Oscillatory decay in a degenerate parabolic equation

Abstract

AbstractThe Cauchy problem in $$\mathbb {R}^n$$ R n , $$n\ge 1$$ n ≥ 1 , for the degenerate parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$ u t = u p Δ u ( ⋆ ) is considered for $$p\ge 1$$ p ≥ 1 . It is shown that given any positive $$f\in C^0([0,\infty ))$$ f ∈ C 0 ( [ 0 , ∞ ) ) and $$g\in C^0([0,\infty ))$$ g ∈ C 0 ( [ 0 , ∞ ) ) satisfying $$\begin{aligned} f(t)\rightarrow + \infty \quad \text{ and } \quad g(t)\rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$ f ( t ) → + ∞ and g ( t ) → 0 as t → ∞ , one can find positive and radially symmetric continuous initial data with the property that the initial value problem for ($$\star $$ ⋆ ) admits a positive classical solution such that $$\begin{aligned} t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow \infty \qquad \text{ and } \qquad \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$ t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) → ∞ and ‖ u ( · , t ) ‖ L ∞ ( R n ) → 0 as t → ∞ , but that $$\begin{aligned} \liminf _{t\rightarrow \infty } \frac{t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{f(t)} =0 \end{aligned}$$ lim inf t → ∞ t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) f ( t ) = 0 and $$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{\Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{g(t)} =\infty . \end{aligned}$$ lim sup t → ∞ ‖ u ( · , t ) ‖ L ∞ ( R n ) g ( t ) = ∞ .