GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE DOUBLY NONLINEAR DIFFUSION EQUATION WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS

Na Chen; Peihe Wang; Fushan Li

doi:10.11948/20230256

2024 Volume 14 Issue 3

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Na Chen, Peihe Wang, Fushan Li. GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE DOUBLY NONLINEAR DIFFUSION EQUATION WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1467-1484. doi: 10.11948/20230256

Citation:

Na Chen, Peihe Wang, Fushan Li. GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE DOUBLY NONLINEAR DIFFUSION EQUATION WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1467-1484. doi: 10.11948/20230256

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE DOUBLY NONLINEAR DIFFUSION EQUATION WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS

1.
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

Author Bio: Email: peihewang@hotmail.com(P. Wang); Email: fushan99@163.com(F. Li)
Corresponding author: Email: nachen93@163.com(N. Chen)

Abstract

Abstract

Under the nonlinear Neumann boundary conditions, an initial boundary value problem for doubly nonlinear diffusion equation is considered in this paper. We establish the new sufficient conditions on nonlinear functions to guarantee that the positive solution $u(\mathit{\boldsymbol{x}}, t)$ exists globally. Under the conditions to guarantee that the positive solution blows up, by establishing the Sobolev inequality in multidimensional space, we obtain upper and lower bounds of the blow-up time $T$.
- Nonlinear porous medium equations /
- nonlinear Neumann boundary condition /
- global existence /
- blow up
MSC: 35B44, 35K51

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References

[1]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Google Scholar
[2]	S. Y. Chung and M. J. Choi, A new condition for the concavity method of blow-up solutions to $p$-Laplacian parabolic equations, Journal of Differential Equations, 2018, 265(12), 6384–6399. doi: 10.1016/j.jde.2018.07.032 CrossRef $p$-Laplacian parabolic equations" target="_blank">Google Scholar
[3]	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, 2020, 481(1), 123439. doi: 10.1016/j.jmaa.2019.123439 CrossRef $p$-Laplacian and logarithmic nonlinearity terms" target="_blank">Google Scholar
[4]	J. Ding and H. Hu, Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions, Journal of Mathematical Analysis and Applications, 2016, 433(2), 1718–1735. doi: 10.1016/j.jmaa.2015.08.046 CrossRef Google Scholar
[5]	J. T. Ding and X. H. Shen, Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source, Electronic Journal of Qualitative Theory of Differential Equations, 2017, 99, 1–15. Google Scholar
[6]	J. T. Ding and X. H. Shen, Blow-up time estimates in porous medium equations with nonlinear boundary conditions, Zeitschrift ür angewandte Mathematik und Physik, 2018, 69(4), 1–13. Google Scholar
[7]	J. T. Ding and X. H. Shen, Blow-up analysis in quasilinear reaction-diffusion problems with weighted nonlocal source, Computers and Mathematics with Applications, 2018, 75(4), 1288–1301. doi: 10.1016/j.camwa.2017.11.009 CrossRef Google Scholar
[8]	L. C. Evans, Partial Differential Equations(Second Edition), Berkeley: Department of Mathematics University of California, 2010. Google Scholar
[9]	M. Giulia and P. Fabio, Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density, Journal of Differential Equations, 2020, 269(10), 8918–8958. doi: 10.1016/j.jde.2020.06.017 CrossRef Google Scholar
[10]	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, 2018, 75(2), 459–469. doi: 10.1016/j.camwa.2017.09.027 CrossRef $p$-Laplacian equation with logarithmic nonlinearity" target="_blank">Google Scholar
[11]	B. Hu, Blow-Up Theories for Semilinear Parabolic Equations, Heidelberg: Lecture Notes in Mathematics, vol. 2018, Springer, 2011. Google Scholar
[12]	N. Irkıla, On the $p$-Laplacian type equation with logarithmic nonlinearity: Existence, decay and blow up, Filomat, 2023, 37(16), 5485–5507. $p$-Laplacian type equation with logarithmic nonlinearity: Existence, decay and blow up" target="_blank">Google Scholar
[13]	V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Journal of Soviet Mathematics, 1978, 10(1), 53–70. doi: 10.1007/BF01109723 CrossRef Google Scholar
[14]	H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+\mathcal{F}(u)$, Transactions of the American mathematical society, 1974, 192, 1–21. $Pu_{tt}=-Au+\mathcal{F}(u)$" target="_blank">Google Scholar
[15]	J. Li and F. Li, Global existence and blow-up phenomena for divergence form parabolic equations with inhomogeneous Neumann boundary, Journal of Mathematical Analysis and Applications, 2012, 385(2), 1005–1014. doi: 10.1016/j.jmaa.2011.07.018 CrossRef Google Scholar
[16]	M. Marras and S. Vernier-Piro, Blow-up time estimates in nonlocal reaction-diffusion systems under various boundary conditions, Boundary Value Problems, 2017, 2, 1–16. Google Scholar
[17]	M. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up time in a parabolic problem with a gradient term under various boundary conditions, Kodai Mathematical Journal, 2014, 37(3), 532–543. Google Scholar
[18]	M. Marras, S. Vernier-Piro and G. Viglialoro, Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term, Journal of Modern Dynamics, 2017, 22(6), 2291–2300. Google Scholar
[19]	L. E. Payne, G. A. Philippin and S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I, Zeitschrift ür angewandte Mathematik und Physik, 2010, 61(6), 999–1007. doi: 10.1007/s00033-010-0071-6 CrossRef Google Scholar
[20]	B. Sabitbek and B. Torebek, Global existence and blow-up of solutins to the nonliear porous medium equation, 2021. arXiv: 2104.06896. Google Scholar
[21]	H. Tian and L. Zhang, Global and blow-up solutions for a nonlinear reaction diffusion equation with Robin boundary conditions, Boundary Value Problems, 2020, 2020(4), 971–978. Google Scholar
[22]	S. Toualbia, A. Zaraï and S. Boulaaras, Decay estimate and non-extinction of solutions of $p$-Laplacian nonlocal heat equations, AIMS Mathematics, 2020, 5(3), 1663–1680. $p$-Laplacian nonlocal heat equations" target="_blank">Google Scholar
[23]	J. L. Vazquez, The Porous Medium Equation: Mathematical Theory, Africa: Oxford University Press, 2006. Google Scholar
[24]	X. C. Wang and R. Z. Xu, Global existence and finite time blow up for a nonlocal semilinear pseudo-parabolic equation, Advances in Nonlinear Analysis, 2021, 10, 261–288. Google Scholar
[25]	J. Zhang and F. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equation with time-dependent coefficient in multidimensional space, Zeitschrift ür angewandte Mathematik und Physik, 2019, 70, 150. Google Scholar