Extinction of solutions in parabolic equations with different diffusion operators

ABSTRACT

In this paper, we study the evolution p, q-Laplacian equations $u_t=\mathrm{d}\mathrm{i}\mathrm{v}\left(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\right)+u^{\alpha }{\int }_{\mathrm{\Omega }}{v}^m\mathrm{d}x$ and $v_t=\mathrm{d}\mathrm{i}\mathrm{v}\left(|\mathrm{\nabla }v{|}^{q-2}\mathrm{\nabla }v\right)+v^{\text{β}}{\int }_{\mathrm{\Omega }}{u}^n\mathrm{d}x$ with 1<p, q<2, subject to homogeneous Dirichlet boundary conditions. If $mn>(p-1-\alpha )(q-1-\text{β})$, there exist suitable initial data such that vanishing solutions exist. If $mn<(p-1-\alpha )(q-1-\text{β})$, we find the explicit scopes of initial data such that the solutions could not vanish, which complete the corresponding classifications of solutions in Math. Methods Appl. Sci. 39 (2016) 1325–1335 and Appl. Math. Comp. 259 (2015) 587–595, respectively. For the critical case $mn=(p-1-\alpha )(q-1-\text{β})$, the solutions vanish in finite time with small initial data.

Keywords:

• p, q-Laplacian equations
• extinction
• nonextinction

MSC (2010):

• 35K55
• 35B40
• 35K15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This paper is Shandong Provincial Natural Science Foundation of China.