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This work addresses a value problem concerning a system of high-order nonlinear equations with viscoelastic terms acting in both equations and homogeneous Dirichlet conditions. Initially, we demonstrate that the system has a weak local solution by combining fixed point theory and Galerkin's method. Under certain conditions on the variable exponents p (.) , as well as conditions related to the kernel functions η (t) ≤ μ (t) , for all t ≥ 0 , we establish that the solution with positive initial energy exhibits exponential growth. In addition, we study some numerical examples to illustrate our theoretical results and demonstrate the effectiveness of our new approach. [ABSTRACT FROM AUTHOR]

In this work, we deal with the higher-order hyperbolic-type equation. Firstly, we prove the global existence and blow up of solutions with the subcritical initial energy. Later, we prove the global existence and blow up of solutions with the critical initial energy. [ABSTRACT FROM AUTHOR]

Partial solutions with a singularity which are expressed in terms of the Wright functions for equations with Riemann–Liouville fractional derivative are constructed. [ABSTRACT FROM AUTHOR]

This paper proposes a higher-order μ -Camassa–Holm equation, which is regarded as a higher-order extension of the μ -Camassa–Holm equation, and preserves some properties of the μ -Camassa–Holm equation. We first show that the equation admits the peaked traveling wave solution, which is given by a Green function of the momentum operator. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established. Finally, the blow-up criterion and wave breaking mechanism for solutions with certain initial profiles are studied. It turns out that all the nonlinearities even the first-order nonlinearity may have the effect on the blow up. [ABSTRACT FROM AUTHOR]

Abstract Under some assumptions on the nonlinearity f, we will study the nonexistence of nontrivial stable solutions or solutions which are stable outside a compact set of R n $\mathbb {R}^{n}$ for the following semilinear higher-order problem: ( − Δ ) k u = f ( u ) in R n , $$\begin{aligned} (-\Delta)^{k} u= f(u) \quad \mbox{in }\mathbb {R}^{n}, \end{aligned}$$ with k = 1 , 2 , 3 , 4 $k=1,2,3,4$ . The main methods used are the integral estimates and the Pohozaev identity. Many classes of nonlinearity will be considered; even the sign-changing nonlinearity, which has an adequate subcritical growth at zero as for example f ( u ) = − m u + λ | u | θ − 1 u − μ | u | p − 1 u $f(u)= -m u +\lambda|u|^{\theta-1}u-\mu |u|^{p-1}u$ , where m ≥ 0 $m\geq0$ , λ > 0 $\lambda>0$ , μ > 0 $\mu>0$ , p , θ > 1 $p, \theta>1$ . More precisely, we shall revise the nonexistence theorem of Berestycki and Lions (Arch. Ration. Mech. Anal. 82:313-345, 1983) in the class of smooth finite Morse index solutions as the well known work of Bahri and Lions (Commun. Pure Appl. Math. 45:1205-1215, 1992). Also, the case when f ( u ) u $f(u)u$ is a nonnegative function will be studied under a large subcritical growth assumption at zero, for example f ( u ) = | u | θ − 1 u ( 1 + | u | q ) $f(u)=|u|^{\theta-1}u(1 + |u|^{q})$ or f ( u ) = | u | θ − 1 u e | u | q $f(u)= |u|^{\theta-1}u e^{|u|^{q}}$ , θ > 1 $\theta>1$ and q > 0 $q>0$ . Extensions to solutions which are merely stable are discussed in the case of supercritical growth with k = 1 $k=1$ .

Under some assumptions on the nonlinearity f, we will study the nonexistence of nontrivial stable solutions or solutions which are stable outside a compact set of $\mathbb {R}^{n}$ for the following semilinear higher-order problem: with $k=1,2,3,4$ . The main methods used are the integral estimates and the Pohozaev identity. Many classes of nonlinearity will be considered; even the sign-changing nonlinearity, which has an adequate subcritical growth at zero as for example $f(u)= -m u +\lambda|u|^{\theta-1}u-\mu |u|^{p-1}u$ , where $m\geq0$ , $\lambda>0$ , $\mu>0$ , $p, \theta>1$ . More precisely, we shall revise the nonexistence theorem of Berestycki and Lions (Arch. Ration. Mech. Anal. 82:313-345, 1983) in the class of smooth finite Morse index solutions as the well known work of Bahri and Lions (Commun. Pure Appl. Math. 45:1205-1215, 1992). Also, the case when $f(u)u$ is a nonnegative function will be studied under a large subcritical growth assumption at zero, for example $f(u)=|u|^{\theta-1}u(1 + |u|^{q})$ or $f(u)= |u|^{\theta-1}u e^{|u|^{q}}$ , $\theta>1$ and $q>0$ . Extensions to solutions which are merely stable are discussed in the case of supercritical growth with $k=1$ . [ABSTRACT FROM AUTHOR]

The representation by the Mittag-Leffler function of the solution to the Cauchy problem for the evolution equation solved for the higher derivative is used in the study of degenerate linear and quasilinear evolution equations under some special constraints on the nonlinear part of the equation. The solvability conditions for the Cauchy problem are simplified in the situation when the generalized Showalter-Sidorov condition is used as the initial condition. These results are applied to studying an initial boundary value problem for the motion equation of the Kelvin-Voigt fluid. [ABSTRACT FROM AUTHOR]

An n -th order delayed differential equation y (n) (t) = f (t , y t , y t ′ , ... , y t (n − 1) ) is considered, where y t (θ) = y (t + θ) , θ ∈ [ − τ , 0 ] , τ > 0 , if t → ∞. A criterion is formulated guaranteeing the existence of a solution y = y (t) in a cone 0 < (− 1) i − 1 y (i − 1) (t) < (− 1) i − 1 φ (i − 1) (t) , i = 1 , ... , n where φ is an n -times continuously differentiable function such that 0 < (− 1) i φ (i) (t) , i = 0 , ... , n. The proof is based on a similar result proved first for a system of delayed differential equations equivalent in a sense. Particular linear cases are considered and an open problem is formulated as well. [ABSTRACT FROM AUTHOR]

In this article we establish Harnack type inequality and derive a result of local Hölder continuity for solutions of degenerate nonlinear higher-order elliptic equations where the degeneracy is of Muckenhoupt kind. [ABSTRACT FROM AUTHOR]

This paper proposes a new symbolic method for solving a class of higher-order equations with an unknown function over the complex domain. Our method exploits the closure property of group structure (for functions) in order to allow an equivalent system of equations to be expressed and solved in the first-order setting.Our work is an initial step towards the relatively unexplored realm of higher-order constraint solving, in general; and higher-order equational solving, in particular. We shall provide some theoretical background for the proposed method, and also prototype an implementation under Mathematica. [Copyright &y& Elsevier]