-1)+\gamma(pq+2p+1),$ $B = p+1.$ From $\beta_{j}\leq B\frac{(pq)^{j}}{pq-1}$, we deduce

where $\widetilde{\theta} = C_{1}(C_{2})^{p}(\frac{B}{pq-1})^{-(p+1)}.$ From (3.16), we have

Choosing $j_{0} = \max\{0, \frac{\log\widetilde{\theta}}{\log(pq)^{p+1}}-\frac{pq}{pq-1}\},$ we derive

for $j > j_{0}$, where $\hat{\theta} = M(pq)^{-\frac{pq(p+1)}{(pq-1)^{2}}}\;\;\widetilde{\theta}^{\frac{1}{pq-1}}.$ Combining (3.11), (3.15) with (3.17), we have

We deduce $2(z-R)\geq R+z$ when $z\geq3R$. Therefore, we obtain

We choose $\varepsilon_{1} = \varepsilon_{1}(n, m, p, q, R, u_{0}, u_{1}, v_{0}, v_{1})$ such that $(\hat{\theta}2^{-\frac{B}{pq-1}}\varepsilon_{1})^{-\Lambda_{GG}^{-1}}\geq4R.$ For all $\varepsilon\in(0, \varepsilon_{1}]$ and $\phi_{m}(t) > (\hat{\theta}2^{-\frac{B}{pq-1}}\varepsilon)^{-\Lambda_{GG}^{-1}},$ we have

Sending $j\rightarrow \infty$ in (3.19), we deduce that $\widetilde{U}(z)$ blows up in finite time. Therefore, we conclude

Similarly, for $\Lambda_{GG}(n, m, q, p) > 0,$ we have

In summary, we obtain the upper bound lifespan estimate

for $\widetilde{\Lambda}_{GG}(n, m, p, q) > 0.$

3.2.   Proof for the critical case

Let $\max\{\widetilde{\Lambda}_{GG}(n, m, p, q), \widetilde{\Lambda}_{GG}(n, m, q, p)\} = 0.$ Without loss of generality, we set $\widetilde{\Lambda}_{GG}(n, m, p, q) = 0 > \widetilde{\Lambda}_{GG}(n, m, q, p).$ We assume

where $l_{j} = 2-2^{-(j+1)}$, $\{D_{j}\}_{j\in \mathbb{N}}$ and $\{E_{j}\}_{j\in \mathbb{N}}$ are suitable sequences of non-negative real numbers. From (3.9) and (3.20), we deduce $D_{0} = M\varepsilon$ and $E_{0} = 0$ when $j = 0$. For $l_{j}\geq l_{0} = \frac{3}{2},$ we have $z\geq\frac{3}{5}(R+z).$ According to (3.10), we derive

Plugging (3.21) into (3.9), we conclude

where we have used the fact $\widetilde{\Lambda}_{GG}(n, m, p, q) = 0$ and $(R+y)^{-1}\geq\frac{3}{5}y^{-1}.$ When $1-\frac{l_{j}}{l_{j+1}}\geq 2^{-(j+3)},$ we obtain

Let

Direct computation implies

Taking advantage of $E_{j}\leq\frac{(pq)^{j}}{pq-1},$ we deduce

where $\widetilde{D} = (pq-1) C_{1}(\frac{3}{5})^{p+1}(C_{2})^{p}2^{-2p}.$ From (3.24), we derive

Let $j_{1} = \max\{0, \frac{\log\widetilde{D}}{\log(2^{p}pq)}-\frac{pq}{pq-1}\}.$ For $j > j_{1}$, we acquire

where $\hat{D} = M(2^{p}pq)^{-\frac{2}{(pq-1)^{2}}}\widetilde{D}^{\frac{1}{pq-1}}.$

Applying (3.20), (3.23) and (3.25), for $j > j_{1}, z\geq l_{j}R$, we deduce

We choose $\varepsilon_{2} = \varepsilon_{2}(n, m, p, q, R, u_{0}, u_{1}, v_{0}, v_{1})$ such that $\exp((\hat{D}\varepsilon_{2})^{-(pq-1)})\geq1.$ For all $\varepsilon\in(0, \varepsilon_{2}],$ $z > 2R\exp((\hat{D}\varepsilon)^{-(pq-1)}),$ we have $\hat{D}\varepsilon(\log\frac{z}{2R})^{\frac{1}{pq-1}} > 1.$ Sending $j\rightarrow \infty,$ we have $\widetilde{U}(z)\rightarrow \infty.$ Therefore, we arrive at

For $\phi_{m}(t) = z+R < 2z,$ we obtain the lifespan estimate

Similarly, we acquire

when $\widetilde{\Lambda}_{GG}(n, m, q, p) = 0 > \widetilde{\Lambda}_{GG}(n, m, p, q)$.

For the case $\widetilde{\Lambda}_{GG}(n, m, p, q) = \widetilde{\Lambda}_{GG}(n, m, q, p) = 0$ when $p = q,$ following a similar deduction, we prove

Employing the fact $-\frac{n-1}{2}(p-1)-\gamma(p+1) = -\frac{n-1}{2}(q-1)-\gamma(q+1) = -1$ when $\widetilde{\Lambda}_{GG}(n, m, p, q) = \widetilde{\Lambda}_{GG}(n, m, q, p) = 0$, we deduce

where $\widetilde{F} = 2^{-(p+1)}C_{1}C_{2}^{p}(p-1)^{p+1}.$

Direct calculation shows

Let $j_{2} = \max\{0, \frac{\log\widetilde{F} }{2(p+1)\log p}-\frac{p^{2}}{p^{2}-1}\}.$ For $j > j_{2},$ we obtain

where $\hat{F} = Mp^{-\frac{2p^{2}(p+1)}{(p^{2}-1)^{2}}}\widetilde{F}^{\frac{1}{p^{2}-1}}.$ Therefore, we acquire

We choose small positive constant $\varepsilon_{3} = \varepsilon_{3}(n, m, p, q, R, u_{0}, u_{1}, v_{0}, v_{1})$ such that

Then, for all $\varepsilon\in(0, \varepsilon_{3}],$ $z > R\exp[(\hat{F}\varepsilon)^{-(p-1)}],$ we have

Therefore, $\widetilde{U}(z)$ blows up in finite time when $j\rightarrow \infty$. It follows that

Bearing in mind $\phi_{m}(t) = z+R\leq2z,$ we have

Analogously, we have the same lifespan estimate for $\widetilde{V}(z)$ in (3.27).

Choosing $\varepsilon_{4} = \min\{\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}\},$ we conclude the lifespan estimates in (1.8). This completes the proof of Theorem 1.2.

4.   Conclusions

In this paper, blow-up results of solutions to coupled system of the Tricomi equations with derivative type nonlinearities are studied. Upper bound lifespan estimates of solutions to the Cauchy problem with small initial values are derived. We illustrate the key results by using the test function method (see Theorem 1.1) and integral representation formula together with iteration argument (see Theorem 1.2), respectively. Our main new contribution is that lifespan estimates of solutions to the problem in the sub-critical and critical cases are connected with the Glassey conjecture. To the best of our knowledge, the results in Theorems 1.1 and 1.2 are new. In addition, we present a comparison for lifespan estimates in Theorems 1.1 and 1.2 in a special case (see Remark 1.1).

Acknowledgments

The author Sen Ming would like to express his sincere thank to Professor Yi Zhou for his guidance and encouragements during the postdoctoral study in Fudan University. The author Sen Ming also would like to express his sincere thank to Professors Han Yang and Ning-An Lai for their helpful suggestions and discussions. The project is supported by Fundamental Research Program of Shanxi Province (No. 20210302123021, No. 20210302123045, No. 20210302123182), Innovative Research Team of North University of China (No. TD201901), Program for the Innovative Talents of Higher Education Institutions of Shanxi Province, Science and Technology Innovation Project of Higher Education Institutions in Shanxi (No. 2020L0277) and Science Foundation of North University of China (No. XJJ201922).

Conflict of interest

This work does not have any conflict of interest.