Mathematische Annalen

, Volume 355, Issue 2, pp 519–549

Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type


DOI: 10.1007/s00208-012-0795-z

Cite this article as:
Winkler, M. Math. Ann. (2013) 355: 519. doi:10.1007/s00208-012-0795-z


This work deals with positive classical solutions of the degenerate parabolic equation
$$u_t=u^p u_{xx} \quad \quad (\star)$$
when p > 2, which via the substitution v = u1−p transforms into the super-fast diffusion equation \({v_t=(v^{m-1}v_x)_x}\) with \({m=-\frac{1}{p-1} \in (-1,0)}\) . It is shown that (\({\star}\)) possesses some entire positive classical solutions, defined for all \({t \in \mathbb {R}}\) and \({x \in \mathbb {R}}\) , which connect the trivial equilibrium to itself in the sense that u(x, t) → 0 both as t → −∞ and as t → + ∞, locally uniformly with respect to \({x \in \mathbb {R}}\) . Moreover, these solutions have quite a simple structure in that they are monotone increasing in space. The approach is based on the construction of two types of wave-like solutions, one of them being used for −∞ < t ≤  0 and the other one for 0 < t <  + ∞. Both types exhibit wave speeds that vary with time and tend to zero as t → −∞ and t → + ∞, respectively. The solutions thereby obtained decay as x → −∞, uniformly with respect to \({t \in \mathbb {R}}\) , but they are unbounded as x → + ∞. It is finally shown that within the class of functions having such a behavior as x → −∞, there does not exist any bounded homoclinic orbit.

Mathematics Subject Classification (2000)

Primary 37C29 Secondary 35K65 35K55 35B40 

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany

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