Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

Abstract

This work is concerned with the gradient flow of absolutely p-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite ($p<2$) or infinite extinction time ($p \ge 2$). We give upper bounds for the finite extinction time and establish sharp convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and non-local versions of PDEs like p-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting. We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here, we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions.

Appendices

Appendix

Absolutely p-homogeneous convex functionals and their subdifferential

Proposition A.1

Let $J:\mathcal {H}\rightarrow \mathbb {R}\cup \{\infty \}$ be absolutely p-homogeneous and convex. Then it holds

• 1. $\mathcal {N}(J):=\{u\in \mathcal {H}\,:\,J(u)=0\}$ and ${\mathrm {dom}}(J):=\{u\in \mathcal {H}\,:\,J(u)<\infty \}$ are linear subspaces of $\mathcal {H}$, referred to as null-space and effective domain of J.

• 2. $\mathcal {N}(J)$ is closed if J is lower semi-continuous.

Proposition A.2

Under the conditions of Proposition A.1 it holds for any $u\in \mathcal {H}$ that $\partial J(u)\subset \mathcal {N}(J)^\perp$.

Proof

Using the definition of the subdifferential (1.2) for $v=\pm v_0\in \mathcal {N}(J)$ together with (1.4) yields

$$\begin{aligned} |\langle \zeta ,v_0\rangle |\le (p-1)J(u),\quad \forall \zeta \in \partial J(u). \end{aligned}$$

However, due to the homogeneity (1.3) it holds $c^{p-1}\zeta \in \partial J(c u)$ for all $c>0$. Replacing $\zeta$ by $c^{p-1}\zeta$ and u by cu in the inequality above yields

$$\begin{aligned} |\langle \zeta ,v_0\rangle |\le c(p-1)J(u) \end{aligned}$$

after using the homogeneity of J and dividing by $c^{p-1}$. Letting $c\searrow 0$ concludes the proof. $\square$

The following Proposition states that the value of the functional J and its subdifferential is invariant under addition of a null-space element.

Proposition A.3

Let $v_0\in \mathcal {N}(J)$, $u\in \mathcal {H}$, and $\zeta \in \partial J(u)$. Then it holds $J(u)=J(u+v_0)$ and $\zeta \in \partial J(u+v_0)$.

Proof

In the case $p=1$ the statement has been proven in [14, 16]. In the case $p>1$ one argues as follows: For every $\zeta \in \partial J(u)$ it holds

$$\begin{aligned} pJ(u)=\langle \zeta ,u\rangle =\langle \zeta ,u+v_0\rangle \le J(u+v_0)+(p-1)J(u) \end{aligned}$$

which implies $J(u)\le J(u+v_0)$. On the other hand, since J equals its bi-conjugate, it holds

$$\begin{aligned} J(u+v_0)=J^{**}(u+v_0)=\sup _{\zeta \in \mathcal {H}}\langle \zeta ,u+v_0\rangle -J^*(\zeta )=: \sup _{\zeta \in \mathcal {H}}H(\zeta ;u+v_0) \end{aligned}$$

where $H(\zeta ;v):=\langle \zeta ,v\rangle -J^*(\zeta )$. For any $\zeta \in \mathcal {H}$ which is a subgradient, we can use the orthogonality from Proposition A.2 together with the definition of the convex conjugate to infer

$$\begin{aligned} H(\zeta ;u+v_0)=\langle \zeta ,u+v_0\rangle -J^*(\zeta )=\langle \zeta ,u\rangle -J^*(\zeta )\le J(u). \end{aligned}$$

This implies that $\sup _{\zeta \in \mathcal {S}}H(\zeta ;u+v_0)\le J(u)$, where $\mathcal {S}\subset \mathcal {H}$ is given by

$$\begin{aligned} \mathcal {S}:=\bigcup _{u\in {\mathrm {dom}}(\partial J)}\partial J(u) \end{aligned}$$

and ${\mathrm {dom}}(\partial J):=\left\{ u\in \mathcal {H}\,:\,\partial J(u)\ne \emptyset \right\}$. Now it is classical [11] that, given the coercivity inequality (2.6) for $p>1$, the operator $\partial J$ is surjective, meaning that $\mathcal {S}=\mathcal {H}$. Hence, we can conclude that

$$\begin{aligned} J(u+v_0)=\sup _{\zeta \in \mathcal {H}}H(\zeta ;u+v_0)=\sup _{\zeta \in \mathcal {S}}H(\zeta ;u+v_0)\le J(u). \end{aligned}$$

Having established $J(u)=J(u+v_0)$ it is straightforward to show $\zeta \in \partial J(u+v_0)$ using the definition of the subdifferential together with the orthogonality from Proposition A.2. $\square$

Gradient flow of absolutely p-homogeneous functionals

We conclude the “Appendix” by listing several important properties of the gradient flow (GF), starting with the existence theorem due to Brezis. To this end, we have to introduce the single-valued operator

$$\begin{aligned} \partial ^0 J(u)=\mathrm {arg}\min \left\{ \left\| \zeta \right\| \,:\,\zeta \in \partial J(u)\right\} , \quad u\in \mathcal {H}, \end{aligned}$$

(B.1)

which gives the subgradient with minimal norm in $\partial J(u)$ and is well defined since the latter is a convex set.

Theorem B.1

(Brezis) Let $J:\mathcal {H}\rightarrow \mathbb {R}\cup \{\infty \}$ be convex and lower semi-continuous and let $f\in \mathcal {H}$. Then, there exists exactly one continuous map $u:[0,\infty )\rightarrow \mathcal {H}$ which is Lipschitz continuous on $[\delta ,\infty ),\;\delta >0$ and right-differentiable on $(0,\infty )$ such that

• $u(0)=f$,

• $\zeta (t):=\partial _t^+ u(t)=-\partial J^0(u(t))$ for all $t>0$,

• $t\mapsto J(u(t))$ is convex, non-increasing and Lipschitz continuous on $[\delta ,\infty ),\;\delta >0$ with

  $$\begin{aligned} \frac{\,\mathrm {d}^+}{\,\mathrm {d}t}J(u(t))=-\left\| \zeta (t) \right\| ^2,\quad t>0, \end{aligned}$$

  (B.2)

where $\,\mathrm {d}^+/\,\mathrm {d}t$ and $\partial _t^+$ denote right-derivatives and will be replaced by standard derivative symbols throughout the rest of this manuscript.

Proposition B.1

(Conservation of mass) Let u(t) solve the gradient flow (GF) with data f and let $\overline{\cdot }:\mathcal {H}\rightarrow \mathcal {N}(J)$ denote the orthogonal projection onto $\mathcal {N}(J)$. Then it holds $\overline{u(t)}=\overline{f}$.

Proof

It holds

$$\begin{aligned} u(t)-f=-\int _0^t\zeta (s)\,\mathrm {d}s \end{aligned}$$

and from Proposition A.2 we deduce $\overline{\zeta (s)}=0$ for all $s>0$. Hence, also $\overline{u(t)-f}=0$ which implies the statement due to linearity of the projection. $\square$

Proposition B.2

Let u(t) solve (GF) with data $f-\overline{f}\in \mathcal {N}(J)^\perp$. Then $v(t):=u(t)+\overline{f}$ solves (GF) with data f.

Proof

The proof reduces to checking whether $-\partial _t v(t)=-\partial _t u(t)\in \partial J(v(t))$, which is true due to Proposition A.3. $\square$

Proposition B.3

Let u(t) denote the solution of the gradient flow (GF) corresponding to the absolutely p-homogeneous functional J and let $f\in \mathcal {N}(J)^\perp$. Then it holds

$$\begin{aligned} u(t)\rightarrow 0,\quad&J(u(t))\rightarrow 0, \quad t\rightarrow \infty , \end{aligned}$$

(B.3)

$$\begin{aligned} \frac{\,\mathrm {d}}{\,\mathrm {d}t}\frac{1}{2}\left\| u(t) \right\| ^2&=-pJ(u(t)),\quad t>0. \end{aligned}$$

(B.4)

Proof

The proof for $u(t)\rightarrow 0$ can be found in [14] and mainly relies on Proposition A.2. The statement $J(u(t))\rightarrow 0$ is classical [11]. For (B.4) one uses the chain rule together with (1.4) to obtain

$$\begin{aligned} \frac{\,\mathrm {d}}{\,\mathrm {d}t}\frac{1}{2}\left\| u(t) \right\| ^2=\left\langle \partial _tu(t),u(t)\right\rangle =-\langle \zeta (t),u(t)\rangle =-pJ(u(t)). \end{aligned}$$

$\square$