This paper investigates the application of the Hardy-Littlewood-Sobolev inequality to analyze the global existence of solutions for a wave equation incorporating distributed delay effects and Hartree-type nonlinearities under suitable analytical conditions.
The analysis of solutions to nonlinear systems is of great importance due to their widespread applications and complex dynamic behavior [1– 4]. Wave equations with nonlocal nonlinearities play a crucial role in describing various physical phenomena, including quantum mechanics, plasma physics, and nonlinear optics [5– 7]. When delay and fractional effects are incorporated, these models become even more realistic, capturing memory and hereditary characteristics of complex systems [8, 9]. However, the presence of distributed delay and fractional operators significantly complicates the analysis of global existence and regularity of solutions. In this work, we address these challenges by employing the Hardy-Littlewood-Sobolev (HLS) inequality, a powerful tool in harmonic analysis that enables the control of nonlocal nonlinear terms through integral estimates. By applying the HLS inequality under suitable analytical conditions, we establish the global existence of weak solutions for a wave equation of Hartree type with distributed delay governed by a fractional condition. The results presented here extend and generalize several existing works on wave equations with nonlocal and delayed interactions, providing deeper insights into the interplay between fractional dynamics and nonlocal nonlinear effects.
In this paper, we considered the following problem of Hartree-type wave equation: \[\left\{ \begin{array}{ll} \vert \mathcal{U}_{t}\vert^{l} \mathcal{U}_{tt}-\Delta \mathcal{U}+b_{1}\mathcal{U}_{t}+\displaystyle\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}b_{2}(\varkappa)\partial_{t}^{\alpha,\delta} \mathcal{U}\left( x, t-\varkappa\right)d\varkappa=\mathbb{F}(\mathcal{U}), & 0<t ~ \text{and} ~x\in \mho, \\ \mathcal{U}(x,t)=0, & 0<t ~\text{and}~ x\in \partial\mho, \\ \mathcal{U}(x,0)=\mathcal{U}_{0}(x),\quad \mathcal{U}_{t}(x,0)=\mathcal{U}_{1}(x), & x\in \mho ,\\ \mathcal{U}_{t}( x,-t) =f_{0}( x,t),& t\in (0, \mathcal{N}_{2}) ~\text{and} ~x\in\mho. \end{array} \right. \label{sys1} \tag{1}\]
The function \(\mathbb{F}\) is defined as: \[\mathbb{F}(\mathcal{U}):=\bigg(\frac{1}{\vert x\vert^{n-2}}\ast\vert \mathcal{U}\vert^{p}\bigg)|\mathcal{U}|^{p-2}\mathcal{U},\] here \[\frac{1}{\vert x\vert^{n-2}}\ast\vert \mathcal{U}\vert^{p}=\int_{\mho}\frac{\vert \mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dy,\] where \(\mathbb{F}(\mathcal{U})\) represents the Hartree term and is given in the general case as follows: \(\bigg(\frac{1}{\vert x\vert}\ast\vert \mathcal{U}\vert^{2}\bigg)u\). Moreover, \(\mho \subseteq\mathbb{R}^{n}\), \(5\leq n\) with a sufficiently smooth boundary \(\partial \mho\) of class \(\mathcal{C}^{2}\). \(b_{1},l>0\), \(\ 1<p<\frac{n+4}{n-4}\), and \(b_{2}\in L^{\infty}\), \(0<\mathcal{N}_{1}\leq \mathcal{N}_{2}\). Also, the term \(\partial _{t}^{\beta ,\delta }\) refers to Caputo fractional derivative, with \(1>\beta >0\) [10, 11], It is defined as: \[\partial _{t}^{\beta ,\delta }\mathcal{U}(t)=\frac{1}{\Gamma (1-\beta )} \int_{0}^{t}(t-s)^{-\beta }e^{-\delta (t-s)}\mathcal{U}_{s}(s)ds,\quad 0 \leq \delta,\] where \[\partial _{t}^{\beta ,\delta }\mathcal{U}(t)=I^{1-\beta ,\delta }\mathcal{U}_{t}(t), \label{1.2} \tag{2}\] and \[I^{\beta ,\delta }\mathcal{U}(t)=\frac{1}{\Gamma (\beta )}\int_{0}^{t}(t-s)^{\beta -1}e^{-\delta (t-s)}\mathcal{U}(s)ds,\quad 0 \leq \delta,\] where \(I^{\beta,\delta }\) is the exponential fractional integro-differential and \(\Gamma\) Euler gamma function.
Hartree-type equations are fundamental in physics, mathematics, and approximate quantum mechanics. Because they describe a quantum particle interacting with its own field, they are widely used in models involving a large number of particles [12, 13]. In physical applications, these equations are employed to study the structure of many-electron atoms and appear in the modeling of plasmas, electron gases, and quantum astrophysical systems to describe stable configurations. They are also utilized in condensed matter physics. From a mathematical perspective, Hartree-type equations arise in nonlocal quasilinear problems and are used to investigate stability properties, blow-up phenomena, and the existence and uniqueness of solutions.
Also, our Eq. (1) is a type of the below equation: \[-\Delta u+V(x)u=\bigg(\frac{1}{\vert x\vert^{v}}\ast\vert u\vert^{p}\bigg)|u|^{p-2}u,\label{abc01} \tag{3}\] where \(\frac{2n-v}{n}\leq p\leq\frac{2n-v}{n-2}.\)
The expression (3), in the special case \((n, p) = (1, 2)\), corresponds to the helium atom; see [14] for more details. Picard [15] (1976) also interpreted this equation as describing the quantum theory of stationary polarons, with related discussions found in the works of Choquard and Lieb [16]. For other values of \(n\) and \(p\), we refer the reader to [17] and the studies of Petrovsky. Such problems and equations naturally arise in various branches of science and physics, including acoustics, optics, and other related fields.
In recent years, many research papers have addressed this type of problem, including [13]. Here, in the present study, the researchers discuss the below equation: \[u_{tt}-\Delta u=\bigg(\frac{1}{\vert x\vert^{n-2}}\ast\vert u\vert^{p}\bigg)|u|^{p-2}u.\label{abc02} \tag{4}\]
They established the well-posedness of the problem using semigroup theory and proved the global existence of solutions. In addition, they derived blow-up results for both nonnegative and negative initial energy cases. Building upon these and related works, we incorporate fractional conditions together with distributed delay effects, leading to a novel and distinct contribution compared to previous studies. This short paper is organized as follows: in the next section, we present the fundamental concepts and key inequalities employed in the analysis, followed by the definition of the corresponding energy functional. §3 is devoted to establishing the main result on the global existence of solutions. Finally, we conclude with several remarks and suggestions for future research.
This section presents several lemmas, fundamental concepts, and inequalities that will be used in further analysis.
Lemma 1. [18] Let \(1\leq p\leq +\infty\) \(\left( N=1,2\right)\) or \(1\leq p\leq \frac{N+2}{N-2}\), \(\left( 3\leq N\right)\), then, there exists \(0 < \mathcal{C}_{*}=B_{p,\mho}\) in a way that \[\Vert \mathcal{U}\Vert _{p+1}\leq \mathcal{C}_{*}\Vert \nabla \mathcal{U}\Vert _{2}, \ \ \forall \mathcal{U}\in H_{0}^{1}(\mho ).\]
The Hardy-Littlewood-Sobolev inequality is a fundamental result in mathematical analysis, partial differential equations (PDEs), and mathematical physics, as it provides an estimate for a nonlocal double integral. In the following lemma, we present the complete statement of the inequality, along with a reference for readers seeking a detailed proof and further discussion.
Lemma 2. [Hardy-Littlewood-Sobolev inequality [19, 20]] Suppose that \(\sigma; p > 1\) and \(n>\vartheta>0\) with \(\frac{1}{\sigma} +\frac{\vartheta}{n}+ \frac{1}{p}= 2\); \(f \in L^{\sigma} (\mathcal{R}^{n})\) and \(h \in L^{s} (\mathcal{R}^{n})\). Thus there exists a constant \(C (\sigma; n;\vartheta; s)\); independent of \(f, h\) so that for all \(\mathcal{U}\in H^{1}(\mathcal{R}^{n})\), we have \[\int_{\mathcal{R}^{n}}\int_{\mathcal{R}^{n}}\frac{f(x)h(y)}{\vert x-y\vert^{\vartheta}}dxdy\leq \mathcal{C} (\sigma, n,\vartheta, s)\Vert f\Vert_{\sigma}\Vert h\Vert_{p}. \tag{5}\]
Also, \(\int_{\mathcal{R}^{n}}\int_{\mathcal{R}^{n}}\frac{\vert \mathcal{U}(x)\vert^{p}\vert \mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dxdy\) with \(\frac{2n-\vartheta}{n}\leq p\leq \frac{2n-\vartheta}{n-2}\). Furthermore, from Lemma 5, for \(\mathcal{U}\in H^{1}_{0}(\mho)\), we deduce \(\mathcal{U}(x)=0\) by \(x\in \mathcal{R}^{n}/\mho\). Therefore \(\mathcal{U}\in H^{2}(\mathcal{R}^{n})\); i.e. for a general field, for \(\mathcal{U} \in L^{\sigma} (\mathcal{R}^{n})\) and \(\varrho \in L^{s} (\mathcal{R}^{n})\), we find the Hardy-Littlewood-Sobolev inequality as: \[\int_{\mho}\int_{\mho}\frac{\mathcal{U}(x)\varrho (y)}{\vert x-y\vert^{\vartheta}}dxdy\leq C (\sigma, n,\vartheta, s,\mho)\Vert \mathcal{U}\Vert_{\sigma}\Vert \varrho \Vert_{p},\label{a4} \tag{6}\] and the integral if \(\frac{2n-\vartheta}{n}\leq p\leq \frac{2n-\vartheta}{n-2}\) then for \(\mathcal{U}\in H^{1}_{0}(\mho)\), \(\int_{\mho}\int_{\mho}\frac{\vert \mathcal{U}(x)\vert^{p} \vert\varrho (y)\vert^{p}}{\vert x-y\vert^{\vartheta}}dxdy\) is well defined.
Hence, for \(\mathcal{U}\in H^{1}_{0}(\mho)\), applying the Sobolev embedding theorem and the inequality given in Eq. (6), gives \[\int_{\mho}\int_{\mho}\frac{\vert \mathcal{U}(x)\vert^{p} \vert \mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dxdy\leq \mathcal{C}_{1} (n,p,\mho)\Vert\mathcal{U}\Vert_{\frac{2np}{n+2}}^{2p}\leq \mathcal{C}_{2} \Vert \nabla \mathcal{U}\Vert^{2p},\label{a5} \tag{7}\] here \(\mathcal{C}_{2}=\mathcal{C}_{1} (n,p,\mho)c_{*}^{2p}\), where \(\mathcal{C}_{1}\) and \(c_{*}\) are the Hardy-Littlewood-Sobolev and Sobolev embedding constants respectively. Next, by applying the Sobolev embedding theorem, \[\frac{n+2}{n-2}>p>\frac{n+2}{n}.\label{a6}\]
Theorem 1. [8] Consider the mapping \(\vartheta\) given by \[\vartheta (\zeta )=|\zeta |^{\frac{(2\beta -1)}{2}},\quad 1>\beta >0, \ \zeta \in\mathcal{R}. \label{2.13} \tag{8}\]
Thus, we obtained \[O=UI^{1-\beta ,\delta }, \label{2.14} \tag{9}\] where it achieves \[\partial _{t}z (x,\varkappa,\zeta ,t)+(\zeta ^{2}+\delta )z (x,\varkappa,\zeta ,t)-U(x,\varkappa,t)\vartheta (\zeta )=0, \ \ 0 \leq \delta, 0<t, \text{ }\zeta \in\mathcal{R}, \label{2.15} \tag{10}\] \[z (x,\varkappa,\zeta ,0)=0,\label{2.16} \tag{11}\] \[O(x,\varkappa,t)=\frac{\sin {(\beta \pi )}}{\pi}\int_{-\infty }^{+\infty }z (x,\varkappa,\zeta ,t)\vartheta (\zeta )d\zeta,\text{ } \zeta \in \mathcal{R},\varkappa\in [\mathcal{N}_{1}, \mathcal{N}_{2}] ,\ t>0. \label{2.17} \tag{12}\]
Lemma 3. [21] \(\forall\eta\in D_{\delta}:=\{\eta\in\mathfrak{C} : \Re e\eta+\delta>0\}\cup\{\eta\in\mathfrak{C} : Im\eta\neq 0\}\), \[A_{\eta}=\int_{-\infty}^{+\infty}\frac{\vartheta^{2}(\zeta)}{\eta+\delta+\zeta^{2}}d\zeta=\frac{\pi}{\sin (\beta\pi)}(\eta+\delta)^{\beta-1}.\label{A2.0} \tag{13}\]
As in [9], we take the new variable as: \[\theta(x,\mu,\varkappa,t)=\mathcal{U}_{t}(x,t-\varkappa\mu),\] in which \[(x,\mu,\varkappa,t)\in\mathcal{D}:= \mho\times(0, 1)\times (\mathcal{N}_{1},\mathcal{N}_{2})\times \mathcal{R}_{+},\] that satisfies \[\left\{ \begin{array}{l} \varkappa \theta_{t}(x,\mu,\varkappa,t)+\theta_{\mu}(x,\mu,\varkappa,t)=0\\ \theta(x,0,\varkappa,t)=\mathcal{U}_{t}(x,t). \end{array} \right.\label{m1} \tag{14}\]
Consequently, by theorem 1 and (2) we obtained \[\left\{ \begin{array}{ll} \vert \mathcal{U}_{t}\vert^{l} \mathcal{U}_{tt}-\Delta \mathcal{U}+b_{1}\mathcal{U}_{t}+a\displaystyle\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }b_{2}(\varkappa)z (x,\zeta,\varkappa ,t)\vartheta (\zeta )d\zeta d\varkappa=\mathbb{F}(\mathcal{U}) , & \\ \partial _{t}z (x,\zeta,\varkappa,t)+(\zeta ^{2}+\delta )z (x,\zeta,\varkappa,t)-\theta(x,1,\varkappa,t)\vartheta (\zeta )=0, & \\ \varkappa \theta_{t}(x,\mu,\varkappa,t)+\theta_{\mu}(x,\mu,\varkappa,t)=0,&\\ \mathcal{U}(x,t)=0, \hspace{1cm}x\in \partial\mho, & \\ \mathcal{U}(x,0)=\mathcal{U}_{0}(x),\quad \mathcal{U}_{t}(x,0)=\mathcal{U}_{1}(x), & \\ \theta(x,\mu,\varkappa,0)=f_{0}(x,\varkappa\mu),\hspace{0.2cm} \varkappa \in (0, \mathcal{N}_{2}),& \end{array} \right. \label{system1} \tag{15}\] where \[\zeta \in \mathbb{R} ~~\text{and}~~(x,\mu,\varkappa,t) \in \mathcal{D}\hspace{0.3cm}\text{with} \hspace{0.3cm} a=\frac{\sin(\beta\pi)}{\pi}.\]
We also introduce the following hypothesis about the distributed delay function \(b_{2}\), which helps us in the rest of the proofs:
(H1) \(b_{2}:[\mathcal{N}_{1}, \mathcal{N}_{2}]\rightarrow\mathbb{R}\) is a bounded function which holds \[b_{1}>2aA_{0}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\vert b_{2}(\varkappa)\vert d\varkappa.\label{A2} \tag{16}\]
Next, we introduce \(E\), the energy functional of the system (15), along with its definition and proof.
Lemma 4. Consider the solution of the above system (15) is \((\mathcal{U},z,\theta)\), then the energy functional is given by the following \[\begin{aligned} E(t)=&\frac{1}{l+2}\Arrowvert \mathcal{U}_{t}\Arrowvert_{l+2}^{l+2}+\frac{1}{2} \Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2}-\frac{1}{2p}\int_{\mho}\int_{\mho}\frac{\vert \mathcal{U}(x)\vert^{p} \vert \mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dxdy\notag\\ &+\frac{a}{2}\int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }\vert b_{2}(\varkappa)\vert|z(x,\zeta ,\varkappa,t)|^{2}d\zeta d\varkappa dx\notag\\ &+aA_{0}\int_{\mho}\int_{0}^{1}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\varkappa\vert b_{2}(\varkappa)\vert \vert \theta\left( x, \mu,\varkappa, t\right)\vert^{2} d\varkappa d\mu dx, \label{2.19} \end{aligned} \tag{17}\] satisfies \[E'(t)\leq -\mathcal{C}_{0}\Vert \mathcal{U}_{t}\Vert_{2}^{2}-\frac{a}{2}\int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty}\vert b_{2}(\varkappa)\vert(\zeta^{2}+\delta )\vert z(x,\zeta ,\varkappa,t)\vert^{2}d\zeta d\varkappa dx \leq 0, \label{2.20} \tag{18}\] in which \[\mathcal{C}_{0}=b_{1}-2aA_{0}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\vert b_{2}(\varkappa)\vert d\varkappa>0.\]
This section is devoted to proving the theorem on the global existence of solutions for the problem (15). We follow well-established methods from the literature, incorporating some essential modifications. To establish the boundedness of the solution, we first define the following functions: \[\mathfrak{J}(t)=\frac{1}{2} \Arrowvert\nabla \mathcal{U}% \Arrowvert_{2}^{2}-\frac{1}{2p}\int_{\mho}\int_{\mho}\frac{\vert \mathcal{U}(x)\vert^{p} \vert \mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dxdy,\label{v00} \tag{19}\] and \[\mathfrak{I}(t)=\Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2}-\int_{\mho}\int_{\mho}\frac{\vert \mathcal{U}(x)\vert^{p} \vert \mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dxdy.\label{v0} \tag{20}\]
Using the last definitions, we get \[\mathfrak{J}(t)=\frac{1}{2p}I\left(t \right)+\left( \frac{p-1}{2p}\right) \Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2},\label{3.3} \tag{21}\] and \[\begin{aligned} \label{v000} E(t)=&\frac{1}{l+2}\Arrowvert \mathcal{U}_{t}\Arrowvert_{l+2}^{l+2}+\frac{a}{2}\int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }\vert b_{2}(\varkappa)\vert|z(x,\zeta ,\varkappa,t)|^{2}d\zeta d\varkappa dx\notag\\ &+bA_{0}\int_{\mho}\int_{0}^{1}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\varkappa\vert b_{2}(\varkappa)\vert \vert \theta\left( x, \mu,\varkappa, t\right)\vert^{2} d\varkappa d\mu dx+\mathfrak{J}(t)\notag \\=&\frac{1}{l+2}\Arrowvert \mathcal{U}_{t}\Arrowvert_{l+2}^{l+2}+\left( \frac{p-1}{2p}\right) \Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2}+\frac{1}{2p}\mathfrak{I}\left(t \right)\notag\\ &+\frac{a}{2}\int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }\vert b_{2}(\varkappa)\vert|z(x,\zeta ,\varkappa,t)|^{2}d\zeta d\varkappa dx\notag\\ &+aA_{0}\int_{\mho}\int_{0}^{1}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\varkappa\vert b_{2}(\varkappa)\vert \vert \theta\left( x, \mu,\varkappa, t\right)\vert^{2} d\varkappa d\mu dx. \end{aligned} \tag{22}\]
Lemma 5. Assume that \(V_{0}=(\mathcal{U}_{0},\mathcal{U}_{1},z_{0},f_{0})^{T} \in \mathcal{H}\) which hold \(0<I(0)\) \(\forall~ t\in[0, T ]\), then \[\varsigma:=\mathcal{C}_{2}\bigg(\frac{2p E(0)}{( p-1)}\bigg)^{p-1}<1.\label{G1} \tag{23}\]
Proof. From \(0<\mathfrak{I}(0)\), we get that \(\exists\) \(T\geq T^{*}> 0\) such that \(0\leq\mathfrak{I}(t)\) \(\forall\) \(t\in[0, T^{*}]\). Also, we define \[\begin{aligned} \mathfrak{J}(t)=&\frac{1}{2p}\mathfrak{I}\left(t \right)+\left( \frac{p-1}{2p}\right) \Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2}\notag \\ \geq&\left( \frac{p-1}{2p}\right) \Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2}\hspace{0.3cm}\forall t\in[0, T^{*}].\label{G3} \end{aligned} \tag{24}\]
Then, by (22), we find \[\begin{aligned} \Vert \nabla \mathcal{U} \Vert_{2}^{2}\leq&\frac{2p}{p-1}\mathfrak{J}(t)\leq\frac{2p}{p-1}E(t)\leq\frac{2p}{p-1}E(0),\hspace{0.2cm}\forall t\in [0, T^{*}] .\label{v1} \end{aligned} \tag{25}\]
The embedding \(H^{1}_{0}(\mho))\hookrightarrow L^{2p}(\mho)\) and (7) gives \[\begin{aligned} \int_{\mho}\int_{\mho}\frac{\vert \mathcal{U}(x)\vert^{p} \vert \mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dxdy \leq \mathcal{C}_{2}\Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2p}. \end{aligned} \tag{26}\]
From Eq. (25), we have \[\begin{aligned} \int_{\mho}\int_{\mho}\frac{\vert \mathcal{U}(x)\vert^{p} \vert\mathcal{U}(y)\vert^{p}}{\vert x-y\vert^{n-2}}dxdy<\mathcal{C}_{2}\bigg(\frac{2p E(0)}{( p-1)}\bigg)^{p-1}\Vert\nabla \mathcal{U}(t)\Vert_{2}^{2}<\varsigma\Vert\nabla \mathcal{U}(t)\Vert_{2}^{2},\label{G2} \end{aligned} \tag{27}\] here \[\varsigma:=\mathcal{C}_{2}\bigg(\frac{2p E(0)}{( p-1)}\bigg)^{p-1}.\]
The following inequality get from Eq. (20), Eq. (23) and Eq. (27): \[\begin{aligned} \mathfrak{I}(t)>& (1-\varsigma)\Vert\nabla \mathcal{U}(t)\Vert_{2}^{2}> 0, \hspace{0.5cm}\forall t\in[0,T^{*}]. \end{aligned} \tag{28}\]
This process can be repeated to extend \(T^{*}\) to \(T\). ◻
Remark 1. According to the suppositions of Lemma 5 and \(0\leq E(t), \mathfrak{J}(t)\), for all \(t\in [0,T]\). Then, from Eqs. (22) and (24) the following results hold \[\begin{aligned} \Vert \mathcal{U}_{t}(t)\Vert_{l+2}^{l+2}\leq& (l+2)E(0),\notag\\ \int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }\vert b_{2}(\varkappa)\vert|z (x,\zeta ,\varkappa,t)|^{2}d\zeta d\varkappa dx\leq&\frac{2}{a} E(0),\notag\\ \int_{\mho}\int_{0}^{1}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\varkappa\vert b_{2}(\varkappa)\vert \vert \theta\left( x, \mu,\varkappa, t\right)\vert^{2} d\varkappa d\mu dx\leq& \frac{1}{aA_{0}}E(0).\label{Remark} \end{aligned} \tag{29}\]
Theorem 2. Assume that the assumptions of Lemma 5 are satisfied. Then, the solution of system (15) exists globally in time and remains bounded.
Proof. To achieve our goal, we prove the following: \[\begin{aligned} \Vert (\mathcal{U},z,\theta)\Vert_{H}:=&\Vert \mathcal{U}_{t}\Vert^{l+2}_{l+2}+\Vert\nabla \mathcal{U}\Vert^{2}_{2}+\int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }\vert b_{2}(\varkappa)\vert|z (x,\zeta ,\varkappa,t)|^{2}d\zeta d\varkappa dx\notag\\ &+\int_{\mho}\int_{0}^{1}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\varkappa\vert b_{2}(\varkappa)\vert \vert \theta\left( x, \mu,\varkappa, t\right)\vert^{2} d\varkappa d\mu dx. \end{aligned}\]
The above equation is independently bounded with respect to time \(t\). By using the Eq. (29) to find \[\begin{aligned} E(0)>&E(t)=\mathfrak{J}(t)+\frac{1}{l+2}\Arrowvert\mathcal{U}_{t}\Arrowvert_{l+2}^{l+2}+\frac{a}{2}\int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }\vert b_{2}(\varkappa)\vert|z (x,\zeta ,\varkappa,t)|^{2}d\zeta d\varkappa dx\notag\\ &+aA_{0}\int_{\mho}\int_{0}^{1}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\varkappa\vert b_{2}(\varkappa)\vert \vert \theta\left( x, \mu,\varkappa, t\right)\vert^{2} d\varkappa d\mu dx\notag \\\geq&\frac{1}{l+2}\Arrowvert\mathcal{U}_{t}\Arrowvert_{l+2}^{l+2}+\left( \frac{p-1}{2p}\right) \Arrowvert\nabla \mathcal{U}\Arrowvert_{2}^{2}+\frac{a}{2}\int_{\mho}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\int_{-\infty }^{+\infty }\vert b_{2}(\varkappa)\vert|z (x,\zeta ,\varkappa,t)|^{2}d\zeta d\varkappa dx\notag\\ &+aA_{0}\int_{\mho}\int_{0}^{1}\int_{\mathcal{N}_{1}}^{\mathcal{N}_{2}}\varkappa\vert b_{2}(\varkappa)\vert \vert \theta\left( x, \mu,\varkappa, t\right)\vert^{2} d\varkappa d\mu dx. \end{aligned} \tag{30}\]
Therefore, \[\Vert (\mathcal{U},z,\theta)\Vert_{H}\leq \mathcal{C}E(0),\] where \(\mathcal{C}(l,p,a,A_{0})>0\). ◻
In this work, by combining distributed delay effects with a fractional condition, we have derived new results for a nonlinear wave equation of Hartree type. These findings establish the global existence of solutions through a rigorous analysis of the associated energy functional under suitable analytical assumptions. Moreover, they extend and generalize several previously known results in the literature. As a continuation of this study, we intend to examine related problems by incorporating additional damping terms within a similar analytical framework.
Boulaaras, S., Jan, R., Choucha, A., Zeraï, A., & Benzahi, M. (2024). Blow-up and lifespan of solutions for elastic membrane equation with distributed delay and logarithmic nonlinearity. Boundary Value Problems, 36.
Boulaaras, S., Choucha, A., Ouchenane, D., Alharbi, A., & Abdalla, M. (2022). Result of local existence of solutions of nonlocal viscoelastic system with respect to the nonlinearity of source terms. Fractals, 30(01), 2240027.
Choucha, A., Boulaaras, S., Ouchenane, D., Alharbi, A., & Abdalla, M. (2022). Result of local existence of solutions of nonlocal viscoelastic system with respect to the nonlinearity of source terms. Fractals, 30(01), 2240027.
Choucha, A., Boulaaras, S., Jan, R., & Alharbi, R. (2024). Blow-up and decay of solutions for a viscoelastic Kirchhoff-type equation with distributed delay and variable exponents. Mathematical Methods in the Applied Sciences, 47(8), 6928-6945.
Ha, T. G., & Park, S. H. (2020). Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity. Advances in Difference Equations, 2020, 235.
Kafini, M., & Messaoudi, S. A. (2018). Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay. Applicable Analysis, 99(3), 530-547.
Kirane, M., & Tatar, N.-E. (2003). Exponential growth for fractionally damped wave equation. Zeitschrift für Analysis und ihre Anwendungen, 22, 167-178.
Mbodje, B. (2006). Wave energy decay under fractional derivative controls. IMA Journal of Mathematical Control and Information, 23, 237-257.
Nicaise, S., & Pignotti, C. (2008). Stabilization of the wave equation with boundary or internal distributed delay. Differential and Integral Equations, 21(9-10), 935-958.
Aounallah, R., Choucha, A., Boulaaras, S., & Zarai, A. (2024). Asymptotic behavior of a viscoelastic wave equation with a delay in internal fractional feedback. Archives of Control Sciences, 34(2), 379-413.
Dai, H., & Zhang, H. (2014). Exponential growth for wave equation with fractional boundary dissipation and boundary source term. Boundary Value Problems, 2014, 138.
Fidan, A., Piskin, E., Adam Saad, S. A., Alkhalifa, L., & Abdalla, M. (2025). Existence and blow up of solution for a Petrovsky equations of Hartree type. Discrete and Continuous Dynamical Systems – Series S.
Zhang, H., & Su, X. (2022). Initial boundary value problem for a class of wave equations of Hartree type. Studies in Applied Mathematics, 149, 798-814.
Menzala, G. P., & Strauss, W. A. (1982). On a wave equation with a cubic convolutions. Journal of Differential Equations, 43, 93-105.
Pekar, S. I. (1954). Untersuchungen Üuber die Elektronentheorie der Kristalle. De Gruyter.
Lieb, E. H. (1977). Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies in Applied Mathematics, 57, 93-105.
Magin, R. L. (2006). Fractional Calculus in Bioengineering. Begell House Publishers, Redding.
Adams, R. A., & Fournier, J. J. F. (2003). Sobolev Spaces. Academic Press.
Lieb, E., & Loss, M. (1997). Analysis. American Mathematical Society, Graduate Studies in Mathematics.
Benaissa, A., & Benkhedda, H. (2018). Global existence and energy decay of solutions to a wave equation with a dynamic boundary dissipation of fractional derivative type. Zeitschrift für Analysis und ihre Anwendungen, 37, 315-339.
