Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

Abstract

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity ut − X ut − X u = u log |u|, where X = (X 1 , X 2 , • • • , Xm) is an infinitely degenerate system of vector fields, and X := m j=1 X 2 j is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincaré inequality, we obtain the global existence and blow-up at +∞ of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

“…For the non-local source case, we refer to some very recent related references, e.g. [24] for the threshold results of the global existence and non-existence for the signchanging weak solutions of thin film equation; [14] and [8] for the Kirchhoff type problem with non-local source 1 |Ω| Ω |u| q−1 udx; [15] for the finite time blow-up of solutions with non-positive initial energy J(u 0 ) and non-local source 1 |Ω| Ω |u| q dx with q > 1; [18] for the well-posedness of pseudo-parabolic equation with singular potential term at three initial energy levels, the logarithmic nonlinearity in [7] and the non-local source 1 |Ω| Ω |u| q−1 udx in [30]. As far as we know, there are few research works concerned with the sign-changing solutions for the mixed pseudoparabolic Kirchhoff equation.…”

The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term

“…Recently, Chen and Xu [11] study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity and obtain the similar results. Nhan and Truong studied parabolic p-Laplacian equation [23] and pseudo-parabolic p-Laplacian equation [24] with logarithmic nonlinearity |u| p−2 u log |u| k where they need the p-Laplacian term to control the logarithmic nonlinearity.…”

On mixed problem of Chaplygin's hodograph equation in hyperbolic domain near the parabolic degenerate line

“…Chen, Wang, and Xu in Ref. 9 considered the behaviors of solutions for infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity.…”