# Blow-up Phenomena For Pseudo-Parabolic Kirchhoff Equations With Logarithmic Nonlinearity

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## Abstract

We consider pseudo-parabolic equations with $r(x)$-Kirchhoff coefficient and

logarithmic nonlinear term subject to Dirichlet boundary condions%

\begin{equation*}

\upsilon _{t}-\mu \Delta \upsilon _{t}-M(\left( \left\Vert \nabla \upsilon

\right\Vert _{r(x)}^{r(x)}\right) \Delta _{r(x)}\upsilon =\left\vert

\upsilon \right\vert ^{s(x)-2}\upsilon \ln \left\vert \upsilon \right\vert .

\end{equation*}%

Using a method based on differential inequalities, we prove that the

solutions become unbounded at a finite time $T$, and, we ascertain an upper

limit for this time in the case of negative initial energy. Additionally, we

determine a lower limit for the time at which blow-up occurs.

## References

G.A. Afrouzi, N.T. Chung and S. Shakeri, Existence of positive solutions for Kirchhoff type equations. Electron. J.

Differential Equations, 180(2013), 8, (2013).

A.B. Al’shin, M.O. Korpusov and A.G. Sveshnikov, Blow-up in nonlinear Sobolev type equations, (Vol. 15). Walter de

Gruyter, (2011).

S.N. Antontsev, J.I. Díaz, S. Shmarev and A.J. Kassab, Energy methods for free boundary problems: Applications to

non linear PDEs and fluid mechanics. Progress in Nonlinear Differential Equations and Their Applications, Vol 48.

Appl. Mech. Rev, 55(4) (2002), B74–B75.

N. Barrouk and S. Mesbahi, Generalized result on the global existence of positive solutions for a parabolic reaction

diffusion model with a full diffusion matrix. Studia Universitatis Babes-Bolyai, Mathematica, 68, 2 (2023).

K. Bartkowski and P. Górka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities. Journal of

Physics A: Mathematical and Theoretical, 41(35), 355201, (2008).

J.D. Barrow and P. Parsons, Inflationary models with logarithmic potentials. Physical Review D, 52(10), 5576, (1995).

I. Bialynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities. Bulletin de l’Academie Polonaise

des Sciences. Serie des Sciences, Mathematiques, Astronomiques et Physiques, 23(4) (1975), 461–466.

Y. Cao and Q. Zhao, Initial boundary value problem of pseudo-parabolic Kirchhoff equations with logarithmic nonlinearity. Authorea Preprints, (2022).

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. Journal of Differential Equations, 258(12) (2015), 4424–4442.

T. Cömert and E. Pişkin, Blow-up of weak solutions for a higher-order heat equation with logarithmic nonlinearity.

Miskolc Mathematical Notes, 24(2), (2023), 749–762.

T. Cömert and E. Piskin, Global existence and stability of solutions for Kirchhoff-type parabolic system with logarithmic source term, Advanced Studies: Euro-Tbilisi Mathematical Journal, Special Issue (10 - 2022) 153–170.

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents. Springer,

(2011).

P. Dai, C. Mu and G. Xu, Blow-up phenomena for a pseudo-parabolic equation with p-Laplacian and logarithmic nonlinearity terms. Journal of Mathematical Analysis and Applications, 481(1), 123439, (2020).

H. Ding and J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity. Journal of Mathematical Analysis and Applications, 478(2), (2019), 393–420.

K. Enqvist and J. McDonald, Q-balls and baryogenesis in the MSSM. Physics Letters B, 425(3-4), (1998), 309-321.

X. Fan and D. Zhao, On the spaces Lp(x)

(W) and Wm,p(x)

(W). J. Math. Anal. Appl. 263, (2001), 424–446.

Y. Fu and M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent.

Applicable Analysis, 95(3), (2016), 524–544.

P. Górka, Logarithmic Klein-Gordon equatio. Acta Physica Polonica B, 40(1), (2009).

M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations:

Time-decay estimates. Journal of Differential Equations, 245(10), (2008), 2979–3007.

A. Hamydy, M. Massar and N. Tsouli, Existence of solutions for p-Kirchhoff type problems with critical exponent.

Electron. J. Differential Equations, 105, (2011), 1–8.

Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Computers and Mathematics with Applications, 75(9), (2018), 3283–3297.

Y. Han, W. Gao, Z. Sun, and H. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with

arbitrary initial energy. Computers and Mathematics with Applications, 76(10), (2018), 2477–2483.

Y. He, H. Gao and H. Wang, Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic

nonlinearity. Computers and Mathematics with Applications, 75(2), (2018), 459–469.

M.O. Korpusov and A.G. Sveshnikov, Three-dimensional nonlinear evolution equations of pseudoparabolic type in

problems of mathematical physics. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 43(12), (2003),

–1869.

M.O. Korpusov and A.G. Sveshnikov, Three-dimensional nonlinear evolutionary pseudoparabolic equations in mathematical physics. II. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 44(11), (2004), 2041–2048.

D. Liu, On a p-Kirchhoff equation via fountain theorem and dual fountain theorem. Nonlinear Analysis: Theory,

Methods and Applications, 72(1), (2010), 302–308.

J. Li and Y. Han, Global existence and finite time blow-up of solutions to a nonlocal p-Laplace equation. Mathematical

Modelling and Analysis, 24(2), (2019), 195–217.

N. Lakshmipriya, S. Gnanavel, K. Balachandran and Y.K. Ma, Existence and blow-up of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity. Boundary Value Problems, 2022(1) (2022), 1–17.

R. Pan, Y. Gao and Q. Meng, Properties of Weak Solutions for a Pseudoparabolic Equation with Logarithmic

Nonlinearity of Variable Exponents. Journal of Mathematics, 2023, (2023).

E. Pişkin and G. Butakin, Blow-up phenomena for a p(x)-biharmonic heat equation with variable exponent. Mathematica

Moravica, 27(2), (2023), 25–32.

E. Pişkin and G. Butakin, Existence and Decay of solutions for a parabolic-type Kirchhoff equation with variable exponents. Journal of Mathematical Sciences and Modelling, 6(1), (2023), 32–41.

E. Piskin and T. Cömert, Qualitative analysis of solutions for a parabolic type Kirchhoff equation with logarithmic

nonlinearity. Open Journal of Discrete Applied Mathematics, 4(2), (2021), 1–10.

S. Toualbia, A. Zaraï and S. Boulaaras, Decay estimate and non-extinction of solutions of p-Laplacian nonlocal heat

equations. AIMS Mathematics, 5(3), (2020), 1663–1680.

C. Yang and Z. Qiuting, Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations.

Electronic Research Archive, 29(6), (2021), 3833–3851.

G. Kirchhoff, Mechanik. Leipzig: Teubner, Germany, 1883

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*7*(2), 115–126. Retrieved from https://nonlinear-analysis.com/index.php/pub/article/view/370

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