Blow up, growth, and decay of solutions for a class of coupled nonlinear viscoelastic Kirchhoff equations with distributed delay and variable exponents

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, distributed delay


Introduction
Our understanding of real-world phenomena and our technology today are largely based on mathematical analysis for partial differential equations (PDEs) [1,2,4,5].This mathematical analysis helps us to visualize and understand different real-world problems [7,8,10,11].The mathematical analysis study of PDEs has also taught us to show a little modesty: we have discovered the impossibility of predicting certain phenomena governed by nonlinear PDEs in the medium term-think of the now famous butterfly effect: a small variation of the initial conditions can lead to very large variations in very long time.On the other hand, we have also learned to "hear the shape of a drum": it has been shown mathematically that the frequencies emitted by a drum during membrane vibration-a phenomenon described by a PDE-allow the drum shape to be perfectly reconstructed.One of the things to keep in mind about PDEs is that you usually do not want to get their solutions explicitly!What mathematics can do, on the other hand, is to say whether one or more solutions exist, and sometimes to very precisely describe certain properties of these solutions.However, the emergence of extremely powerful computers today makes it possible to obtain approximate solutions for partial derivative equations, even very complicated.This is what happens, for example, when you look at the weather forecast, or when we see the moving images of a simulation of airflow on the wing of airplane.The role of mathematicians is then to build approximation schemes and to demonstrate the relevance of the simulations by establishing a priori estimates on the made errors.When did EDP appear?They likely originated in the early days of rational mechanics in the seventeenth century, with figures like Newton and Leibniz playing crucial roles.As scientific disciplines, especially physics, advanced in energy functional, fluid mechanics equations, Navier-Stokes equations, where they contributed to the expansion of partial differential equations (PDEs).
To highlight a few key contributors, Euler's name stands out, as well as Navier and Stokes for fluid mechanics equations, Fourier for heat equations, Maxwell for electromagnetism equations, and Schrödinger, Heisenberg, and Einstein for quantum mechanics and the theory of relativity PDEs, respectively (see e.g.[1,6,9] and the references therein).Nevertheless, the systematic examination of partial differential equations (PDEs) is relatively recent, with mathematicians embarking on this endeavor only in the twentieth century.A significant leap occurred with Schwartz's formulation of the theory of distributions in the 1950s, and comparable progress emerged through Hörmander's work on pseudodifferential calculus in the early 1970s.Importantly, the study of PDEs remains highly active as we progress into the twenty-first century [12][13][14][15][16]. Mathematics serves as a potent tool in both scientific inquiry and engineering applications, enabling precise modeling, analysis, and solution exploration of complex mathematical systems fundamental to advancing our understanding of the natural world and optimizing technological innovations [17][18][19][21][22][23].This research not only influences applied sciences but also plays a crucial role in the ongoing evolution of mathematics itself, particularly in the domains of geometry and analysis.In this work, the following problem is addressed: in which η ≥ 0 for N = 1, 2 and 0 < η ≤ 2 N-2 for N ≥ 3, and h i (.) : R + → R + (i = 1, 2) represents positive relaxation functions, which will be specified later.The term -(.)tt denotes the dispersion term, and M(σ ) is a nonnegative locally Lipschitz function for γ , σ ≥ 0 such that M(σ ) = α1 + α 2 σ γ .Specifically, we choose α 1 = α 2 = 1, and In this context, we consider nonnegative constants τ 1 < τ 2 such that βi : [τ 1, τ 2 ] → R, where i = 2, 4 represents the time delay in the distributive case.Furthermore, q(.), m(.), and s(.) are variable exponents defined as measurable functions on in the following manner: where with This research is organized into distinct sections.In the following section, we present the hypotheses, concepts, and lemmas essential for our study.Section 2 is dedicated to proving the blow-up result, followed by the derivation of exponential growth of solutions.In Sect.4, we establish the general decay when f 1 = f 2 = 0.The paper concludes with a comprehensive summary in the final section.

Fundamental theory
The importance of studying the blow-up of solutions in various systems lies in its ability to reveal critical thresholds, instabilities, and singularities that can significantly impact the behavior and evolution of dynamic processes [27][28][29][30].Here, we will present some related theory and will define suitable assumptions for the proof of blow-up result.(A1) Take a decreasing and differentiable function h i : R + → R + in a manner that a 1 |v + w| 2(q(y)+2) + 2b 1 |vw| q(y)+2 ≥ 0, in which Here, consider a 1 = b 1 = 1 for convenience.
In the upcoming step, the energy functional is introduced.

Blow-up
Here, we establish the blow-up result for the solution of (2.8).Initially, we introduce the functional as follows: 3) hold and assume E(0) < 0, then the solution of (2.8) blows up in finite time.
Lemma 3.2 Let ∃c > 0 in a way that any solution of (2.8) we have Hence, we get According to (3.5), we have Hence Similarly, we find The adding of (3.11) and (3.12) gives us (3.6).

Growth of solution
Here, the exponential growth of solution of problem (2.8) will be established.For this, the functional is defined as follows: 3) are satisfied, and suppose E(0) < 0, then Then the solution of problem (2.8) grows exponentially.
Therefore, we deduce that the solution experiences exponential growth in the L 2(p + +2) norm.This concludes the proof.

Conclusion
In this research, we investigated the blow-up and growth of solutions in a coupled nonlinear viscoelastic Kirchhoff-type system with sources, distributed delay, and variable exponents.Additionally, we obtained a general decay result when f 1 = f 2 = 0 by leveraging an integral inequality introduced by Komornik [20].Such problems are commonly encountered in various mathematical models of real-world problems.In future research, we plan to apply this approach to address similar problems, incorporating additional damping effects such as Balakrishnan-Taylor damping and logarithmic terms.We will also try to prove the general decay result in the case (f 1 , f 2 = 0).
which 0 < λ, this relies only on β and c.Further simplification of (4.31) leads to