The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions Articles uri icon

publication date

  • March 2015

start page

  • 807

end page

  • 827

issue

  • 3

volume

  • 35

international standard serial number (ISSN)

  • 1078-0947

electronic international standard serial number (EISSN)

  • 1553-5231

abstract

  • Aparece tambien en Arxiv.com : arXiv:1407.5554v1 Abstract: Fundamental global similarity solutions of the standard form u gamma(x,t) = t(-alpha)gamma(f gamma)(y), with the rescaled variable y = x/t(beta gamma), beta(gamma) = 1 n alpha gamma/10, where alpha gamma > 0 are real nonlinear evenvalues (gamma is a multiindex in R-N) of the tenth-order thin film equation (TFE-10) u(t) = del . (vertical bar u vertical bar(n) del Delta(4)u) in R-N x R+, n > 0, (0.1) (0, 1) are studied. The present paper continues the study began in [1]. Thus, the following questions are also under scrutiny: (I) Further study of the limit n -> 0, where the behaviour of finite interfaces and solutions as y > infinity are described. In particular, for N = 1, the interfaces are shown to diverge as follows: vertical bar x(0)(t)vertical bar similar to 10 (1/n sec (4 pi/9)) (9/10) t(1/10) -> infinity as n -> 0+. (II) For a fixed n is an element of (0, 9/8), oscillatory structures of solutions near interfaces. (III) Again, for a fioxed n is an element of (0, 9/8), global structures of some nonlinear and analytical methods

keywords

  • thin film equation; the cauchy problem; source-type global similarity solutions of changing sign; parabolic equations