Blow-up of energy solutions for the semilinear generalized Tricomi equation with nonlinear memory term

1.   Introduction

In the present paper, we will investigate blow-up conditions for the following semilinear generalized Tricomi equation with a time-dependent speed of propagation:

with $p > 1$, where $\ell$ is a positive parameter and $g = g(t)$ denotes the relaxation function in the memory term (or the so-called memory kernel). Moreover, the right-hand side of the equation in the initial value problem (1) is given by

in which the relaxation function satisfies some assumptions that will be specified later. Roughly speaking, our main purpose in this paper is to understand the interplay effect of the parameter $\ell$ and the relaxation function $g(t)$ to the blow-up condition (i.e. energy solutions for the Cauchy problem (1) blows up in finite time) for the power exponent $p$.

In order to introduce the background related to our model (1), let us recall some results for the limit case $\ell = 0$ in the Cauchy problem (1), i.e.

Concerning the special case (the Riemann-Liouville fractional integral of $1-\gamma$ order)

where $\Gamma$ stands for the Euler integral of the second kind. The completed blow-up result has been firstly derived by Chen-Palmieri ^[6]. They proved blow-up of energy solutions to the semilinear wave equation with nonlinear memory carrying (3) if

and $1 < p < \infty$ for $n = 1$. Furthermore, $p_0(n, \gamma)$ is the greatest root of the equation

Here, we should mention that the exponent $p_0(n, \gamma)$ is a natural generalization of the so-called Strauss exponent, since when $\gamma\uparrow 1$ the exponent $p_0(n, \gamma)$ tends to the Strauss exponent. Here, the Strauss exponent $p_{\mathrm{Str}}(n)$ was proposed by Strauss ^[23] as critical exponent for the semilinear wave equation with power nonlinearity and it is the greatest root of

In the same year, Chen-Palmieri ^[5] investigated the blow-up condition of local in time solution when $1 < p < p_{\mathrm{Str}}(n)$ for the semilinear Moore-Gibson-Thompson (MGT) equation in the conservative case, which is somehow a particular case (the relaxation function $g(t)$ is an exponential decay function) of the Cauchy problem (2). More detailed explanations on the relation between the semilinear MGT equation and semilnear hyperbolic-like (or wave) model also shown in Chen-Ikehata ^[4]. Therefore, one may conjecture some influence of the relaxation function on the blow-up condition. Quite recently, Chen ^[3] answered this conjecture and developed a generalization of the result in ^[5] for a wider class of relaxation functions of the wave equation and it corresponding weakly coupled system with general nonlinear memory terms, where some new blow-up results combined with a new threshold in the subcritical case were derived. Although the global (in time) existence condition for the semilinear wave equation with nonlinear memory term is a completely open problem, there are some ideas and hints that would suggest the likelihood of the critical condition. All these results are based on the constant propagation speed. Clearly, when the model owns variable propagation speed, the shape of the light-cone depends on the speed of propagation. Concerning other studies for nonlinear memory in evolution equations, we refer readers to ^[2,9,10,11].

Let us recall the results for the semilinear generalized Tricomi equation

In recent year, this problem caught a lot of attention. According to a series of work on global existence and blow-up from He-Witt-Yin ^[12,13,14] and Lin-Tu ^[18], one may conjecture the reasonable critical exponent for the Cauchy problem (5) is the greatest real root $p_1(n, \ell)$ of the quadratic equation

Obviously, when $\ell = 0$ in the last equation, the greatest real root is the Strauss exponent. Therefore, it is a natural generalization of the Strauss exponent. In some sense, the power nonlinearity can be treated thanks to Hölder's inequality. However, when the nonlinear part has the memory effect, we need to find a suitable way to deal with relaxation function under suitable assumption. Other works for the semilinear Tricomi equation with derivative-type or mixed-type nonlinearity are refereed to Chen-Lucente-Palmieri ^[7], Lucente-Palmieri ^[19] and references therein.

From the above introduction, we found that these two generalized Strauss exponent are different, and they have been influenced by the property of the relaxation function or the time-dependent coefficient. Nevertheless, to the best of author's knowledge, the combined influence of the relaxation function and the time-dependent coefficient on the wave equation (especially, generalized Tricomi equation) is still unknown. In this paper, our purpose is to investigate the effects on the blow-up range for $p$ due to the combined presence of the speed of propagation $t^{\ell}$ and of the relaxation function $g$. Moreover, we will consider a general assumption on the relaxation function whose form is not only including the Riemann-Liouville type but also some oscillations. For these reasons, we will face the treatment of a general memory term. Actually, we cannot use the classical Kato's lemma (see Kato ^[16]). To overcome this difficulty, we found that the iteration procedure (see, for example, Agemi-Kurokawa-Takamura ^[1], Lai-Takamura ^[17], Palmieri-Takamura ^[20,21,22], Chen-Reissig ^[8] and reference therein) will provide us a strong tool to deal with it.

After stating our main result, we will give some examples. For one thing, we will investigate the relation between our model and the memoryless case or the constant coefficient case. For another, we will show the wide applications of our result.

Throughout this paper, we denote $B_R(0)$ as the ball around the origin with radius $R$ in the whole space $\mathbb{R}^n$; $f\lesssim g$ as $f\leq C g$ with a positive constant $C$; $f\gtrsim g$ as $f\geq C g$ with a positive constant $C$; $\mathbb{N}_0: = \mathbb{N}\cup\{0\}$.

2.   Main result

To begin with, following the idea of Chen-Palmieri ^[6], let us introduce a suitable definition of an energy solution for our main problem.

Definition 2.1. Let $u_0\in H^1(\mathbb{R}^n)$ and $u_1\in L^2(\mathbb{R}^n)$. We say $u = u(t, x)$ is an energy solution to the Cauchy problem (1) if

satisfies $u(0, \cdot) = u_0\in H^1(\mathbb{R}^n)$ and the next identity is fulfilled:

for any test function $\psi\in C_0^{\infty}([0, T)\times \mathbb{R}^n)$ and any $t\in(0, T)$.

Applying integration by parts in (7), we get

Thanks to Definition 2.1, we can show the main result of the present paper. Before doing this, we introduce the exponent $p_2(n, \ell, \theta)$ given by the greatest root of the quadratic equation

We feel the combined effect of the speed of propagation and of the relaxation function in (8). Precisely, we denote

The influence from the relaxation function (represented by $\theta$) and the time-dependent coefficient (represented by $\ell$) plays different crucial roles on the determination of the exponent $p_2(n, \ell, \theta)$. Indeed, this effect cannot be easily observed from the beginning of the model.

Theorem 2.1. Let us consider the positive relaxation function $g = g(t)$ such that

and the monotonously decreasing or constant function $h = h(t)$ has the lower bound

where $\theta\geq 0$. Let us suppose that $u_0\in H^1(\mathbb{R}^n)$ and $u_1\in L^2(\mathbb{R}^n)$ are nonnegative and compactly supported functions with supports contained in a ball $B_R(0)$ with some $R > 0$ such that $u_0, u_1$ are not identically zero. Let $u = u(t, x)$ be the local (in time) energy solution to the Cauchy problem (1) according to Definition 2.1. Providing that

then a solution to (1) according to Definition 2.1 that satisfies

blows up in finite time, i.e. $T < \infty$. Here, $\phi(t): = t^{\ell+1}/(\ell+1)$ denotes the speed of propagation.

Remark 2.1. The condition (9) may be relaxed by some approaches developed in ^[3], where the trick slicing method need to be introduced. For our model, we need to investigate a new sequence in slicing procedure. In the forthcoming future, this relaxation of condition will be studied.

Remark 2.2. Let us take the first example when $g(t)$ is chosen so that $g\ast|u|^p$ is the Riemann-Liouville fractional integral of $1-\gamma$ order, i.e. (3). Clearly,

where $h(t)$ is a monotonously decreasing function with $\gamma\in(0, 1)$ and it satisfies

So, the nontrivial energy solution to (1) blows up in finite time provided $1 < p < p_2(n, \ell, \gamma)$. Let us consider the limit case $\ell\downarrow0$. We observe from (4) and (8) that

for any $n\geq 1$ and $\gamma\in(0, 1)$. We recall

Thus, the blow-up result seems reasonable due to the fact that

for any $n\geq 1$ and $\ell > 0$. One may see (6) and (8). In this sense, we believe our result is reasonable.

Remark 2.3. Let us consider another example, in which we will show that the blow-up result not only holds for the polynomial decay relaxation function. We set $g(t) = (2+\sin\omega(t))t^{-\gamma}$ with $\gamma\in(0, 1)$ and a continuous function $\omega(t)$. Then, we may find

Thus, we still can prove blow-up of energy solutions if $1 < p < p_2(n, \ell, \gamma)$ for $\ell > 0$ and $\gamma\in(0, 1)$.

Remark 2.4. To determine the critical exponent for the Cauchy problem (1), even in the special case of the Riemann-Liouville fractional integral type (3), we still need to investigate the situation when $p\geq p_2(n, \ell, \gamma)$. Honestly, we conjecture that for the supercritical case $p > p_2(n, \ell, \gamma)$, under some suitable assumptions on small data, the solution exists globally in time. Concerning the limit case $p = p_2(n, \ell, \gamma)$, we expect every nontrivial solution blows up in finite time. Nevertheless, these questions are still open problems.

Remark 2.5. Concerning lifespan estimates, we consider initial data $(u, u_t)(0, x) = \varepsilon(u_0, u_1)(x)$ with the positive parameter $\varepsilon > 0$. Therefore, by following the same approach, the estimate (17) will be changed into

By denoting the power of $t$ in the exponential function by

we arrive at upper bound estimate for lifespan

to the model (1) in $(0, T_{\varepsilon})$. The derivation of lifespan in this case is standard, and one may see, for example, the end of Section 2 of ^[6].

3.   Proof of the main theorem

3.1.   Iteration frame and first lower bound estimate

Recalling some results from the paper He-Witt-Yin ^[12], we may define a test function $\Psi = \Psi(t, x)$ such that

where the time-dependent test function $\lambda = \lambda(t)$ is given by

with the positive constant $C_{\ell}$ such that $\lambda(0) = 1$, moreover, $K_{\alpha}(t)$ stands for the modified Bessel function of the second kind with the order $\alpha$. The space-dependent test function was introduced by Yordanov-Zhang ^[24], namely,

where $S^{n-1}$ denotes the $n-1$ dimensional sphere. Therefore, it holds that

Due to the fact that $K_{\alpha}(\tau)$ with positive variable $\tau$ is the solution to the next differential equation of second-order:

we may find from (11) that

carrying its value $\Psi(0, x) = \Phi(x)$ and $\Psi(\infty, x) = 0$. In other words, the test function in (11) solves the adjoint equation of the homogeneous generalized Tricomi equation.

We now introduce two suitable functionals as follows:

where the test function $\Psi(t, x)$ was defined in (11). We should emphasize that the functional $F_0(t)$ plays a crucial role in the proof, and if $F_0(t)$ blows up then $u(t, x)$ will blow up in finite time. Our aim will be to show that $F_0(t)$ blows up in finite time. The auxiliary functional $F_1(t)$ will provide a first lower bound for $F_0(t)$.

Due to the support assumption in the statement of Theorem 2.1, we have

With the help of the previous statement, one may choose the test function in (7) by $\psi(\tau, x)\equiv 1$ for $|x|\leq R+\phi(\tau)$, namely,

Integrating the last one with respect to the time variable, it yields

The inequality (12) implies two facts. For one thing, the nonnegativity of $F_0(t)$ for any $t\geq0$ can be asserted because of the assumption on initial data. For another, by using Hölder's inequality and the assumption on relaxation function (9), we get

Employing the next estimate from He-Witt-Yin ^[12]:

we see

At the same time, the application of Hölder's inequality associated with nonnegativity of $F_0(t)$ shows

with a positive constant $C_0$, which provides the iteration frame for the functional $F_0$.

With the aim of deriving a first lower bound of $F_0(t)$, it is necessary for us to turn to the lower bound of $F_1(t)$. Thanks to nonnegativity of the nonlinear term, we may directly follow the same approach of Lemma 2.3 in He-Witt-Yin ^[12]. Then, there exists a constant $t_0 > 0$ such that

Here, we have used the nontrivial assumption on $u_0$ and $u_1$ in deriving the last estimate. Thus, combining (13) with (15), we have

with a positive constant $C_1 > 0$, for any $t\geq t_0$. In other words, we have deduced the first lower bound estimate

where

3.2.   Iteration procedure

In the previous section, a first lower bound for the functional $F_0(t)$ has been established, and therefore, we will investigate a sequence of lower bounds for it by making use of the frame of iteration (14). To do that, we establish the sequence of lower bounds

where $\{Q_j\}_{j\in\mathbb{N}_0}$, $\{\alpha_j\}_{j\in\mathbb{N}_0}$, $\{\beta_j\}_{j\in\mathbb{N}_0}$ and $\{\gamma_j\}_{j\in\mathbb{N}_0}$ are sequences of nonnegative real numbers which will be determined in the forthcoming part.

Let us assume that (16) holds for $j$. We need to prove that the lower bound estimate also holds for $j+1$. Let us plug (16) into the nonlinear integral inequality (14). Then, it holds

for any $t\geq t_0$. So, the lower bound estimate (16) holds for $j+1$ providing that

We now determine the sequence for $\alpha_j, \beta_j, \gamma_j$ firstly. We know that if

with a constant $m$, then the sequence can be deduced by

Hence, the elements in all sequences can be determined

One derives

which implies

In order to estimate $Q_j$ from below, we apply the logarithmic function to both sides of the last inequality, obtaining

In the last chain, we used

Let us choose $j\geq j_0$ such that

Here, the positive integer $j_0$ can be fixed by

Fixing

in conclusion for $j\geq j_0$, the lower bound for $F_0(t)$ can be expressed by

According to our assumption on the relaxation function, it is valid that

Concerning $j\geq j_0$ and $t\geq\max\{ 2t_0, R\}$, we arrive at

where

The coefficient of $\log t$ is

if and only if

that is equivalent to our assumption $1 < p < p_2(n, \ell, \theta)$. All in all, for $t\geq\max\{ 2t_0, R\}$ and letting $j\to \infty$, we can claim that the lower bound for the functional $F_0(t)$ blows up in finite time. The proof is complete.

4.   Conclusions

In this paper, we found a new blow-up result for the semilinear generalized Tricomi equation with a general nonlinear memory term. Particularly, the combined effect from the "decay" property of the relaxation function and the propagation speed on the blow-up condition was investigated. Finally, it seems also interesting to derive interplay effect for the corresponding nonlinear system

where $\ell_1, \ell_2 > 0$ and $g_1, g_2$ satisfy suitable assumptions. The crucial point of interesting is the blow-up condition $\Omega(p, q) = \Omega(p, q;\ell_1, \ell_2, g_1, g_2) > 0$. The weakly coupled system of generalized Tricomi equations with power nonlinearities is studied by ^[15] recently. In the forthcoming future, we will study this problem.

Acknowledgments

The work was supported by the Foundation for natural Science in Higher Education of Guangdong (Grant No. 2019KZDXM042) and the Science Foundation of Huashang College Guangdong University of Finance and Economics (Grant No.2019HSDS28). The authors thank the anonymous referees for carefully reading the paper and giving some useful suggestions.