Abstract:

This paper is devoted to the well-posedness and long-time behavior of initial-boundary value problems for a class of nonlinear parabolic equations associated with the Grushin operator $$u_t - \Delta_\alpha u = |u|^{p-2} u \log |u|,$$ where $\Delta_\alpha = \frac{\partial^2}{\partial x^2} + |x|^{2\alpha} \frac{\partial^2}{\partial y^2}$ is the Grushin operator, and $p > 2$ satisfies a subcritical growth condition. Via the semigroup theory in the framework of weighted Sobolev spaces, the existence and uniqueness of local solutions is proved. Subsequently, using the potential well method, the global dynamics of solutions is established. More precisely, when the initial energy satisfies $J(u_0) \leq d$ and the Nehari functional $I(u_0) > 0$, the equation admits a global solution whose energy decays exponentially;when the initial energy $J(u_0) \leq d$ and $I(u_0) < 0$, the solution blows up in finite time. For the case when the initial energy $J(u_0) > d$, by defining relevant invariant sets and functionals, the conditions are clarified under which the solution exists globally or blows up in finite time.

Key words: Grushin operator, degenerate parabolic equation, local existence, potential well method, global existence and blow-up