Critical Blow-Up and Extinction Exponents for Non-Newton Polytropic Filtration Equation With Source

Abstract

Abstract. 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 ∈ (0, +∞) with q 1 < q 2 . In other words, when q belongs to different intervals (0, q 1 ), (q 1 , q 2 ), (q 2 , +∞), 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 ∈ (q 2 , +∞), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval (q 1 , +∞), while for q ∈ (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 λ will play an important role. In other words, when λ belongs to different interval (0, λ 1 ) or (λ 1 , +∞), where λ 1 is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties.

Extinction and non-extinction for a polytropic filtration system with non-linear sources

“…(II) By the results of scalar problems (see [16,17]) we conjecture that (1.1) should admit at least one non-extinction solution for any non-negative initial data (u 0 , v 0 ) when 1 < p, q and αβ < mn( p − 1)(q − 1).…”

“…Motivated mainly by [18,19], we shall study the extinction properties of solutions to (1.1) for any N ≥ 1 and give some criteria for the solutions to vanish in finite time or not, extending some results obtained in [16,17,19] to (1.1). It is known that one of the most frequently used tools in studying the extinction phenomena of evolutionary equations is super and subsolution method (see [20]).…”

“…There are also some papers concerning the extinction for fast diffusive equations. For instance, in [1,12], the authors studied the extinction and positivity for the evolutionary p-Laplacian equation without sources; in [13], the authors investigated the critical extinction exponents for a fast diffusive filtration equation with non-linear sources; in [14,15], the critical extinction exponents of solutions for a fast diffusive p-Laplacian equation with non-linear source term were determined; and in [16,17], the above-mentioned results were extended to non-Newtonian polytropic filtration equations.…”

Extinction behavior of solutions for a quasilinear parabolic system with nonlocal sources

“…For the critical case p ¼ m, the first eigenvalue of ÀD in X plays a crucial role. Similar results to the p-Laplacian equations or the doubly degenerate equations, we refer to [16,27,30,33] and the references therein.…”

Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

“…The first result concerning the extinction of a solution for the general heat equation with absorption was established in. In recent years, there were many works on the extinction properties of solutions for different kinds of evolution equations (see and the references therein ). In particular, Liu studied the extinction properties of solutions to the following problem with the nonlocal source and absorption where q , p ∈ (0,1), d , k > 0, , and obtained the critical extinction exponent q = p .…”