The fractional porous medium equation on the hyperbolic space

Abstract

We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $\Delta _{\mathbb {H}^{N}}$, or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.

Appendix A. Green function estimates in \(\mathbb {H}^{N}\)

This section is devoted to provide proper estimates of Green function $\mathbb {G}_{\mathbb {H}^N}^s(x, y)$ for the fractional laplacian on the hyperbolic space for $0< s < 1.$ It is well-known that the Green function is given by the following explicit formula

$$\begin{aligned} \mathbb {G}_{\mathbb {H}^N}^s(x, y) = \int _0^{+\infty } \frac{k_{\mathbb {H}^{N}}(t,x,y)}{t^{1-s}}\,dt, \end{aligned}$$

where $k_{\mathbb {H}^{N}}(t,x,y)$ is the heat kernel for $- \Delta _{\mathbb {H}^{N}}.$ Furthermore, $k_{\mathbb {H}^{N}}$ satisfies the following estimates (see [21]): for $N \ge 2,$ there exist some positive constants $A_N$ and $B_N$ such that

$$\begin{aligned} A_N h_N(t, x, y) \le k_{\mathbb {H}^{N}}(t, x, y) \le B_N h_N(t, x, y) \quad \text{ for } \text{ all } \ t > 0 \ \text{ and } \ x, y \in \mathbb {H}^{N}, \end{aligned}$$

(A.1)

where $h_N (t, x, y)$ is given by

$$\begin{aligned} h_N(t, x, y) := h_N(t, r) = (4 \pi t)^{-\frac{N}{2}} e^{- \frac{(N-1)^2 t}{4} - \frac{(N-1) r}{2} - \frac{r^2}{4t}} ( 1 + r + t)^{\frac{(N-3)}{2}}(1 + r), \end{aligned}$$

(A.2)

where $r := r(x, y) = \mathrm{dist}(x, y)$.

Using (A.1) we shall obtain the following Green functions estimates:

Lemma A.1

Let $N \ge 3$ and $0< s < 1$. Then there exist $R_1, R_2, {\underline{C}}_1, {{\overline{C}}}_1, {\underline{C}}_2, {{\overline{C}}}_2>0$, only depending on N and s, such that

$$\begin{aligned} \frac{{\underline{C}}_1}{(r(x, y))^{N -2s}} \le \mathbb {G}_{\mathbb {H}^N}^s(x, y) \le \frac{{{\overline{C}}}_1}{(r(x, y))^{N -2s}} \quad \text { for all }(x, y) \in \mathbb {H}^{N}: 0< r(x, y) \le R_1; \end{aligned}$$

(A.3)

and

$$\begin{aligned} \frac{{\underline{C}}_2\,e^{-(N-1) r(x, y)}}{(r(x, y))^{1-s}} \le \mathbb {G}_{\mathbb {H}^N}^s(x, y) \le \frac{{{\overline{C}}}_2\, e^{-(N-1) r(x, y)}}{(r(x, y))^{1-s}} \quad \text { for all }(x, y) \in \mathbb {H}^{N}:r(x, y) \ge R_2. \end{aligned}$$

(A.4)

Proof

Let $h_{N}(t,r)$ be as defined in (A.2), we compute

$$\begin{aligned}&\int _0^{+\infty } \frac{h_{N}(t,r)}{t^{1-s}}\,dt = \int _{0}^{\infty } \frac{(4 \pi t)^{N/2} e^{-\frac{(N-1)^2}{4}t - \frac{(N-1)r}{2}- \frac{r^2}{4t}}(1 + r + t)^{(N-3)/2}(1 + r)}{t^{1 -s}}\,\mathrm{d}t \nonumber \\&\quad = \frac{e^{-(N-1)r/2 }(1 + r)}{(4 \pi )^{N/2}} \int _{0}^{\infty } t^{-N/2 - 1 + s} (1 + r + t)^{(N-3)/2} e^{-(\frac{(N-1) \sqrt{t}}{2} - \frac{r}{2 \sqrt{t}})^2} e^{-(N-1)r/2 } \, \mathrm{d}t \nonumber \\&\quad = \frac{e^{-(N-1) r}(1 + r)}{(4 \pi )^{N/2}} \int _{0}^{\infty } t^{\frac{-N + 2s -2}{2}} (1 + r + t)^{(N-3)/2} e^{-(\frac{(N-1) \sqrt{t}}{2} - \frac{r}{2 \sqrt{t}})^2} \, \mathrm{d}t. \end{aligned}$$

(A.5)

Now making the following substitution one can write

$$\begin{aligned} z = \frac{r}{2\sqrt{t}} - \frac{(N-1)\sqrt{t}}{2} \ \Leftrightarrow \sqrt{t} = \frac{- z + \sqrt{z^2 + (N-1) r}}{N-1}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \mathrm{d}t = -\frac{2}{(N-1)^2} \frac{( \sqrt{z^2 + (N-1)r} - z)^2}{\sqrt{z^2 + (N-1) r}} \, \mathrm{d}z. \end{aligned}$$

Further substituting back in (A.5) yields

$$\begin{aligned}&\int _0^{+\infty } \frac{h_{N}(t,r)}{t^{1-s}}\,dt = \frac{ e^{-(N-1)r}(1 + r)}{(N-1)^{2s - 3}(4\pi )^{N/2}} \times \\&\quad \left\{ \int _{-\infty }^{\infty } \frac{(1 + r + ( \sqrt{z^2 + (N-1)r} -z)^2 )^{(N-3)/2} (\sqrt{z^2 + (N-1)r} - z)^2 e^{-z^2}}{(-z + \sqrt{z^2 + (N-1)r})^{N - 2s +2} \sqrt{z^2 + (N-1) r}} \, \mathrm{d}z \right\} . \end{aligned}$$

Now for $r \rightarrow \infty ,$ one can look for the leading term as follows

$$\begin{aligned}&\int _0^{+\infty } \frac{h_{N}(t,r)}{t^{1-s}}\,dt = C_N e^{-(N-1)r}(1 + r) r^{\frac{-N + 2s -2}{2} + \frac{N-3}{2} + \frac{1}{2}} \times \\&\quad \underbrace{ \left\{ \int _{-\infty }^{\infty } \frac{((1/r + 1)(N-1)^2 + ( \sqrt{z^2/r + (N-1)} -z/\sqrt{r})^2 )^{(N-3)/2} (\sqrt{z^2/r + (N-1)} - z/\sqrt{r})^2 e^{-z^2}}{(-z/r + \sqrt{z^2 + (N-1)})^{N - 2s +2} \sqrt{z^2/r + (N-1)}} \, \mathrm{d}z \right\} }_= I_{N} . \end{aligned}$$

The finiteness of $I_N$ follows easily from the fact that $e^{-z^2}$ is the leading term at infinity. Therefore, recalling (A.1), we obtain

$$\begin{aligned} \int _0^{+\infty } \frac{h_{N}(t,r)}{t^{1-s}}\,dt = C_N I_N e^{-(N-1)r} r^{s-1} + \circ (e^{-(N-1)r} r^{s-1}) \quad \text {as } r \rightarrow \infty . \end{aligned}$$

Hence, we obtain (A.4). Now we turn to prove (A.3). To see this we can make the change of variable as $\alpha = \frac{r^2}{4t}$ and therefore we can write

$$\begin{aligned} \int _0^{+\infty } \frac{h_{N}(t,r)}{t^{1-s}}\,dt = C_N \frac{e^{-(N-1)r/2} (1 + r)}{r^{(N-2s)}} \underbrace{ \int _{0}^{\infty } {\alpha }^{\frac{N-2s -2}{2}} e^{\frac{-(N-1)^2 r^2}{16 \alpha } - \alpha } \left( 1 + r + \frac{r^2}{4 \alpha }\right) ^{\frac{(N-3)}{2}} \, \mathrm{d}\alpha }_= J_N. \end{aligned}$$

Again, as before, the integral is finite for $0< s < 1$ and hence

$$\begin{aligned} G_{s}(r, t) = C_N J_N r^{-(N - 2s)} + o(r^{-(N - 2s)})\quad \text {as } r \rightarrow 0^+. \end{aligned}$$

This proves the lemma. $\square$

From Lemma A.1 we derive

Corollary A.2

Let $N \ge 3$ and $0< s < 1$. Then there exist positive constants ${\underline{C}}_3,{{\overline{C}}}_3, {\underline{C}}_4, {{\overline{C}}}_4$, only depending on N and s, such that

$$\begin{aligned} {\underline{C}}_3 \frac{1}{(r(x, y))^{N-2s}} \le \mathbb {G}_{\mathbb {H}^N}^s(x, y) \le {{\overline{C}}}_3 \frac{1}{(r(x, y))^{N-2s}} \quad \text { for all }(x, y) \in \mathbb {H}^{N}: 0<r(x, y) \le 1\,, \end{aligned}$$

(A.6)

$$\begin{aligned} {\underline{C}}_4 \frac{ e^{-(N-1) r(x, y)}}{(r(x, y))^{1-s}} \le \mathbb {G}_{\mathbb {H}^N}^s(x, y) \le {{\overline{C}}}_4 \frac{ e^{-(N-1) r(x, y)}}{(r(x, y))^{1-s}} \quad \text { for all }(x, y) \in \mathbb {H}^{N}: r(x, y) \ge 1\,.\nonumber \\ \end{aligned}$$

(A.7)

Furthermore, the following global estimate holds:

$$\begin{aligned} \mathbb {G}_{\mathbb {H}^N}^s(x, y) \le \frac{{\overline{C}}_5}{(r(x, y))^{N-2s}} \quad \text { for all }(x, y) \in \mathbb {H}^{N}: r(x, y) > 0\,, \end{aligned}$$

(A.8)

where ${\overline{C}}_5=\min \{{{\overline{C}}}_3,{{\overline{C}}}_4\}$.

Proof

The proof of this corollary is a straightforward consequence of Lemma A.1. We give a detailed proof of (A.7), the proof of (A.6) can be achieved with similar arguments. On the other hand, the proof of (A.8) follows by noting that $e^{-(N-1)r}\le r^{-N+s+1}$ for all $r>1$ and then combining (A.6) with (A.7). $\square$

Proof of (A.7)

We may assume $R_2 > 1,$ where $R_2$ is as given in Lemma A.1 (otherwise, there is nothing to prove). Next we set

$$\begin{aligned} g(r):= \frac{e^{-(N-1)r}}{r^{1-s}} \quad \text {with }r>0 \, \end{aligned}$$

(A.9)

and we note that

$$\begin{aligned} g(r(x, y)) \ge g(R_2) \quad \text {for all }1 \le r(x, y) \le R_2\,. \end{aligned}$$

Set $M : = {\mathrm{max}}_{1 \le r(x, y) \le R_2} \mathbb {G}_{\mathbb {H}^N}^s(x, y)$, then

$$\begin{aligned} \mathbb {G}_{\mathbb {H}^N}^s(x, y) \le M = \frac{M}{g(R_2)} g(R_2) \le \frac{M}{g(R_2)} g(r(x, y)) \quad \text {for all } 1 \le r(x, y) \le R_2. \end{aligned}$$

By taking ${{\overline{C}}}_4: = {\mathrm{max}} \{ \frac{M}{g(R_2)}, \overline{C}_2 \}$ we get the upper bound in (A.7).

Now, to obtain the lower bound, we note that

$$\begin{aligned} g(r(x, y)) \le g(1) \quad \text {for all }1 \le r(x, y) \le R_2\,. \end{aligned}$$

Then, we set $m : = { \mathrm min}_{1 \le r(x, y) \le R_2} \mathbb {G}_{\mathbb {H}^N}^s(x, y)$ and we conclude

$$\begin{aligned} \mathbb {G}_{\mathbb {H}^N}^s(x, y) \ge m =\frac{m}{g(1)} g(1) \ge \frac{m}{g(1)} g(r(x, y)) \quad \text {for all } 1 \le r(x, y) \le R_2. \end{aligned}$$

Taking ${\underline{C}}_4 := {\mathrm{min}} \{ \frac{m}{g(1)}, \underline{C}_2 \},$ we obtain the lower bound in (A.7). $\square$

It can be useful to check that, if $\psi \in C^1_c(0,T; \text{ L}_c^{\infty }(\mathbb {H}^{N}))$ with $\psi \ge 0$, then $(- \Delta _{\mathbb {H}^{N}})^{-s}(\psi )$ behaves like $G_{s}(x_0,x)$ near infinity:

Lemma A.3

For all $\psi \in C^1_c(0,T; \text{ L}_c^{\infty }(\mathbb {H}^{N}))$ with $\psi \ge 0$, there exist ${{\hat{R}}} > 0$ and two positive functions ${\underline{C}},{{\overline{C}}} \in C^1_c(0,T)$ such that

$$\begin{aligned} {\underline{C}}(t)\,\frac{ e^{-(N-1) r(x_0, x)}}{(r(x_0, x))^{1-s}} \le (-\Delta _{\mathbb {H}^{N}})^{-s}\,\psi (x,t) \le {{\overline{C}}}(t)\, \frac{ e^{-(N-1) r(x_0, x)}}{(r(x_0, x))^{1-s}} \quad \text { for } r(x_0,x)\ge {{\hat{R}}}\,. \end{aligned}$$

(A.10)

Furthermore, for all $R>0$, there exists a positive function ${{\overline{D}}} \in C^1_c(0,T)$ such that

$$\begin{aligned} (-\Delta _{\mathbb {H}^{N}})^{-s}\,\psi (x,t) \le {{\overline{D}}}(t) \quad \text { for } r(x_0,x)\le R\,. \end{aligned}$$

(A.11)

Proof

Proof of (A.10). We first prove the upper bound. Let $g=g(r)$ be as in (A.9) and denote with $K\subset \mathbb {H}^{N}$ the compact support of $\psi$ with respect to the space variable. Then we can write

$$\begin{aligned} (-\Delta _{\mathbb {H}^{N}})^{-s}\,\psi (x,t)&= \int _{K} \mathbb {G}_{\mathbb {H}^N}^s(x, y) \psi (y,t) \, \mathrm{d}\mu _{\mathbb {H}^N}(y) \nonumber \\&= g(r(x_0, x)) \int _{K} \frac{ \mathbb {G}_{\mathbb {H}^N}^s(x, y)}{g(r(x_0, x))} \, \psi (y,t) \, \mathrm{d}\mu _{\mathbb {H}^N}(y)\,. \end{aligned}$$

(A.12)

Furthermore, by Corollary A.2 there exist constants ${\underline{C}}_4$ and ${{\overline{C}}}_4$ such that there holds

$$\begin{aligned} {\underline{C}}_4\, g(r(x, y)) \le \mathbb {G}_{\mathbb {H}^N}^s(x, y) \le {{\overline{C}}}_4 \, g(r(x, y)) \quad \text{ for } \ r(x, y) > 1. \end{aligned}$$

Let $\gamma > 0$ be such that $r(x_0, y) \le \gamma$ for all $y \in K$ and assume $r(x_0, x) \ge 1 + \gamma ,$ we have

$$\begin{aligned} r(x,y) \ge r(x_0, x) - r(x_0, y) > 1 + \gamma - \gamma = 1. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\mathbb {G}_{\mathbb {H}^N}^s(x, y)}{g(r(x_0, x))} \le {{\overline{C}}}_4 \frac{g(r(x,y))}{g(r(x_0, x))}\quad \text{ for } \ r(x_0, x) \ge 1 + \gamma . \end{aligned}$$

Now, using the fact that the function g is decreasing, for $r(x_0, x) \ge 1 + \gamma$ and for all $y\in K$ we have

$$\begin{aligned} \frac{\mathbb {G}_{\mathbb {H}^N}^s(x, y)}{g(r(x_0, x))}&\le {{\overline{C}}}_4 \, \frac{g(r(x, y))}{g(r(x_0, x))} \le {{\overline{C}}}_4 \, \frac{g(r(x_0, x) - r(x_0, y))}{g(r(x_0, x))} \\&= {{\overline{C}}}_4 \, e^{(N-1)r(x_0, y)} \left( \frac{r(x_0, x)}{r(x_0, x) - r(x_0, y)} \right) ^{1-s} \\&= {{\overline{C}}}_4 \, e^{(N-1)r(x_0, y)} \frac{1}{\left( 1- \frac{r(x_0, y)}{r(x_0, x)} \right) ^{1-s}} \le {{\overline{C}}}_4 \frac{e^{(N-1)r(x_0, y)}}{\left( 1- \frac{r(x_0, y)}{1 + \gamma } \right) ^{1-s}}\,, \end{aligned}$$

where the latter term is well defined since $r(x_0, y) \le \gamma$ for all $y \in K.$ Now, plugging the above estimate into (A.12) we conclude that

$$\begin{aligned} (-\Delta _{\mathbb {H}^{N}})^{-s}\,\psi (x,t) \le g(r(x_0, x)) \underbrace{\int _{K} {{\overline{C}}}_4 \, \frac{e^{(N-1)r(x_0, y)}}{\left( 1- \frac{r(x_0, y)}{1 + \gamma } \right) ^{1-s}} \, \psi (y,t)\, \mathrm{d}\mu _{\mathbb {H}^N}(y)}_{= {{\overline{C}}}(t)\in C^1_c(0,T)} \quad \text {for all } r(x_0, x) \ge 1+\gamma \,. \end{aligned}$$

Next we turn to prove lower the bound in (A.10). For $r(x_0, x) \ge 1+\gamma$, invoking again Corollary A.2, we have

$$\begin{aligned} \int _{K} \frac{\mathbb {G}_{\mathbb {H}^N}^s(x, y)}{g(r(x_0, x))} \, \psi (y,t) \, \mathrm{d}\mu _{\mathbb {H}^N}(y)&\ge {\underline{C}}_4 \int _{K} \frac{g(r(x, y))}{g(r(x_0, x))} \, \psi (y,t) \, \mathrm{d}\mu _{\mathbb {H}^N}(y) \\&= {\underline{C}}_4 \int _{K} e^{(N-1) (r(x_0, x) - r(x, y))} \left( \frac{r(x_0, x)}{r(x, y)} \right) ^{1-s} \, \psi (y,t) \, \mathrm{d}\mu _{\mathbb {H}^N}(y) \\&\ge {\underline{C}}_4 \int _{K} e^{-(N-1) r(x_0, y)} \left( \frac{r(x_0, x)}{r(x, y)} \right) ^{1-s} \, \psi (y,t) \, \mathrm{d}\mu _{\mathbb {H}^N}(y)\,, \end{aligned}$$

where the last estimate follows from

$$\begin{aligned} r(x_0, x) - r(x, y) \ge - r(x_0, y)\,. \end{aligned}$$

Assume now that $r(x_0, x) > 3\gamma$, recalling that $r(x_0, y) \le \gamma ,$ we infer

$$\begin{aligned} r(x, y) \ge r(x_0, x) - r(x_0, y)\ge 2\gamma \end{aligned}$$

which yield

$$\begin{aligned} \left( 1 - \frac{r(x_0, y)}{r(x, y)} \right) ^{1-s} \ge \frac{1}{2^{1-s}}. \end{aligned}$$

Inserting this into (A.12) we finally obtain

$$\begin{aligned} (-\Delta _{\mathbb {H}^{N}})^{-s}\,\psi (x,t) \ge g(r(x_0, x)) \underbrace{\int _{K} {{\overline{C}}}_4 \, \frac{e^{(N-1)r(x_0, y)}}{2^{1-s}} \, \psi (y,t)\, \mathrm{d}\mu _{\mathbb {H}^N}(y)}_{= {\underline{C}}(t)\in C^1_c(0,T)} \quad \text {for all } r(x_0, x) > 3\gamma . \end{aligned}$$

Summarising, the proof of (A.10) follows by taking ${{\hat{R}}}:=\max \{1+\gamma , 3\gamma \}$.

Proof of (A.11). As above, let K be the compact support of $\psi$ with respect to x and let $\gamma > 0$ be such that $r(x_0, y) \le \gamma$ for all $y \in K$. Let $R>0$, for all $r(x_0,x)\le R$ we have $r(x,y)\le r(x,x_0)+r(x_0,y)\le R +\gamma$. Hence, by exploiting the estimate (A.8), we infer

$$\begin{aligned} (-\Delta _{\mathbb {H}^{N}})^{-s}\,\psi (x,t)&= \int _{K} \mathbb {G}_{\mathbb {H}^N}^s(x, y) \psi (y,t) \mathrm{d}\mu _{\mathbb {H}^N}(y) \\&\le \Vert \psi (x,t)\Vert _{L^{\infty }(\mathbb {H}^{N})} \int _{B_{R +\gamma }(x)} \frac{{\overline{C}}_5}{(r(x, y))^{N-2s}} \, \mathrm{d}\mu _{\mathbb {H}^N}(y) \end{aligned}$$

and the proof follows by setting $D(t):= \Vert \psi (x,t)\Vert _{L^{\infty }(\mathbb {H}^{N})} \omega _N \int _{0}^{R +\gamma } \frac{{\overline{C}}_5}{r^{N-2s}} \,(\sinh (r))^{N-1}\, \mathrm{d}r$, where $\omega _N$ is the volume of the N dimensional unit sphere. $\square$