Carathéodory properties of Gaussian hypergeometric function associated with differential inequalities in the complex plane

1.   Introduction and position of problem

We consider, for $x \in {\mathbb R}^{n}, \ t > 0, \ j = 1, 2, \dots, m$, the following system of $m$ equations

where $a \in \mathbb{R}$, $\omega > 0$, $n \geq 3$.

Various non-linear sources have been combined as follows, we combine all two consecutive equations together and of course the last equation with the first one, which get the whole system closely linked by the strong nonlinear sources. The functions $f_j(u_1, u_2, \dots, u_m) \in (\mathbb{R}^m, \mathbb{R})$ are given for $j = 1, 2, \dots, m-1$, by

and

with $d, e > 0, \ p > 3$. For simplicity reason, we take $d = e = 1$.

There exists a function $\mathcal{F}\in C^1(\mathbb{R}^3, \mathbb{R})$ such that

such that

In order to use Poincare's inequality which is a key in calculus for the PDEs, we will study the problem (1.1) in the presence of a density function $\theta$ to find a generalized formula for Poincare's inequality that can be used in unbounded domain $\mathbb{R}^n$. The function $\Theta (x) > 0$ for all $x \in {\mathbb R}^{n}$ is a density and $(\Theta)^{-1}(x) = 1 /\Theta (x) \equiv\theta(x)$ such that

We define a new space related to the nature of our system, taking into account the boundless of space $\mathbb{R}^n$. The function spaces ${\mathcal H}$ is defined as the closure of $C_{0}^{\infty}({\mathbb R}^{n})$, as in ^[20], we have

with respect to the norm $\left\Vert v \right\Vert_{{\mathcal H}} = \left(v, v \right)_{{\mathcal H}}^{1/2}$ for the inner product

and $L_{\theta}^{2}({\mathbb R}^{n})$ as that to the norm $\left\Vert v \right\Vert_{L_{\theta}^{2}} = \left(v, v \right)_{L_{\theta}^{2}}^{1/2}$ for

For general $r \in [1, +\infty)$

is the norm of the weighted space $L_{\theta}^{r}({\mathbb R}^{n})$.

The following references in connection to our system for a single equation ^[6] and ^[7]. The work ^[6] was the pioneer in the literature for the single equation, source of inspiration of several works, while the work ^[7] is a recent generalization of ^[6] by introducing less dissipative effects (See ^{[8,9,19,24,26]}). With regard to the study of this type of systems without viscoelasticity, with the existence of both weak damping $u_t$ and strong damping $\Delta u_t$, under condition (3.2), we mention the work recently published in one equation in ^[14]

where $\Omega$ is a bounded domain of $\mathbb{R}^n, \ n\geq 1$ with a smooth boundary $\partial\Omega$. The aim goal was mainly on the local existence of weak solution by using contraction mapping principle and of course the authors showed the global existence, decay rate and infinite time blow up of the solution with certain conditions on initial energy.

In the case of non-bounded domain $\mathbb{R}^n$, we mention the paper recently published by T. Miyasita and Kh. Zennir in ^[18], where they considered equation as follows

with initial data

The authors succeeded in highlighting the existence of unique local solution and they continued to expand it to be global in time. The rate of the decay for solution was the main result, for more results related to decay rate of solution of this type of problems, please see ^{[15,23,25,28]}.

Regarding the study of the coupled system of two nonlinear wave equations, we mention the work done by Baowei Feng et al. which was considered in ^[12], a coupled system for viscoelastic wave equations with nonlinear sources in bounded domain with smooth boundary as follows

Under appropriate hypotheses, they established a general decay result by multiplication techniques to extend some existing results for a single equation to the case of a coupled system.

There are several results in this direction, notably the generalization made by Shun in a complicate nonlinear case with degenerate damping term in ^[21]. The IBVP for a system of nonlinear viscoelastic wave equations in a bounded domain was considered in the problem

where $\Omega$ is a bounded domain with a smooth boundary. Given certain conditions on the kernel functions, degenerate damping and nonlinear source terms, they got a decay rate of the energy function for some initial data.

In $n-$equations, paper in ^[1] considered a system

where the absence of global solutions with positive initial energy was investigated. Next, a nonexistence of global solutions for system of three semilinear hyperbolic equations was introduced in ^[3]. A coupled system of semilinear hyperbolic equations was investigated by many authors and many results were obtained with the nonlinearities in the form $f_1 = |u|^{p-1}|v|^{q+1}u, f_2 = |v|^{p-1}|u|^{q+1}v$. (Please, see ^[2,16,22])

2.   Preliminaries

We introduce the Sobolev embedding and generalized Poincaré inequalities.

Lemma 2.1. ^[13,18] Let $\theta$ satisfy (1.4). For positive constants $C_{\tau} > 0$ and $C_{P} > 0$ depending only on $\theta$ and $n$, we have

and

hold for $v \in {\mathcal H}$. Here $\tau = 2n/(2n-rn+2r)$ for $1 \leq r \leq 2n / (n-2)$.

In the 1950s and 1970s, the linear theory of viscoelasticity was extensively developed and now, it becomes widely used to represent this term using several improvements to the nature of decreasing the kernel function. We assume that the kernel functions $\varpi_j \in C^1(\mathbb{R}^{+}, \mathbb{R}^{+})$ satisfying

We mean by $\mathbb{R}^{+}$ the set $\{ \tau \mid \tau \geq 0 \}$. Noting by

and

We assume that there is a function $\chi \in C^1(\mathbb{R}^{+}, \mathbb{R}^{+})$ which is linear or is strictly convex $C^2$ function on $(0, \varepsilon_0), \varepsilon_0\leq\varpi_j(0)$, with $\chi(0) = \chi'(0) = 0$ and a positive nonincreasing differentiable function $\xi:[0, \infty) \to [0, \infty)$, such that the novel properties

satisfied for any $\varrho \geq 0$.

We note that, if $\chi$ is a strictly increasing convex $C^2-$ function on $(0, \tau]$ with $\chi(0) = \chi'(0) = 0$, then $\chi$ has an extension $\bar{\chi}$, which is strictly increasing and strictly convex $C^2-$function on $(0, \infty)$. For example, $\bar{\chi}$ can be given by

Hölder and Young's inequalities give

Thanks to Minkowski's inequality to give

Then, there exist $\eta > 0$ such that

We need to define positive constants $\lambda_{0}$ and $\mathcal{E}_{0}$ by

The mainly aim of the present paper is to obtain a novel decay rate of solution from the convexity property of the function $\chi$ given in Theorem 3.4.

We denote, as in ^[18], an eigenpair $\left\{ (\lambda_{i}, e_{i}) \right\}_{i \in {\mathbb N}} \subset {\mathbb R} \times {\mathcal H}$ of

for any $i \in {\mathbb N}$. Then

holds and $\left\{ e_{i} \right\}$ is a complete orthonormal system in ${\mathcal H}$.

Definition 2.2. The vectors $(u_1, u_2, \dots, u_m)$ is said a weak solution to (1.1) on $[0, T]$ if satisfies for $x \in {\mathbb R}^{n}$

for all test functions $\varphi_j \in {\mathcal H}, j = 1, 2, \dots, m$ for almost all $t \in [0, T]$.

3.   Statement of main results

The local solution (in time $[0, T]$) is given in next Theorem.

Theorem 3.1. (Local existence) Assume that

Let $(u_{10}, u_{20}, \dots u_{m0}) \in {\mathcal H}^m$ and $(u_{11}, u_{21}, \dots, u_{m1}) \in [L_{\theta}^{2}({\mathbb R}^{n})]^m$. Under the assumptions (1.4)-(1.3) and (2.1)-(2.4), suppose that

Then (1.1) admits a unique local solution $(u_1, u_2, \dots, u_m)$ such that

for sufficiently small $T > 0$.

Remark 3.2. The constant $\lambda_{1}$ introduced in (3.2) being the first eigenvalue of the operator $-\Delta$.

We will show now the global solution in time established in Theorem 3.3. Let us introduce the potential energy $J:\mathcal H^m \to \mathbb{R}$ defined by

The modified energy is defined by

here

for any $w \in L^2({\mathbb R}^{n}), j = 1, 2, \dots, m$.

Theorem 3.3. (Global existence) Let (1.4)-(1.3) and (2.1)-(2.4) hold. Under (3.1), (3.2) and for sufficiently small $(u_{10}, u_{11}), (u_{20}, u_{21}), \dots, (u_{m0}, u_{m1}) \in {\mathcal H} \times L_{\theta}^{2}({\mathbb R}^{n})$, problem (1.1) admits a unique global solution $(u_1, u_2, \dots, u_m)$ such that

The decay rate for solution is given in the next Theorem.

Theorem 3.4. (Decay of solution) Let (1.4)-(1.3) and (2.1)-(2.4) hold. Under conditions (3.1), (3.2) and

there exists $t_{0} > 0$ depending only on $\varpi_j$, $a$, $\omega$, $\lambda_{1}$ and $\chi'(0)$ such that

holds for all $t \geq t_{0}$.

In particular, by the positivity of $\mu$ in (2.2), we have, as in ^[17],

for a single wave equation.

Lemma 3.5. For $(u_1, u_2, \dots, u_m)\in {\mathcal X}^m_{T}$, the functional $\mathcal{E}(t)$ associated with problem (1.1) is a decreasing energy.

Proof. For $0 \leq t_{1} < t_{2} \leq T$, we have

owing to (2.1)-(2.4).

We define an inner product as

and the associated norm is given by

$\forall v, w \in {\mathcal H}$. By (3.2), we get

The following Lemma yields.

Lemma 3.6. Let $\theta$ satisfy (1.4). Under condition (3.2), we get

for $v \in {\mathcal H}$.

4.   Proof of existence results

We give here the outline of the proof for local solution by a standard procedure (See ^[15,28]).

Proof. (Of Theorem 3.1.) Let $(u_{10}, u_{11}), (u_{20}, u_{21}), \dots, (u_{m0}, u_{m1}) \in {\mathcal H} \times L_{\theta}^{2}({\mathbb R}^{n})$. The presence of the nonlinear terms in the right hand side of our problem (1.1) gives us negative values of the energy. For this purpose, for any fixed $(u_1, u_2, \dots, u_m) \in {\mathcal X}^m_{T}$, we can obtain first, a weak solution of the related system

The Faedo-Galerkin's method consist to construct approximations of solutions $(z_{1n}, z_{2n}, \dots, z_{mn}, )$ for (4.1), then we obtain a prior estimates necessary to guarantee the convergence of approximations. In the last step we pass to the limit of the approximations by using the compactness of some embedding in the Sobolev spaces. The uniqueness is obtain by letting two solutions for (4.1) and then, after ordinary calculations, we find that the solutions are equal.

Some details regarding the transition to ODE systems are given, for this end let $\{e_{i}\}$ be the Galerkin basis and let

Given initial data $u_{j0} \in {\mathcal H}$, $u_{j1} \in L_{\theta}^{2}({\mathbb R}^{n})$, we define the approximations

which satisfy the following approximate problem

with initial conditions

which satisfies

Taking $e_{ji} = g_{ji}$ in (4.3) yields the following Cauchy problem for a ordinary differential equation with unknown $g^n_{ji}$.

By using the Caratheodory Theorem for standard ordinary differential equations theory, the problem (4.3)-(4.4) has a solutions $(g_{1in}, g_{2in}, \dots, g_{min})_{i = 1, n}\in (H^{3}[0, T])^{m}$ and by using the embedding $H^{m}[0, T]\rightarrow C^{m}[0, T]$, we deduce that the solution $(g_{1in}, g_{2in}, \dots, g_{min})_{i = 1, n}\in (C^{2}[0, T])^{4}$. In turn, this gives a unique $(z_{1n}, z_{2n}, \dots, z_{mn})$ defined by (4.2) and satisfying (4.3).

To return to the problem (1.1), we should find a solution map

from ${\mathcal X}^m_{T}$ to ${\mathcal X}^m_{T}$. We are now ready to show that $\top$ is a contraction mapping in an appropriate subset of ${\mathcal X}^m_{T}$ for a small $T > 0$. Hence $\top$ has a fixed point

which gives a unique solution in ${\mathcal X}^m_{T}$.

We will show the global solution. For this end, by using conditions on functions $\varpi_j$, we have

here $\beta^2 = J(u_1, u_2, \dots, u_m)$, for $t \in [0, T)$, where

Noting that $\mathcal{E}_0 = G(\lambda_{0})$, given in (2.7). Then

Moreover, $\lim\limits_{\xi \to +\infty}G(\xi) \to -\infty$. Then, we have the following Lemma

Lemma 4.1. Let $0 \leq \mathcal{E}(0) < \mathcal{E}_{0}$.

$(i)$ If $\sum\limits_{j = 1}^{m}\left\Vert u_{j0} \right\Vert^2_{\mathcal H} < \lambda^2_{0}$, then local solution of (1.1) satisfies

$(ii)$ If $\sum\limits_{j = 1}^{m}\left\Vert u_{j0} \right\Vert^2_{\mathcal H} > \lambda^2_{0}$, then, local solution of (1.1) satisfies

Proof. Since $0 \leq \mathcal{E}(0) < \mathcal{E}_{0} = G(\lambda_{0})$, there exist $\xi_{1}$ and $\xi_{2}$ such that $G(\xi_{1}) = G(\xi_{2}) = \mathcal{E}(0)$ with $0 < \xi_{1} < \lambda_{0} < \xi_{2}$.

The case $(i)$. By (4.7), we have

which implies that $J(u_{10}, u_{20}, \dots u_{m0}) \leq \xi^2_{1}.$ Then, we claim that $J(u_1, u_2, \dots, u_m) \leq \xi^2_{1}, \ \forall t \in [0, T)$. Moreover, there exists $t_{0} \in (0, T)$ such that

Then

by Lemma 3.5, which contradicts (4.7). Hence we have

The case $(ii)$. We can now show that $\sum\limits_{j = 1}^m\left\Vert u_{j0} \right\Vert^2_{\mathcal H} \geq \xi^2_{2}$ and that $\sum\limits_{j = 1}^m\left\Vert u_j \right\Vert^2_{\mathcal H} \geq \xi^2_{2} > \lambda^2_{0}$ in the same way as $(i)$.

Proof. (Of Theorem 3.3.) Let $(u_{10}, u_{11}), (u_{20}, u_{21}), \dots, (u_{m0}, u_{m1}) \in {\mathcal H} \times L_{\theta}^{2}({\mathbb R}^{n})$ satisfy both $0 \leq \mathcal{E}(0) < \mathcal{E}_{0}$ and $\sum\limits_{j = 1}^m\left\Vert u_{j0} \right\Vert^2_{\mathcal H} < \lambda^2_{0}$. By Lemma 3.5 and Lemma 4.1, we have

This completes the proof.

5.   Proof of general decay result

Let

Lemma 5.1. Let $(u_1, u_2, \dots, u_m)$ be the solution of problem (1.1). If

Then, under condition (3.6), the functional $\Pi(u_1, u_2, \dots, u_m) > 0, \ \forall t > 0.$

Proof. By (5.1) and continuity, there exists a time $t_1 > 0$ such that

Let

Then, by (5.1), we have for all $(u_1, u_2, \dots, u_m)\in Y$,

Owing to (3.4), it follows for $(u_1, u_2, \dots, u_m)\in Y$

By (2.6), (3.6) we have

hence $\Pi(u_1(t_0), u_2(t_0), \dots, u_m(t_0)) > 0$ on $Y$, which contradicts the definition of $Y$ since $\Pi(u_1(t_0), u_2(t_0), \dots, u_m(t_0)) = 0$. Thus $\Pi(u_1, u_2, \dots, u_m) > 0, \ \forall t > 0.$

We are ready to prove the decay rate.

Proof. (Of Theorem 3.4.) By (2.6) and (5.3), we have for $t \geq 0$

Let

where $\mu$ and $\mu_0$ defined in (2.2) and (2.3).

Noting that $\lim \limits_{t \to +\infty}\mu(t) = 0$ by (2.1)-(2.3), we have

Then, we take $t_{0} > 0$ such that

with (2.4) for all $t > t_{0}$. Due to (3.4), we have

Then, by definition of $I(t)$, we have

and Lemma 3.5, we have for all $t_1, t_2\geq 0$

then, by generalized Poincaré's inequalities, we get

Finally, by (5.6), $\forall t \geq t_0$, we have

by the convexity of $\chi$ and (2.4), we have

Then

which completes the proof.

6.   Conclusions

The paper deals with a kind of $m-$nonlinear wave equations with viscoelastic structures. We considered the local existence, global existence and exponential decay rate of solution. We discussed the effects of weak and strong damping terms on decay rate. The methods are standard for local existence and we extended the local solution to a global one by appropriate energy estimates. At last, We obtained a novel decay rate of solution from the convexity property of the function which extends the results in [Math. Meth. Appl. Sci., 43(3), 1138 (2020); Mathematics 8(2), 203 (2020)]. The treatment of Cauchy problem for a family of effectively damped single wave models with a nonlinear memory on the righthand side, that is for $x\in \mathbb{R}^n$

where $\omega > 0, p > 1, r\in (-1, 1)$ and $\gamma\in(0, 1)$, remains as an open problem, which will be our next work, based on ^[4,5,10,11].

Acknowledgments

The author would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.