Critical blow-up and extinction exponents for non-Newton polytropic filtration equation with source
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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 q1 < q2. In other words, when q belongs to different intervals (0, q_1), (q_1, q_2), (q2,+∞), 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.
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 q1 < q2. In other words, when q belongs to different intervals (0, q_1), (q_1, q_2), (q2,+∞), 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.
참고문헌 (13)
- K. Deng, "The role of critical exponents in blow-up theorems: the sequel", J. Math. Anal. Appl, 243(1), pp.85~126 (2000)
- E. DiBenedetto, "Degenerate Parabolic Equations" (1993), Springer-Verlag
- H. Fujita, "On the blowing up of solutions to the cauchy problems for ut = ¢u + u1+®", Journal of the Faculty of Science University of Tokyo Section 1Mathematics Astronomy Physics Chemistry, 13, pp.109~124 (1996)
- C. H. Jin, "Critical exponents and non-extinction for a fast diffusive polytropic filtration equation with nonlinear boundary sources", Nonlinear Anal, 67(7), pp.2217~2223 (2007)
- A. S. Kalashnikov, "Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations", Uspekhi Mat. Nauk, 42(2), pp.135~176 (1987)
- Y. C. Kwong, "Boundary behavior of the fast diffusion equation", Trans. Amer. Math. Soc, 322(1), pp.263~283 (1990)
- H. A. Levine, "The role of critical exponents in blowup theorems", SIAM Rev, 32(2), pp.262~288 (1990)
- J. L. V´azquez, "The Porous Medium Equations: Mathematical Theory" (2007), Oxford Univ. Press
- Z. J. Wang, "Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition", Appl. Math. Lett, 20(2), pp.142~147 (2007)
- M. Winkler, "A strongly degenerate diffusion equation with strong absorption", Math. Nachr, 277, pp.83~101 (2004)
- Z. Q. Wu, "Nonlinear Diffusion Equations" (2001), orld Scientific Publishing Co., Inc., River Edge
- H. J. Yuan, "Extinction and positivity for the evolution p-Laplacian equation", J. Math. Anal. Appl, 196(2), pp.754~763 (1995)
- H. J. Yuan, "Extinction and positivity for the evolution p-Laplace equation in RN", Nonlinear Anal, 60(6), pp.1085~1091 (2005)