Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2022
Volume
Vol. 42
Number
No. 2
Description
Special Issue - Advances in Nonlinear Partial Differential Equations
Journal Volume
Opuscula Mathematica
Vol. 42 (2022)
Projects
Pages
Articles
Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms
(Wydawnictwa AGH, 2022) Di, Huafei; Song, Zefang
Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with $t$. Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time $T^\ast$. Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span $T^\ast$ is derived by the means of integro-differential inequality techniques.
Ground states for fractional nonlocal equations with logarithmic nonlinearity
(Wydawnictwa AGH, 2022) Guo, Lifeng; Sun, Yan; Shi, Guannan
In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by $\begin{cases}\mathcal{L}_{K}u(x)+u\log|u|+|u|^{q-2}u=0, & x\in\Omega,\\ u=0, & x\in\mathbb{R}^{n}\setminus\Omega,\end{cases}$
where $2\lt q\lt 2^{*}_s$, $L_{K}$ is a non-local operator, $\Omega$ is an open bounded set of $\mathbb{R}^{n}$ with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.
The d-bar formalism for the modified Veselov-Novikov equation on the half-plane
(Wydawnictwa AGH, 2022) Hwang, Guenbo; Moon, Byungsoo
We study the modified Veselov-Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural $(2+1)$-dimensional generalization of the $(1+1)$-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov-Veselov equation is related to the Korteweg-de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the $d$-bar formalism, also known as Pompieu's formula. In addition, we characterize the $d$-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.
Blowup phenomena for some fourth-order strain wave equations at arbitrary positive initial energy level
(Wydawnictwa AGH, 2022) Lin, Qiang; Luo, Yongbing
In this paper, we study a series of fourth-order strain wave equations involving dissipative structure, which appears in elasto-plastic-microstructure models. By some differential inequalities, we derive the finite time blow up results and the estimates of the upper bound blowup time with arbitrary positive initial energy. We also discuss the influence mechanism of the linear weak damping and strong damping on blowup time, respectively.
Exponential decay of solutions to a class of fourth-order nonlinear hyperbolic equations modeling the oscillations of suspension bridges
(Wydawnictwa AGH, 2022) Liu, Yang; Yang, Chao
This paper is concerned with a class of fourth-order nonlinear hyperbolic equations subject to free boundary conditions that can be used to describe the nonlinear dynamics of suspension bridges.