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Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity

Cordeiro Sebastião, Raposo Carlos, Ferreira Jorge, Rocha Daniel, Shahrouzi Mohammad

Opuscula Mathematica

2024

Vol. 44, no. 1

19--47

This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma.

Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms

Di Huafei, Song Zefang

Opuscula Mathematica

2022

Vol. 42, no. 2

119--155

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∗. 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∗ is derived by the means of integro-differential inequality techniques.