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 $u_t-\triangle_{X} u_t-\triangle_{X} u=u\log|u|$, where $X=(X_1, X_2, ··· , X_m)$ is an infinitely degenerate system of vector fields, and $\triangle_{X}:=\sum^{m}_{j=1}X^{2}_{j}$ 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.