Oscillating Solutions and Large ω-limit Sets in a Degenerate Parabolic Equation

Abstract

The paper deals with positive solutions of the initial-boundary value problem for $u_t=f(u) (\Delta u+\lambda_1 u) \qquad (*)$ with zero Dirichlet data in a smoothly bounded domain $\Omega \subset \mathbb{R}^{n}, n\ge 1$. Here $f \in C^{0}([0,\infty)) \cap C^{1}((0,\infty))$ is positive on (0,∞) with f(0) = 0, and λ₁ is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence of a sufficiently strong degeneracy. More precisely, writing $H(s):=\int_{1}^{s} \frac{d\sigma}{f(\sigma)}$, it is shown that if $\int_{0} sH(s)ds=-\infty$ then there exist global classical solutions of (*) satisfying $\limsup_{t\to\infty} \|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty$ and $\liminf_{t\to\infty} \|u(\cdot,t)\|_{L^\infty(\Omega)}=0$. Under the additional structural assumption $\frac{sf'(s)}{f(s)}\ge\kappa > 0$, s > 0, this result can be sharpened: If $\int_0 sH(s)ds=-\infty$ then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition $\int_0 sH(s)ds > -\infty$ ensures that all solutions of (*) will stabilize to a single equilibrium.