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

Fundamental global similarity solutions of the standard form u_\g(x,t)=t^{-\a_\g} f_\g(y), with the rescaled variable y= x/{t^{\b_\g}}, \b_\g= \frac {1-n \a_\g}{10}, where \a_\g>0 are real nonlinear eigenvalues (\g is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} = \nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The present paper continues the study began by the authors in the previous paper P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem for a tenth-order thin film equation I. Bifurcation of self-similar oscillatory fundamental solutions, Mediterranean Journal of Mathematics, No. 4, Vol. 10 (2013), 1759-1790. Thus, the following questions are also under scrutiny: (I) Further study of the limit n \to 0, where the behaviour of finite interfaces and solutions as y \to infinity are described. In particular, for N=1, the interfaces are shown to diverge as follows: |x_0(t)| \sim 10 \left( \frac{1}{n}\sec\left( \frac{4\pi}{9} \right) \right)^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+. (II) For a fixed n \in (0, \frac 98), oscillatory structures of solutions near interfaces. (III) Again, for a fixed n \in (0, \frac 98), global structures of some nonlinear eigenfunctions \{f_\g\}_{|\g| \ge 0} by a combination of numerical and analytical methods

Classification of global and blow-up sign-changing solutions of a semilinear heat equation in the subcritical fujita range:Second-order diffusion

AbstractIt is well known from the seminal paper by Fujita [22] for 1 p0there exists a class of sufficiently "small" global in time solutions. This fundamental result from the 1960-70s (see also [39] for related contributions), was a cornerstone of further active blow-up research. Nowadays, similar Fujita-type critical exponents p0, as important characteristics of stability, unstability, and blow-up of solutions, have been calculated for various nonlinear PDEs. The above blow-up conclusion does not include solutions of changing sign, so some of them may remain global even for p ≤ p0. Our goal is a thorough description of blow-up and global in time oscillatory solutions in the subcritical range in (0.1) on the basis of various analytic methods including nonlinear capacity, variational, category, fibering, and invariant manifold techniques. Two countable sets of global solutions of changing sign are shown to exist. Most of them are not radially symmetric in any dimension N ≥ 2 (previously, only radial such solutions in ℝNor in the unit ball B1⊂ℝNwere mostly studied). A countable sequence of critical exponents, at which the whole set of global solutions changes its structure, is detected:, l = 0, 1, 2, ... .. See [47, 48] for earlier interesting contributions on sign changing solutions