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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.

Ground states for fractional nonlocal equations with logarithmic nonlinearity

Guo Lifeng, Sun Yan, Shi Guannan

Opuscula Mathematica

2022

Vol. 42, no. 2

157--178

In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by [formula] where 2 < q < 2∗s, LK is a non-local operator, Ω is an open bounded set of Rn 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.