The study of nonlinear boundary conditions has opened new avenues in the analysis of pseu-doparabolic and pseudohyperbolic equations, revealing complex dynamics that were previously overlooked in the linear framework. Researchers have begun to investigate the impact of these nonlinearities on the existence, uniqueness, and regularity of solutions. Notably, the work [5] demonstrates that nonlinearities can significantly influence the qualitative behavior of solutions, leading to phenomena such as blow-up or the loss of smoothness under certain conditions. Fur-thermore, the exploration of mixed boundary conditions has gained traction, prompting a reeval-uation of classical approaches. In particular, cases where Dirichlet and Neumann conditions interplay in nonlinear contexts have raised intriguing questions about stability and bifurcation. The coupling of these boundary conditions forces a deeper understanding of the interaction be-tween the solutions and the domain’s geometry, leading to a richer theoretical framework. In light of these developments, ongoing research continues to address the numerical aspects of these equations. The implementation of advanced computational techniques is essential for capturing the intricate patterns that arise from nonlinear boundary conditions. Approximative methods and simulations serve as crucial tools for validating theoretical predictions and for exploring parameter spaces where traditional analytical techniques falter. Ultimately, as the field evolves, addressing the challenges posed by nonlinear Dirichlet and Neumann boundary conditions will remain a central focus. The implications of these studies extend beyond pure mathematics, in-fluencing applied domains such as fluid dynamics, materials science, and biological systems, underscoring the interdisciplinary nature of current mathematical research.

To begin our investigation, we define the mixed problem more rigorously, specifying the boundary conditions that govern the behavior of our system. The presence of third-order non-linearity introduces complexities that warrant careful handling, particularly in terms of the reg-ularity and continuity of solutions. We identify an appropriate function space, incorporating both spatial and temporal dimensions, to rigorously analyze the implications of our boundary constraints. Using Lemma Levine, we establish a framework for evaluating the growth rates of solutions in association with their derivatives. This framework enables us to express the con-ditions under which the solutions may exhibit blow-up phenomena. By examining the derived inequalities, we ascertain the relationship between initial conditions and the eventual behavior of the solutions. Further, we explore the implications of various parameter configurations, show-casing how changes in nonlinearity influence the solution’s stability. The resultant analysis not only sheds light on critical thresholds that necessitate consideration but also assists in estab-lishing a more comprehensive understanding of the dynamics involved. Ultimately, the findings underscore the delicate interplay between initial conditions and mixed boundary conditions in exacerbating or mitigating blow-up scenarios.

Consider the following problem

utt − Di(|Diu| Diu) − α∆ut + f(u) = 0, (x, t) ∈ Ω × [0, T] (2.1)

u(x, 0) = u0(x), ut(x, 0) = u0(x), x ∈ Ω, (2.2)

p−2 t

(|Diu| Diu) cos(xi, υ) + α = g(u), (x, t) ∈ ∂Ω × [0, T], (2.3)

where Ω ⊂ R , n ≥ 2 is bounded domain with smooth boundary ∂Ω, u0(x) ∈ W (Ω), u1(x) ∈ L2(Ω) are given functions, f(u) and g(u)- are some nonlinear functions, α > 0-are ∂ X ∂ ∂ some number, p ≥ 2, Di = , i = 1, 2, ..., n, ∆ = -is Laplace operator, -is deriva- ∂xi ∂x ∂n

i=1 i

tive on the external normal at ∂Ω.

Still now, problems with linear and homogeneous Dirichlet and Neumann boundary condi-tions for various pseudoparabolic and pseudohyperbolic equations have been well studied (see [1]-[8] and the literature cited therein). In recent years, results have been obtained when the boundary conditions are nonlinear (for example, in [5], the question of the destruction of solu-tions for the Aller equation was considered in the case when p = 2).

In this paper, we study the question of blow up of solutions for a nonlinear pseudohyperbolic equation forp ≥ 2. For the problem (2.1)-(2.3) it is proved the following theorem

Theorem 3.1. Let for any u ∈ R and for the functions f(t) and g(τ) satisfies following condi-tions Zu 2(2α + 1)F(u) − uf(u) ≥ 0, F(u) = f(s)ds, (3.1)

Zu ug(u) − 2(2α + 1)G(u) ≥ 0, G(u) = g(τ)dτ, (3.2)

p − 2 α ≥ . (3.3)

Then if F(u0)dx− G(u0)ds+ |Diuo| dx ≤ 0, (u0, u1) > 0 and for every solution

1 2 2

u(x, t) ∈ W (0, T; W (Ω)) W (0, T; L2(Ω)) of the problem (2.1)-(2.3), there exists t0 < ∞

2 2 2

such that  

lim ∥u(x, t)∥ + α ∥∇(x, τ)∥ dτ = ∞.

Proof. Multiply both sides of the equation by ut and integrate over the domain Ω, we get

1 d X

2 p−2

∥ut∥ − Di(|Diu| Diu)utdx − ∆ututdx + g(u)utdx = 0.

2 dt

Ω Ω Ω

Applying integration by parts to the second and third terms and taking into account condition (2.3) and notations (3.1) and (3.2), after simple transformations we obtain

Z n Z Z Z d 1 1 X

2 p 2

 ∥u ∥ + |D u| dx + F(u)dx − G(u)ds + (∇u ) dx = 0, (3.4)

t i t

dt 2 p

Ω Ω ∂Ω Ω

Let’s denote by

E(t) = ∥ut∥ + |Diu| dx + F(u)dx − G(u)ds.

Ω Ω ∂Ω

Then equality (3.4) can be we rewritten as

E(t) + ∥∇ut∥ = 0, (3.5) dt Integrating equality (3.5) over from zero to t, we obtain

E(t) − E(0) + ∥∇uτ (τ)∥ dτ = 0, (3.6)

If we take into account the value of E(t), and after moving E(0)the equality to the right side, we get

Z Z Z Zt 1 1 X

2 p 2

∥ut∥ + |Diu| dx + F(u)dx − G(u)ds + ∥∇uτ (τ)∥ dτ = M(0), (3.7)

Ω Ω ∂Ω 0

where 1 1 X E(0) = ∥u0∥ + |Diu0| dx + F(u0)dx − G(u0)ds.

Ω Ω ∂Ω

Consider the function

F(t) = ∥u(x, t)∥ + ∥∇u(x, σ)∥ dσ+(l−t) ∥∇u0∥ , where l is some sufficiently large number.

Obviously

F(t) = 2(u, ut) + 2 (∇u, ∇uτ )dτ,

F (t) = 2 ∥ut∥ + 2(u, utt) + 2(∇u, ∇uτ ).

Taking into account here that, u is the solution to problem (2.1)-(2.3), we obtain

F (t) = 2 ∥ut∥ + 2(u, utt)+

+2(u, Di(|Diu| Diu)) + 2(u, ∆ut) − 2(u, f(u)) + 2(∇u, ∇ut).

Applying integration by parts to the second and third terms and using condition (2.3), we have ′′ 2 p−2

F (t) = 2 ∥ut∥ + 2 |Diu| Diu) cos(xi, n)ds−

Z n Z

p−2 2 t

−2 |Diu| (Diu) dx + 2 uds − 2(∇u, ∇ut)−

Ω ∂Ω

2 p−2 t

−2(u, f(u) + 2(∇u, ∇ut) = 2 ∥ut∥ + 2 ( |Diu| Diu) cos(xi, n) + )uds−

−2 |Diu| dx − 2(u, f(u) = 2 ∥ut∥ +

+2 ug(u)ds − 2 uf(u)dx + 2 |Diu| dx.

∂Ω Ω Ω

Thus

F (t) = 2 ∥ut∥ + 2 ug(u)ds − 2 uf(u)dx + 2 |Diu| dx. (3.8)

∂Ω Ω Ω

Multiplying both sides of equality (3.6) by 4(2β + 1), we get

2(2β + 1) ∥ut∥ +

+ |Diu| dx − 4(2β + 1) G(u)ds + 4(2β + 1) F(u)dx+

Zt Zt +4(2β + 1) ∥∇uτ (τ)∥ dτ = 4(2β + 1)E(0)2 (∇u, ∇uτ )dτ.

Let’s rewrite this equality in the form

0 = 2(2β + 1) ∥ut∥ +

+ |Diu| dx − 4(2β + 1) G(u)ds + 4(2β + 1) F(u)dx+

+4(2β + 1) ∥∇uτ (τ)∥ dτ = 4(2β + 1)E(0).

Let’s add this equality to (3.8), after which we get

F (t) = 4(2β + 1) ∥ut∥ + 4(2β + 1) ∥∇uτ (τ)∥ dτ+

+2 (ug(u) − 2(2β + 1)G(u))ds + 2 (2(2β + 1)F(u) − uf(u))dx+

∂Ω Ω

+ − 2 |Diu| dx − 4(2β + 1)E(0).

Taking into account conditions (3.1)-(3.3) of the theorem here, we will have

F (t)(t) ≥ 4(β + 1) ∥u ∥ + ∥∇u (τ)∥ dτ .

′ ′′

Given the values F(t)and F (t) the estimates F (t) in the difference

′′ ′

F(t)F (t)) − (1 + β)(F (t)) after some transformations we get

′′ ′

F(t)F (t)) − (1 + β)(F (t)) ≥ 0.

F(0) Then, by Lemma [2], we can approve that there exists t0 = ′ such that, βF (0) there is  

lim ∥u(x, t)∥ + α ∥∇u (x, τ)∥ dτ = ∞.

Theorem was proved.

The study of nonlinear problems associated with third-order equations is a compelling area of mathematical analysis, particularly due to the complexity and diverse applications these equa-tions present. Third-order differential equations often arise in the modeling of physical phenom-ena, such as fluid dynamics, beam deflection, and wave propagation. The nonlinear nature of these equations poses significant challenges in obtaining analytical solutions, necessitating the development of robust numerical techniques and innovative analytical methods. One of the piv-otal aspects of dealing with nonlinear third-order equations is the exploration of boundary value problems. The uniqueness and existence of solutions in such contexts hinge on the boundary conditions imposed. Various methods, including perturbation techniques and fixed-point theo-rems, can be leveraged to derive approximate solutions or to analyze the stability of solutions. Additionally, the integration of qualitative theory provides insight into the behavior of solutions and the influence of nonlinearity on their properties. Moreover, technological advancements have enabled the use of computational tools to tackle these complex equations, allowing for simulations that were previously infeasible. This intersection of mathematics and computational science opens new avenues for research and application, especially in engineering and physics. As we continue to delve into these nonlinear problems, the potential for discovering novel solu-tions and theoretical insights remains vast and promising.

In conclusion, the author expresses his sincere gratitude to the reviewers for their useful advice.