BLOW UP AT INFINITY OF WEAK SOLUTIONS FOR A HIGHER-ORDER PARABOLIC EQUATION WITH LOGARITHMIC NONLINEARITY

The main goal of this work is to study the inital boundary value problem for a higher-order parabolic equation with logarithmic source term u_{t}+(-\Delta )^{m}u=uln (u). We obtain blow-up at infinity of weak solutions, by employing potential well technique. This improves and extends some previous studies.

Keywords:

Higher-order parabolic equation Blow up, Logarithmic nonlinearity,

PDF

___

R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Academic Press, (2003).
H. Chen, P. Luo, G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, Journal of Mathematical Analysis and Applications, 422(1), 84-98, (2015).
H. Chen, S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, Journal of Differential Equations, 258, 4424-4442, (2015).
V.A. Galaktionov, Critical global asymptotics in higher-order semilinear parabolic equations, International Journal of Mathematics and Mathematical Sciences, 60, 3809-3825, (2003).
Y. Han, Blow-up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity, Journal of Mathematical Analysis and Applications, 471, 513-517, (2019).
Y. He, H. Gao, H. Wang, Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity, Computers & Mathematics with Applications, 75, 459-469, (2018).
K. Ishige, T. Kawakami, S. Okabe, Existence of solutions for a higher-order semilinear parabolic equation with singular initial data, Annales de l'Institut Henri Poincare C, Analyse Nonlineaire, 37, 1185-1209, (2020).
P. Li, C. Liu, A class of fourth-order parabolic equation with logarithmic nonlinearity, Journal of Inequalities and Applications, 328, 1-21, (2018).
L.C. Nhan, L.X. Truong, Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity, Computers & Mathematics with Applications, 73, 2076-2091, (2017).
J. Peng, J. Zhou, Global existence and blow-up of solutions to a semilinear heat equation with logarithmic nonlinearity, Applicable Analysis, 1-21, (2019).
E. Pişkin, N. Polat, On the decay of solutions for a nonlinear higher-order Kirchhoff-type hyperbolic equation, Journal of Advanced Research in Applied Mathematics, 5(2), 107-116, (2013).
E. Pişkin, Blow up solutions for a class of nonlinear higher-order wave equation with variable exponents, Sigma Journal of Engineering and Natural Sciences, 10(2), 149-156, (2019).
L. Xiao, M. Li, Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations, Boundary Value Problems, 5, 1-24, (2021).
Y. Ye, Existence and asymptotic behaviour of global solutions for a class of nonlinear higher-order wave equation, Journal of Inequalities and Applications, 1-14, (2010).
J. Zhou, X. Wang, X. Song, C. Mu, Global existence and blowup of solutions for a class of nonlinear higher-order wave equations, Zeitschrift für Angewandte Mathematik und Physik, 63, 461-473, (2012).