Convergence to Strong Nonlinear Diffusion Waves for Solutions of p-System with Damping

By Huijiang Zhao. 2001

76 citations recorded by the source corpus.

This paper is concerned with the p-system with frictional damping and our main purpose is two-fold: First, we show that for a certain class of given large initial data (nu (0)(x), u(0)(x)), the Cauchy problem (1.1), (1.2) admits a unique global smooth solution (nu (t, x), u(t, x)) and such a solution tends time-asymptotically, at the optimal L-P(2 less than or equal to p less than or equal to infinity) decay rates, to the corresponding nonlinear diffusion wave ( )over bar>)(t, x), (u) over bar (t, x)) governed by the classical Darcy's law provided that the corresponding prescribed initial error function (V-0(x), U-0(x)) ties in (H-3 x H-2) (R) boolean AND (L-1 x L-1)(R). Compared with former results in this direction obtained by L. Hsiao and T.-P. Liu (1992, Comm. Math. Phys. 143, 599-605), K. Nishihara (1996, J. Differential Equations 131, 171-188), and K. Nishihara, W.-K. Wang, and T. Yang (2000, J. Differential Equations 161, 191-218), our main novelty lies in the facts that the nonlinear diffusion wave ( )over bar>(t, x), (u) over bar (t, x)) need not to be weak and (V-0(x), U-0(x)) can be chosen arbitrarily large. Secondly, we show that the nonlinear diffusion waves ( )over bar>(t, x), (u) over bar (t, x)) are nonlinear stable provided that the strength or the nonlinear diffusion waves is weak and that the initial disturbance (V-0(x), U-0(x)) satisfies the assumption that parallel toV(0xx)(x)parallel to (L infinity) + parallel toU(0x)(x)parallel to (L infinity) is Sufficiently small. We also show that the smallness assumption imposed oil the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by L. Hsiao and T.-P. Liu (1992, Comm. Math. Phys. 143, 599-605), the smallness conditions we imposed on the initial disturbance are much more weaker. (C) 2001 Academic Press.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/27ccee5fd2a5cd8323e02c65ad2de5fd5082fac1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Feimin Huang
  2. Zhen Wang
  3. Ronghua Pan
  4. Ming Mei
  5. Bruno Rubino
  6. Pierangelo Marcati
  7. V. Shelkovich
  8. Juhi Jang

In Mathematical Physics

  1. Zhen Wang
  2. Feimin Huang
  3. Ronghua Pan
  4. Ming Mei
  5. Pierangelo Marcati
  6. Juhi Jang
  7. Nader Masmoudi
  8. Huihui Zeng

Citations in the corpus