A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation
Abstract: We establish a Gagliardo-Nirenberg-type inequality in R n for functions which decay fast as |x| → ∞. We use this inequality to derive upper bounds for the decay rates of solutions of a degenerate parabolic equation. Moreover, we show that these upper bounds, hence also the Gagliardo-Nirenberg-type inequality, are sharp in an appropriate sense.Theorem 1.2 If p ≥ 1 and u 0 ∈ q 0 >0 L q 0 (R n ), then for any δ > 0 one can find C(δ) > 0 such that for the minimal solution u of (1.6) we have u(·, t) L ∞ (R n ) ≤ C(… Show more
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Cited by 7 publications
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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 < u 0 < 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 > 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 < lim inf t → ∞ { t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) } ≤ lim sup t → ∞ { t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) } < ∞ . Under a somewhat complementary hypothesis, particularly fulfilled if $$\{u_0>0\}$$ { u 0 > 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.
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 < u 0 < 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 > 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 < lim inf t → ∞ { t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) } ≤ lim sup t → ∞ { t 1 p ‖ u ( · , t ) ‖ L ∞ ( R n ) } < ∞ . Under a somewhat complementary hypothesis, particularly fulfilled if $$\{u_0>0\}$$ { u 0 > 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.
We improve the Gagliardo-Nirenberg inequalityFila and M. Winkler: A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation, Adv. Math., 357 (2019), https://doi.org/10.1016/j.aim.2019.106823] for rapidly decaying functions (ϕ ∈ W 1,r (R n ) \ {0} with finite K = R n L(ϕ)) by specifying the dependence of C on K and by allowing arbitrary r ≥ 1. Mathematics Subject Classification (MSC 2010): 35A23; 26D10
Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory
We consider the degenerate parabolic equation with nonlocal source given bywhich has been proposed as model for the evolution of the density distribution of frequencies with which different strategies are pursued in a population obeying the rules of replicator dynamics in a continuous infinite-dimensional setting.Firstly, for all positive initial data u 0 ∈ C 0 (R n ) satisfying u 0 ∈ L p (R n ) for some p ∈ (0, 1) as well as R n u 0 = 1, the corresponding Cauchy problem in R n is seen to possess a global positive classical solution with the property that R n u(·, t) = 1 for all t > 0.Thereafter, the main purpose of this work consists in reavealing a dependence of the large time behavior of these solutions on the spatial decay of the initial data in a direction that seems unexpected when viewed against the background of known behavior in large classes of scalar parabolic problems. In fact, it is shown that all considered solutions asymptotically decay with respect to their spatial H 1 norm, so thatalways grows in a significantly sublinear manner in thatthe precise growth rate of E, however, depends on the initial data in such a way that fast decay rates of u 0 enforce rapid growth of E. To this end, examples of algebraical and certain exponential types of initial decay are detailed, inter alia generating logarithmic and arbitrary sublinear algebraic growth rates of E, and moreover indicating that (0.1) is essentially optimal.
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