Bulletin of the
Korean Mathematical Society BKMS

This paper deals with the critical blow-up and extinction exponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents $q_1,q_2\in(0,+\infty)$ with $q_1 < q_2$. In other words, when $q$ belongs to different intervals $(0, q_1), (q_1, q_2), (q_2,+\infty)$, the solution possesses complete different properties. More precisely speaking, as far as the blow-up exponent is concerned, the global existence case consists of the interval $(0, q_2]$. However, when $q \in (q_2,+\infty)$, there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval $(q_1,+\infty)$, while for $q\in (0, q_1)$, there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case $q = q_1$ is concerned, the other parameter $\lambda$, will play an important role. In other words, when $\lambda$ belongs to different interval $(0, \lambda_1)$ or $(\lambda_1,+\infty)$, where $\lambda_1$ is the first eigenvalue of $p$-Laplacian equation with zero boundary value condition, the solution has ompletely different properties.

Keywords: non-Newtonian polytropic equation, critical blow-up exponent, critical extinction exponent

MSC numbers: 35K50, 35K55, 35K65, 35B33