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

Abstract

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.