Critical blow-up and extinction exponents for non-newton polytropic filtration equation with source

TLDR: In this article, the critical blow-up and extinction exponents for the non-Newton polytropic filtration equation were analyzed and two critical exponents q1,q2 2 (0,+1) with q1 < q2 were revealed.

Abstract: This paper deals with the critical blow-up and extinction ex- ponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents q1,q2 2 (0,+1) with q1 < q2. In other words, when q belongs to dierent intervals (0,q1),(q1,q2),(q2,+1), the solution possesses complete dierent prop- erties. More precisely speaking, as far as the blow-up exponent is con- cerned, the global existence case consists of the interval (0,q2). However, when q 2 (q2,+1), there exist both global solutions and blow-up so- lutions. As for the extinction exponent, the extinction case happens to the interval (q1,+1), while for q 2 (0,q1), there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case q = q1 is concerned, the other parameter ‚ will play an im- portant role. In other words, when ‚ belongs to dierent interval (0 ,‚1) or (‚1,+1), where ‚1 is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely dierent properties.