**4. Non-Existence of Global in Time Solutions for Problem** (1)**–**(2)**–**(3)

We assume that


We first establish the following mass conservation law.

**Proposition 1.** *Suppose that for some* <sup>0</sup> <sup>&</sup>lt; *<sup>T</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>∞</sup>*,* (*η*, *<sup>u</sup>*) <sup>∈</sup> *<sup>C</sup>*1([0, *<sup>T</sup>*0] <sup>×</sup> [0, *<sup>L</sup>*]) <sup>×</sup> *<sup>C</sup>*1([0, *<sup>T</sup>*0] <sup>×</sup> [0, *<sup>L</sup>*])*, η* ≥ 0*, is a solution of problem* (1)*–*(2)*–*(3)*. Then*

$$\int\_{0}^{L} \mathcal{K}\_{L}(\mathbf{x})^{\beta - 1} \eta(t, \mathbf{x}) \psi'(\mathbf{x}) \, d\mathbf{x} = \int\_{0}^{L} \mathcal{K}\_{L}(\mathbf{x})^{\beta - 1} \eta\_{0}(\mathbf{x}) \psi'(\mathbf{x}) \, d\mathbf{x} := m\_{0}, \quad 0 \le t \le T\_{0}. \tag{25}$$

*where*

$$
\mathcal{K}\_L(\mathbf{x}) = \psi(L) - \psi(\mathbf{x}).\tag{26}
$$

**Proof.** From the first equation in (1), one has

$$-\partial\_{0|t}^{\alpha,\psi}\eta(t,x) = \partial\_{0|x}^{\beta,\psi}(\eta\mu)(t,x), \quad (t,x) \in (0,T\_0] \times (0,L)\_\*$$

whereupon

$$-\partial\_{0|t}^{\alpha,\psi} \left( l\_0^{\otimes,\psi} \eta \left( t, \cdot \right) \right)(L) = \left( l\_0^{\otimes,\psi} \partial\_{0|x}^{\otimes,\psi} (\eta \mu)(t, \cdot) \right)(L).$$

Using Lemma 4 and the boundary conditions (3), one obtains

$$\left( I\_0^{\beta,\psi} \partial\_{0|\underline{x}}^{\beta,\psi} (\eta \underline{u})(t, \cdot) \right)(L) = \eta(t, L)\underline{u}(t, L) - \eta(t, 0)\underline{u}(t, 0) = 0.$$

Hence, it holds

$$
\partial\_{0|t}^{\alpha,\psi} \left( I\_0^{\otimes,\psi} \eta(t,\cdot) \right)(L) = 0,
$$

i.e.,

$$
\partial\_{0|t}^{\alpha\prime\#} \int\_0^L \mathcal{K}\_L(\mathbf{x})^{\beta - 1} \psi'(\mathbf{x}) \eta(t, \mathbf{x}) \, d\mathbf{x} = 0,
$$

which implies that

$$I\_0^{\alpha,\psi} \partial\_{0|t}^{\alpha,\psi} \int\_0^L \mathcal{K}\_L(\mathbf{x})^{\beta - 1} \psi'(\mathbf{x}) \eta(t, \mathbf{x}) \,d\mathbf{x} = 0.$$

Again, using Lemma 4, one deduces that

$$\int\_0^L \mathcal{K}\_L(\mathbf{x})^{\beta - 1} \Psi'(\mathbf{x}) \eta(t, \mathbf{x}) \, d\mathbf{x} - \int\_0^L \mathcal{K}\_L(\mathbf{x})^{\beta - 1} \Psi'(\mathbf{x}) \eta(0, \mathbf{x}) \, d\mathbf{x} = 0, 1$$

which yields (25).

Our principal result is the following.

**Theorem 1.** *Suppose that for some* <sup>0</sup> <sup>&</sup>lt; *<sup>T</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>∞</sup>*,* (*η*, *<sup>u</sup>*) <sup>∈</sup> *<sup>C</sup>*1(Q) <sup>×</sup> *<sup>C</sup>*1(Q)*,* <sup>Q</sup> = [0, *<sup>T</sup>*0] <sup>×</sup> [0, *<sup>L</sup>*]*, <sup>η</sup>* <sup>≥</sup> <sup>0</sup>*, is a solution of problem* (1)*–*(2)*–*(3)*. Let*

$$T\_{\text{max}} := \sup \left\{ \tau > 0 : (\eta, u) \in \mathbb{C}^1([0, \tau) \times [0, L]) \times \mathbb{C}^1([0, \tau) \times [0, L]) \text{ is a solution of (1) -- (2) -- (3)} \right\}.$$

*If*

$$J(0) := \int\_0^L \eta\_0(\mathbf{x}) u\_0(\mathbf{x}) \mathcal{K}\_L(\mathbf{x})^{\frac{\theta}{2} - 1} \psi'(\mathbf{x}) \, d\mathbf{x} > 0,\tag{27}$$

*where* K*<sup>L</sup> is given by* (26)*, then*

$$T\_0 \le T\_{\max} \le \psi^{-1} \left( \psi(0) + M(a, a) \right) < \infty,\tag{28}$$

*where M*(*a*, *α*) *is given by* (11) *(with θ* = *α),*

$$a = 2\frac{\Gamma\left(1 + \frac{\beta}{2}\right)}{\Gamma\left(1 - \frac{\beta}{2}\right)} \frac{\kappa\_L(0)^{-\frac{\beta}{2}}}{m\_0}$$

*and m*<sup>0</sup> *is given by* (25)*.*

**Proof.** We introduce the function

$$
\varphi(\mathbf{x}) = \mathcal{K}\_L(\mathbf{x})^{\frac{\beta}{2} - 1} \boldsymbol{\psi}'(\mathbf{x}), \quad 0 \le \mathbf{x} < L. \tag{29}
$$

Multiplying the second equation in (1) by *ϕ*(*x*) and integrating over (0, *L*), one obtains

$$\begin{aligned} &\frac{1}{2}\int\_{0}^{L}\varrho(\mathbf{x})\frac{1}{\mathfrak{p}'(t)}\partial\_{t}(\eta u)(t,\mathbf{x})\,d\mathbf{x}+\frac{1}{2}\int\_{0}^{L}\varrho(\mathbf{x})\partial\_{0|\mathfrak{t}}^{a,\mathfrak{p}}(\eta u)(t,\mathbf{x})\,d\mathbf{x}+\int\_{0}^{L}\varrho(\mathbf{x})\partial\_{0|\mathbf{x}}^{\mathfrak{f},\mathfrak{p}}(\eta u^{2})(t,\mathbf{x})\,d\mathbf{x} \\ &+\int\_{0}^{L}\varrho(\mathbf{x})\partial\_{0|\mathbf{x}}^{\mathfrak{f},\mathfrak{p}}(\eta^{2})(t,\mathbf{x})\,d\mathbf{x}=0,\quad 0$$

which yields

$$\left(\frac{1}{2\overline{\rho}'(t)}l'(t) + \frac{1}{2}\left(\partial\_{0|t}^{\mu,\mathfrak{q}}l\right)\right)(t) = -\int\_0^L \rho(\mathbf{x})\partial\_{0|\mathbf{x}}^{\mathbf{f},\mathfrak{q}}(\eta\mathbf{u}^2)(t,\mathbf{x})\,d\mathbf{x} - \int\_0^L \rho(\mathbf{x})\partial\_{0|\mathbf{x}}^{\mathbf{f},\mathfrak{q}}(\eta^2)(t,\mathbf{x})\,d\mathbf{x}, \quad 0 < t < T\_{\mathbf{0}}.\tag{30}$$

where

$$J(t) = \int\_0^L \wp(\mathbf{x}) (\eta \mu)(t, \mathbf{x}) \, d\mathbf{x}, \quad 0 \le t \le T\_{0-}$$

On the other hand, using (8), one has

$$\int\_0^L \varphi(\mathbf{x}) \partial\_{0|\mathbf{x}}^{\otimes,\mathfrak{p}}(\eta \mu^2)(t,\mathbf{x}) \, d\mathbf{x} = \int\_0^L \left( I\_0^{1-\beta,\mathfrak{p}} \frac{\partial\_{\mathbf{x}}(\eta \mu^2)(t,\cdot)}{\boldsymbol{\upupup}'} \right)(\mathbf{x}) \frac{\boldsymbol{\upupup}(\mathbf{x})}{\boldsymbol{\upupup}'(\mathbf{x})} \boldsymbol{\upupup}'(\mathbf{x}) \, d\mathbf{x}.$$

Hence, by Lemma 3, one obtains

$$\begin{split} \int\_{0}^{L} \boldsymbol{\varrho}(\boldsymbol{x}) \partial\_{0|\boldsymbol{x}}^{\boldsymbol{\beta},\boldsymbol{\varphi}} (\boldsymbol{\eta}\boldsymbol{u}^{2})(\boldsymbol{t},\boldsymbol{x}) \, d\boldsymbol{x} &= \int\_{0}^{L} \frac{\partial\_{\boldsymbol{x}}(\boldsymbol{\eta}\boldsymbol{u}^{2})(\boldsymbol{t},\boldsymbol{x})}{\boldsymbol{\upvarphi}^{\boldsymbol{\prime}}(\boldsymbol{x})} \left(\boldsymbol{I}\_{L}^{1-\boldsymbol{\beta},\boldsymbol{\varphi}} \frac{\boldsymbol{\upvarphi}}{\boldsymbol{\upvarphi}^{\boldsymbol{\prime}}}\right)(\boldsymbol{x}) \boldsymbol{\upvarphi}^{\boldsymbol{\prime}}(\boldsymbol{x}) \, d\boldsymbol{x} \\ &= \int\_{0}^{L} \boldsymbol{\eth}\_{\boldsymbol{x}}(\boldsymbol{\eta}\boldsymbol{u}^{2})(\boldsymbol{t},\boldsymbol{x}) \left(\boldsymbol{I}\_{L}^{1-\boldsymbol{\beta},\boldsymbol{\varphi}} \frac{\boldsymbol{\upvarphi}}{\boldsymbol{\upvarphi}^{\boldsymbol{\prime}}}\right)(\boldsymbol{x}) \, d\boldsymbol{x}. \end{split}$$

Next, using an integration by parts and the boundary conditions (3), one deduces that

$$\int\_{0}^{L} \boldsymbol{\varrho}(\mathbf{x}) \partial\_{0|\mathbf{x}}^{\boldsymbol{\beta},\boldsymbol{\Psi}} (\eta \boldsymbol{u}^{2})(t, \mathbf{x}) \, d\mathbf{x} = -\int\_{0}^{L} \boldsymbol{\eta}(t, \mathbf{x}) \boldsymbol{u}^{2}(t, \mathbf{x}) \partial\_{\mathbf{x}} \left( \boldsymbol{l}\_{L}^{1-\boldsymbol{\beta},\boldsymbol{\Psi}} \frac{\boldsymbol{\varrho}\boldsymbol{\Psi}}{\boldsymbol{\Psi}'} \right)(\mathbf{x}) \, d\mathbf{x}.\tag{31}$$

Similarly, one has

$$\int\_{0}^{L} \boldsymbol{\varrho}(\mathbf{x}) \partial\_{0|\mathbf{x}}^{6,\Upph}(\boldsymbol{\eta}^{2})(\mathbf{t}, \mathbf{x}) \, d\mathbf{x} = -\int\_{0}^{L} \boldsymbol{\eta}^{2}(\mathbf{t}, \mathbf{x}) \partial\_{\mathbf{x}} \left( I\_{L}^{1-\not p,\upphup} \frac{\boldsymbol{\varrho}\boldsymbol{\upphup}}{\boldsymbol{\upupupup}} \right)(\mathbf{x}) \, d\mathbf{x}.\tag{32}$$

It follows from (30)–(32) that

$$\frac{1}{2\overline{\varphi}'(t)}l'(t) + \frac{1}{2}\left(\widehat{\sigma}\_{0|t}^{a,\emptyset}l\right)(t) = \int\_0^L (\eta u^2)(t,x)\partial\_x\left(l\_L^{1-\beta,\emptyset}\frac{\varrho}{\overline{\varphi}'}\right)(x)\,dx + \int\_0^L \eta^2(t,x)\partial\_x\left(l\_L^{1-\beta,\emptyset}\frac{\varrho}{\overline{\varphi}'}\right)(x)\,dx.\tag{33}$$

Next, using (29), for *x* ∈ (0, *L*), an elementary calculation gives us that

$$\left(I\_L^{1-\beta,\psi}\frac{\mathcal{P}}{\psi'}\right)(\varkappa) = \frac{\Gamma\left(\frac{\beta}{2}\right)}{\Gamma\left(1-\frac{\beta}{2}\right)}\mathcal{K}\_L(\varkappa)^{-\frac{\beta}{2}}\dots$$

Hence, it holds

$$\partial\_{\mathbf{x}} \left( I\_L^{1-\beta,\psi} \frac{\varphi}{\Psi'} \right)(\mathbf{x}) = \frac{\Gamma\left(1 + \frac{\beta}{2}\right)}{\Gamma\left(1 - \frac{\beta}{2}\right)} \mathcal{K}\_L(\mathbf{x})^{-\frac{\beta}{2} - 1} \psi'(\mathbf{x}) > 0, \quad 0 < \mathbf{x} < L. \tag{34}$$

It follows from (33) and (34) that

$$\frac{1}{2\Psi'(t)}f'(t) + \frac{1}{2}\left(\delta\_{0|t}^{\mu,\psi}f\right)(t) \ge \int\_0^L (\eta \, u^2)(t, \mathbf{x}) \partial\_x \left(l\_L^{1-\beta,\psi} \frac{\varrho}{\Psi'}\right)(\mathbf{x}) \, d\mathbf{x}.\tag{35}$$

On one hand, by Hölder's inequality, one has

$$\begin{split} &\int\_{0}^{L} \left(\int\_{0}^{L} \eta(t,x) |u(t,x)| \, \upvarphi(x) \,, dx\right)^{2} \\ &= \left(\int\_{0}^{L} \sqrt{\eta(t,x)} |u(t,x)| \sqrt{\vartheta\_{x}\left(I\_{L}^{1-\beta,\mathfrak{p}}\frac{\mathfrak{p}}{\mathfrak{p}'}\right)(x)} \sqrt{\frac{\eta(t,x)}{\partial\_{x}\left(I\_{L}^{1-\beta,\mathfrak{p}}\frac{\mathfrak{p}}{\mathfrak{p}'}\right)(x)}} \, q(x) \, dx\right)^{2} \\ &\leq \left(\int\_{0}^{L} (\eta \mu^{2})(t,x) \eth\_{x}\left(I\_{L}^{1-\beta,\mathfrak{p}}\frac{\mathfrak{p}}{\mathfrak{p}'}\right)(x) \, dx\right) \left(\int\_{0}^{L} \frac{\eta(t,x)}{\partial\_{x}\left(I\_{L}^{1-\beta,\mathfrak{p}}\frac{\mathfrak{p}}{\mathfrak{p}'}\right)(x)} \, q^{2}(x) \, dx\right). \end{split} \tag{36}$$

On the other hand, using (29) and (34), one obtains

$$\begin{split} &\int\_{0}^{L} \frac{\eta\left(t,\mathbf{x}\right)}{\partial\_{x}\left(I\_{L}^{1-\beta,\mathfrak{p}}\frac{\varphi}{\mathfrak{p}'}\right)(\mathbf{x})} \mathfrak{q}^{2}(\mathbf{x}) \,d\mathbf{x} \\ &= \frac{\Gamma\left(1-\frac{\beta}{2}\right)}{\Gamma\left(1+\frac{\beta}{2}\right)} \int\_{0}^{L} \mathcal{K}\_{L}(\mathbf{x})^{\beta-1} \mathfrak{p}'(\mathbf{x}) \eta\left(t,\mathbf{x}\right) \mathcal{K}\_{L}(\mathbf{x})^{\frac{\beta}{2}} \,d\mathbf{x} \\ &\leq \frac{\Gamma\left(1-\frac{\beta}{2}\right)}{\Gamma\left(1+\frac{\beta}{2}\right)} \mathcal{K}\_{L}(0)^{\frac{\beta}{2}} \int\_{0}^{L} \mathcal{K}\_{L}(\mathbf{x})^{\beta-1} \mathfrak{p}'(\mathbf{x}) \eta\left(t,\mathbf{x}\right) \,d\mathbf{x}. \end{split}$$

Furthermore, using the mass conservation law (25), one deduces that

$$\int\_{0}^{L} \frac{\eta\left(t, \mathbf{x}\right)}{\partial\_{\mathbf{x}}\left(I\_{L}^{1-\beta, \mathfrak{P}} \frac{\mathbf{q}}{\Psi'}\right)(\mathbf{x})} \boldsymbol{\varrho}^{2}(\mathbf{x}) \, d\mathbf{x} \leq \frac{\Gamma\left(1 - \frac{\beta}{2}\right)}{\Gamma\left(1 + \frac{\beta}{2}\right)} \mathcal{K}\_{L}(0)^{\frac{\beta}{2}} m\_{0}. \tag{37}$$

Next, (36) and (37) yield

$$\int\_{0}^{L} \left(\eta \mu^{2}\right)(t, x) \partial\_{x} \left(l\_{L}^{1-\beta, p} \frac{\mathcal{P}}{\Psi'}\right)(x) \, dx \geq \frac{\Gamma\left(1 + \frac{\beta}{2}\right)}{\Gamma\left(1 - \frac{\beta}{2}\right)} \frac{\mathcal{K}\_{L}(0)^{-\frac{\beta}{2}}}{m\_{0}} l^{2}(t), \quad 0 < t < T. \tag{38}$$

It follows from (35) and (38) that

$$2\frac{1}{\Psi'(t)}f'(t) + \left(\delta\_{0|t}^{a,\psi}f\right)(t) \ge 2\frac{\Gamma\left(1+\frac{\beta}{2}\right)}{\Gamma\left(1-\frac{\beta}{2}\right)}\frac{\mathcal{K}\_L(0)^{-\frac{\beta}{2}}}{m\_0}f^2(t), \quad 0 < t < T\_0.$$

Hence, using (27) and Lemma 5, the estimate (28) follows.

**Example 1.** *Consider the system*

$$\begin{cases} \ ^{\mathbb{C}}D\_{0|t}^{\mathfrak{a}}\eta + \ ^{\mathbb{C}}D\_{0|\mathbf{x}}^{\mathfrak{b}}(\eta\mu) &= 0, \quad t > 0, 0 < \mathbf{x} < L, \\\ \frac{1}{2}\left[\partial\_{t}(\eta\mu) + \ ^{\mathbb{C}}D\_{0|t}^{\mathfrak{a}}(\eta\mu)\right] + \ ^{\mathbb{C}}D\_{0|\mathbf{x}}^{\mathfrak{b}}(\eta\mu^{2}) + \ ^{\mathbb{C}}D\_{0|\mathbf{x}}^{\mathfrak{b}}(\eta^{2}) &= 0, \quad t > 0, 0 < \mathbf{x} < L \end{cases} \tag{39}$$

*under the initial and boundary conditions* (2) *and* (3)*. Here CD<sup>α</sup>* <sup>0</sup>|*<sup>t</sup> is the Caputo derivative in time of fractional order* 0 < *α* < 1 *and CD<sup>β</sup>* <sup>0</sup>|*<sup>x</sup> is the Caputo derivative in space of fractional order* <sup>0</sup> <sup>&</sup>lt; *<sup>β</sup>* <sup>&</sup>lt; <sup>1</sup>*. System* (39) *is a special case of* (1) *with <sup>ψ</sup>*(*s*) = *s. Hence, by Theorem 1, one deduces that if* (*η*, *<sup>u</sup>*) <sup>∈</sup> *<sup>C</sup>*1([0, *<sup>T</sup>*0] <sup>×</sup> [0, *<sup>L</sup>*]) <sup>×</sup> *<sup>C</sup>*1([0, *<sup>T</sup>*0] <sup>×</sup> [0, *<sup>L</sup>*]) *is a solution of problem* (39)*–*(2)*–*(3) *for some* <sup>0</sup> <sup>&</sup>lt; *<sup>T</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>∞</sup>*, and*

$$J(0) := \int\_0^L \eta \rho(\mathbf{x}) \mu\_0(\mathbf{x}) (L - \mathbf{x})^{\frac{\beta}{2} - 1} d\mathbf{x} > 0,$$

*then*

$$T\_0 \le T\_{\max} \le M(a, a) < \infty,$$

*where M*(*a*, *α*) *is given by* (11) *(with θ* = *α) and*

$$a = 2\frac{\Gamma\left(1 + \frac{\beta}{2}\right)}{\Gamma\left(1 - \frac{\beta}{2}\right)} L^{\frac{-\beta}{2}} \left(\int\_0^L (L - x)^{\beta - 1} \eta\_0(x) \, dx\right)^{-1}.$$

#### **5. Conclusions**

A fractional in time and space shallow-water system is investigated in this paper. The considered fractional derivative depends of a function *<sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1([0, <sup>∞</sup>)), and generalizes Caputo fractional derivative, which corresponds to the case *ψ*(*t*) = *t*. Using the test function method, it is shown that under certain conditions imposed on the initial data, the system admits no global in time solutions. Furthermore, an upper bound of the lifespan is obtained.

**Author Contributions:** Investigation, M.J., M.K. and B.S. M.J., M.K. and B.S. contributed equally to this work.

**Funding:** M. Jleli is supported by Researchers Supporting Project number (RSP-2019/57), King Saud University, Riyadh, Saudi Arabia. M. Kirane is supported by "RUDN University program 5-100".

**Conflicts of Interest:** The authors declare no conflict of interest.
