Approaching Critical Decay in a Strongly Degenerate Parabolic Equation - Winkler, Michael (Journal of Dynamics and Differential Equations, 2020)
Abstract The Cauchy problem in $${\mathbb {R}}^n$$ R n , $$n\ge 1$$ n ≥ 1 , for the parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$ u t = u p Δ u ( ⋆ ) is considered in the strongly degenerate regime $$p\ge 1$$ p ≥ 1 . The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies $$\begin{aligned} t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \rightarrow \infty \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$ t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) → ∞ as t → ∞ . The first result of this study complements this by asserting that given any positive $$f\in C^0([0,\infty ))$$ f ∈ C 0 ( [ 0 , ∞ ) ) fulfilling $$f(t)\rightarrow +\infty $$ f ( t ) → + ∞ as $$t\rightarrow \infty $$ t → ∞ one can find a positive nondecreasing function $$\phi \in C^0([0,\infty ))$$ ϕ ∈ C 0 ( [ 0 , ∞ ) ) such that whenever $$u_0\in C^0({\mathbb {R}}^n)$$ u 0 ∈ C 0 ( R n ) is radially symmetric with $$0< u_0 < \phi (|\cdot |)$$ 0 <</mml:mo> u 0 <</mml:mo> ϕ ( | · | ) , the corresponding minimal solution u satisfies $$\begin{aligned} \frac{t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}}{f(t)} \rightarrow 0 \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$ t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) f ( t ) → 0 as t → ∞ . Secondly, ( $$\star $$ ⋆ ) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data $$u_0$$ u 0 . It is shown that if the connected components of $$\{u_0>0\}$$ { u 0 <mml:mo>> 0 } comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to ( $$\star $$ ⋆ ) satisfies $$\begin{aligned} 0< \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} \le \limsup _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} <\infty . \end{aligned}$$ 0 <</mml:mo> lim inf t → ∞ { t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) } ≤ lim sup t → ∞ { t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) } <</mml:mo> ∞ . Under a somewhat complementary hypothesis, particularly fulfilled if $$\{u_0>0\}$$ { u 0 <mml:mo>> 0 } contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.
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