Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source
Abstract
We investigate the initial-boundary value problem for the nonlinear viscoelastic Kirchhoff-type wave equation
\begin{equation*}
u_{tt} - \left(1 + \|\nabla u(t)\|^2 \right) \Delta u + \int_0^t g(t - s) \Delta u(s) \, ds + \alpha u_t = |u|^{p-1} \ln |u|, \quad x \in \Omega, t > 0,
\end{equation*}
with homogeneous Dirichlet boundary conditions. Local existence and uniqueness of weak solutions are established via a Banach fixed-point argument. Moreover, we prove a finite-time blow-up result: if the initial energy is positive and the data satisfy a compatibility condition, the solution’s $H_0^1$-norm becomes unbounded in finite time. This work extends existing results by capturing the combined effects of Kirchhoff-type nonlinearity, viscoelastic damping, and a logarithmic source.
Keywords
References
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Details
Primary Language
English
Subjects
Partial Differential Equations
Journal Section
Research Article
Authors
Publication Date
February 23, 2026
Submission Date
August 2, 2025
Acceptance Date
November 7, 2025
Published in Issue
Year 2026 Volume: 18 Number: 1
APA
Çalışkan Desova, B. (2026). Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source. Turkish Journal of Mathematics and Computer Science, 18(1), 175-182. https://doi.org/10.47000/tjmcs.1757179
AMA
1.Çalışkan Desova B. Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source. TJMCS. 2026;18(1):175-182. doi:10.47000/tjmcs.1757179
Chicago
Çalışkan Desova, Begüm. 2026. “Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation With Logarithmic Power Source”. Turkish Journal of Mathematics and Computer Science 18 (1): 175-82. https://doi.org/10.47000/tjmcs.1757179.
EndNote
Çalışkan Desova B (February 1, 2026) Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source. Turkish Journal of Mathematics and Computer Science 18 1 175–182.
IEEE
[1]B. Çalışkan Desova, “Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source”, TJMCS, vol. 18, no. 1, pp. 175–182, Feb. 2026, doi: 10.47000/tjmcs.1757179.
ISNAD
Çalışkan Desova, Begüm. “Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation With Logarithmic Power Source”. Turkish Journal of Mathematics and Computer Science 18/1 (February 1, 2026): 175-182. https://doi.org/10.47000/tjmcs.1757179.
JAMA
1.Çalışkan Desova B. Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source. TJMCS. 2026;18:175–182.
MLA
Çalışkan Desova, Begüm. “Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation With Logarithmic Power Source”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 1, Feb. 2026, pp. 175-82, doi:10.47000/tjmcs.1757179.
Vancouver
1.Begüm Çalışkan Desova. Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source. TJMCS. 2026 Feb. 1;18(1):175-82. doi:10.47000/tjmcs.1757179