Long time behavior of solutions of degenerate parabolic equations with inhomogeneous density on manifolds

TL;DR

This paper studies the long-term behavior of solutions to degenerate parabolic equations with inhomogeneous density on manifolds, revealing how geometry and coefficients influence solution decay, propagation, and blow-up phenomena.

Contribution

It establishes new estimates on solution decay, propagation speed, and blow-up conditions for degenerate parabolic equations on manifolds with inhomogeneous densities.

Findings

Solutions exhibit finite speed of propagation in subcritical ranges.

Universal bounds and finite-time blow-up are proven in supercritical ranges.

The geometry of the manifold and decay of the weight function critically affect solution behavior.

Abstract

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at infinity. We connect the behavior of non-negative solutions to the interplay between such coefficient and the geometry of the manifold, obtaining, in a suitable subcritical range, estimates of the vanishing rate for long times and of the finite speed of propagation. In supercritical ranges we obtain universal bounds and prove blow up in a finite time of the (initially bounded) support of solutions.

Equations

$$\rho (x)u_t-\mathrm{div}(u^{m-1}|\nabla u{|}^{p-2}\nabla u)$$

$$u(x,0)$$

$$N>p>1,p+m>3,$$

$$|\partial U{|}_{N-1}\ge g(\mu (U)),\text{for all open bounded Lipschitz U⊂M.}$$

$$\omega (v)=\frac{v^{\frac{N-1}{N}}}{g(v)},v>0,\omega (0)=\underset{v\to 0+}{\lim }\omega (v),$$

$$V(2R)\le CV(R),$$

$$g(V(R))\ge c\frac{V(R)}{R},\text{i.e.,}\omega (V(R))\le c^{-1}\frac{R}{V(R{)}^{\frac{1}{N}}},$$

$${\int }_0^s\frac{\text{d}\tau }{V^{(-1)}(\tau {)}^p}\text{d}\mu \le C\frac{s}{V^{(-1)}(s{)}^p},s>0,$$

$$\rho (2R)\ge C^{-1}\rho (R),$$

$$c\tilde{\rho }(x)\le \rho (x)\le c^{-1}\tilde{\rho }(x),x\in M,$$

$$\mathrm{supp}{u}_0\subset B_{R_0},\text{for a given R0>0.}$$

$$\mathrm{supp}u(t)\subset B_{\bar{R}},$$

$$\Vert u(t)\rho {\Vert }_{L^1(M)}=\Vert u_0\rho {\Vert }_{L^1(M)},0<t<\bar{t}.$$

$$\Vert u(t)\rho {\Vert }_{L^1(M)}\le \gamma \Vert u_0\rho {\Vert }_{L^1(M)},0<t<+\infty .$$

$$R^p\rho (R{)}^{p+m-2}\mu (B_R{)}^{p+m-3}\ge \gamma t\Vert u_0\rho {\Vert }_{L^1(M)}^{p+m-3},$$

$$R\mapsto R^p\rho (R{)}^{p+m-2}\mu (B_R{)}^{p+m-3},R>\bar{R},$$

$$R^p\rho (R{)}^{p+m-2}\mu (B_R{)}^{p+m-3}=\gamma t\Vert u_0\rho {\Vert }_{L^1(M)}^{p+m-3},$$

$${\int }_{B_R}\rho (x)\text{d}\mu \le C\mu (B_R)\rho (R),$$

$$\psi (s)=s^p\rho (s),s\ge 0,$$

$$\psi (s)\le C\psi (t),\text{for all t>s>0.}$$

$$\rho (cs)\le C{c}^{-\alpha }\rho (s),\text{for all s>0 and 1>c>0.}$$

$$\mathrm{supp}u(t)\subset B_{Z(t)},t>0,$$

$$\lVert u(t)\rVert_{L^{\infty}(M)}\leq\gamma\Big{(}\frac{Z(t)^{p}\rho(Z(t))}{t}\Big{)}^{\frac{1}{p+m-3}}\,,\qquad t>0\,.$$

$$\Vert u(t){\Vert }_{L^{\infty }(M)}\le \gamma t^{-\frac{1}{p+m-3}},t>0.$$

$${\int }_{M}d(x{)}^{\frac{p}{p+m+\theta -3}}\rho (x{)}^{\frac{p+m+\theta -2}{p+m+\theta -3}}\text{d}\mu <+\infty ,$$

$${\int }_{M}d(x{)}^{\frac{p(1+\theta )}{p+m-3}}\rho (x{)}^{\frac{p+m+\theta -2}{p+m-3}}\text{d}\mu <+\infty .$$

$$\psi (s{)}^{\frac{1}{p+m+\theta -3}}\rho (s)\ge {\gamma }_0\psi (s{)}^{\frac{1+\theta }{p+m-3}}\rho (s),$$

$$\alpha <{\alpha }_{\ast }:=\frac{N(p+m-3)+p}{p+m-2},$$

$$Z_0(t)=\gamma (t\Vert u_0\rho {\Vert }_{L^1(M)}^{p+m-3}{)}^{\frac{1}{(N-\alpha )(p+m-3)+p-\alpha }}.$$

$$\Vert u(t){\Vert }_{L^{\infty }(M)}\le \gamma t^{-\frac{N-\alpha }{(N-\alpha )(p+m-3)+p-\alpha }}\Vert u_0\rho {\Vert }_{L^1(M)}^{\frac{p-\alpha }{(N-\alpha )(p+m-3)+p-\alpha }}.$$

$$\frac{1}{\sqrt{\det(g_{ij})}}\sum_{i,j=1}^{N}\frac{\partial}{\partial{x^{i}}}\Big{(}\sqrt{\det(g_{ij})}g^{ij}u^{m-1}\lvert\operatorname{\nabla}u\rvert^{p-2}\frac{\partial u}{\partial x^{j}}\Big{)}\,,$$

Full text

The first author is member of the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM)

The second author was supported by Sapienza Grant C26V17KBT3

Keywords: Doubly degenerate parabolic equation, noncompact Riemannian manifold, inhomogeneous density, interface blow up, optimal decay estimates.

AMS Subject classification: 35K55, 35K65, 35B40.

1. Introduction

1.1. Statement of the problem and general assumptions

We consider the Cauchy problem

[TABLE]

Here $M$ is a Riemannian manifold of topological dimension $N$, with infinite volume. We always assume we are in the degenerate case, that is

[TABLE]

and that $u\geq 0$. The inhomogeneous density $\rho$ is assumed to be a globally bounded, strictly positive and nonincreasing function of the distance $d$ from a fixed point $x_{0}\in M$. With a slight abuse of notation we still denote $\rho(d(x,x_{0}))=\rho(x)$. In the following all balls $B_{R}\subset M$ are understood to be centered at $x_{0}$, and we denote $d(x)=d(x,x_{0})$, $V(R)=\mu(B_{R})$.

Let us briefly explain the interest of this problem; in the case when $M=\boldsymbol{R}^{N}$ with the Euclidean metric, the first results on the qualitative surprising properties of solutions to the porous media equation with inhomogeneous density are due to [11], [18] (in cases reduced to dimension $1$). The interface blow up in the same setting was discovered in [9] and proved in [21] for a general class of doubly degenerate parabolic equations.

In the Euclidean case, where we assume that $\rho(x)=(1+\lvert x\rvert)^{-\alpha}$, $x\in\boldsymbol{R}^{N}$, for a given $0<\alpha\leq N$, the behavior of solutions depends sharply on the interplay between the nonlinearities appearing in the equation. Specifically, two different features concern us here: the form taken by sup bounds for solutions, and the property of finite speed of propagation (which is actually connected to conservation of mass), see [21] for the following results; see also [7].

If $\alpha\leq p$ one can prove sup estimates similar in spirit to those valid for the standard doubly nonlinear equation with coefficients independent of $x$, though different in the details of functional dependence on the parameters of the problem. That is, a decay as a negative power law of time, multiplied by a suitable power of the initial mass. But, if $\alpha>p$ a universal bound holds true, that is the initial mass does not appear in the estimate anymore.

If the initial data is compactly supported, the evolution of the support of the solution differs markedly in the case $\alpha<\alpha_{*}$ and $\alpha>\alpha_{*}$, where $\alpha_{*}\in(p,N)$ is an explicit threshold. In the subcritical case the support is bounded for all times, and mass is conserved accordingly. In the supercritical case both properties fail after a finite time interval has elapsed.

A more detailed comparison with the Euclidean case is presented in Subsection 1.5 below. Before passing to our results, we quote the following papers dealing with parabolic problems in the presence of inhomogeneous density: [13], [14] where blow up phenomena are investigated; [17], [10] for an asymptotic expansion of the solution of the porous media equation; [15], [8] where the critical case is dealt with.

The main goal of the present paper is to find a similar characterization of the possible behavior of solutions in terms of the density function $\rho$, the nonlinearities in the equation, and of course the Riemannian geometry of $M$. See also [1] for the Euclidean case; we employ the energy approach of [2, 4, 5, 6]. We prove new embedding results which we think are of independent interest, besides allowing us to achieve the sought after precise characterization of the solutions to our problem.

The geometry of $M$ enters our results via the nondecreasing isoperimetric function $g$ such that

[TABLE]

Here $\mu$ denotes the Riemannian measure on $M$, and $\lvert\cdot\rvert_{N-1}$ the corresponding $(N-1)$-dimensional Haussdorff measure. The properties of $g$ are encoded in the function

[TABLE]

which we assume to be continuous and nondecreasing; in the Euclidean case $\omega$ is constant. We also assume that for all $R>0$, $\gamma>1$,

[TABLE]

for a suitable constant $C>1$. In some results we need the following natural assumption on $\omega$, or on $g$ which is the same:

[TABLE]

for $R>0$, where $c>0$ is a given constant. In fact, one could see that the converse to this inequality follows from the assumed monotonic character of $\omega$; thus (1.6) in practice assumes the sharpness of such converse. Finally we require

[TABLE]

which clearly places a restriction on $p$ depending on the growth of $V$.

The density function $\rho$ is assumed to satisfy for all $R>0$

[TABLE]

for a suitable $C>1$.

Remark 1.1*.*

It follows without difficulty from our arguments that the radial character and the assumptions on $\rho$ can be replaced by analogous statements on a radial function $\tilde{\rho}$ such that

[TABLE]

for a given $0<c<1$.

All the assumptions stated so far will be understood in the following unless explicitly noted.

1.2. Conservation of mass

Since $\rho$ is globally bounded, the concept of weak solution is standard. We need the following easy a priori result. Note that it holds regardless of other assumptions on the parameters, whenever standard finitely supported cutoff test functions can be used in the weak formulation (see the proof in Section 2).

We assume for our first results that

[TABLE]

Theorem 1.2**.**

Let $u$ be a solution to (1.1)–(1.2), with $\rho u_{0}\in L^{1}(M)$ satisfying (1.9). Assume that for $0<t<\bar{t}$

[TABLE]

for some $\bar{R}>R_{0}$. Then

[TABLE]

Remark 1.3*.*

At least in the subcritical case of Subsection 1.3, a solution $u$ to (1.1)–(1.2) can be obtained as limit of a sequence to Dirichlet problems with vanishing boundary data on $B_{R}$ with $R\to+\infty$. Since we can limit the $L^{1}(M)$ norm of each such approximation only in terms of the initial mass, passing to the limit we infer

[TABLE]

Notice that this bound follows without assuming finite speed of propagation.

However, known results [16] imply uniqueness in the class of solutions satisfying finite speed of propagation. Below we prove for the constructed solution exactly this property, so that our results apply to the unique such solution. Perhaps more general results of uniqueness follow from arguments similar to the ones quoted, but we do not dwell on this problem here.

1.3. The subcritical cases

In this Subsection we gather results valid in subcritical cases, where however we consider two different notions of subcriticality, the first one being the increasing character of the function in (1.14), the second one being condition (1.18). The latter is stronger in practice, see Subsection 1.5.

We give first our basic result about finite speed of propagation.

Theorem 1.4**.**

Let (1.9), (1.12) be fulfilled. For any given $t>0$ we have that $\operatorname{supp}u(t)\subset B_{R}$ provided

[TABLE]

and $R\geq 4R_{0}$.

Next result follows immediately from Theorem 1.4.

Corollary 1.5**.**

Let (1.9), (1.12) be fulfilled. Assume also that the function

[TABLE]

is strictly increasing for some $\bar{R}>0$, and it becomes unbounded as $R\to+\infty$. For large $t>0$ define $Z_{0}(t)$ as the solution of

[TABLE]

where $\gamma$ is the same as in (1.13). Then $\operatorname{supp}u(t)\subset B_{Z_{0}(t)}$ for all large $t>0$.

Then we proceed to state a sup bound which assumes finite speed of propagation, and is independent of our results above. We need the following property of $\rho$

[TABLE]

for a suitable constant $C>0$. For example (1.16) rules out $\rho$ which decay too fast. We also define the function

[TABLE]

and assume that there exists $C\geq 1$ such that

[TABLE]

We need in the following that for given $C>0$, $0<\alpha<p$,

[TABLE]

Essentially (1.19) implies that $\rho(s)$ decays no faster than $s^{-\alpha}$ as $s\to+\infty$.

Theorem 1.6**.**

Let (1.16)–(1.19) be fulfilled, and assume (1.9). Assume also that $u$ is a solution to (1.1)–(1.2), satisfying

[TABLE]

for a positive nondecreasing $Z\in C([0,+\infty))$. Then

[TABLE]

Clearly, we can combine Corollary 1.5 and Theorem 1.6 to infer an explicit sup bound for $u$.

Theorem 1.7**.**

Let the assumptions of Corollary 1.5 and of Theorem 1.6 be fulfilled. Then (1.21) holds true for large $t$ with $Z$ replaced by $Z_{0}$ as in Corollary 1.5.

1.4. The supercritical cases

We drop in our first result below the assumption that $u_{0}$ be of bounded support.

Theorem 1.8**.**

Let the metric in $M$ be Euclidean, i.e., $\omega$ be constant. Assume that that $\psi_{\alpha}(s)=s^{\alpha}\rho(s)$ is nonincreasing for $s>s_{0}$, for some given $s_{0}>0$, $N\geq\alpha>p$. Let $\rho u_{0}\in L^{1}(M)$, $u_{0}\geq 0$. Then

[TABLE]

Theorem 1.9**.**

Let $u_{0}\in L^{1}(M)$ with bounded support, and assume that for some $\theta>0$

[TABLE]

Let $u$ be a solution to (1.1)–(1.2). Then the law of conservation of mass and the boundedness of the support of $u(t)$ fail over $(0,\bar{t})$ for a sufficiently large $\bar{t}>0$.

Remark 1.10*.*

If $\psi(s)=s^{p}\rho(s)$ is bounded, then

[TABLE]

so that in this case (1.23) implies (1.24).

1.5. Examples

The simplest case is probably the one where $M=\boldsymbol{R}^{N}$, $\rho(x)=(1+\lvert x\rvert)^{-\alpha}$, $\alpha\geq 0$. It is easily seen that our general assumptions of Subsection 1.1 are satisfied. Let us state the conditions corresponding to the ones in our main results.

The subcritical case where $\psi$ is nondecreasing and (1.18) holds true corresponds to $\alpha\leq p$.

The function in (1.14) giving the correct finite speed of propagation is strictly increasing to $+\infty$ as required in Corollary 1.5 since this condition corresponds to

[TABLE]

and $N>\alpha_{*}>p$ according to our restriction $p<N$. Furthermore,

[TABLE]

Finally the subcritical sup estimate can be proved under condition (1.19) which clearly corresponds to $\alpha<p$; it reads

[TABLE]

The supercritical sup estimate of Theorem 1.8 corresponds to $N\geq\alpha>p$.

The assumptions needed for interface blow up i.e., (1.23)–(1.24) correspond to $N\geq\alpha>\alpha_{*}$.

Other examples may be obtained essentially as revolution surfaces in the spirit of [5].

Remark 1.11*.*

In local coordinates, denoted by $x^{i}$, the divergence term in the equation (1.1) is written as

[TABLE]

where $(g_{ij})$ denotes the Riemannian metric, $(g^{ij})=(g_{ij})^{-1}$ so that $\,\textup{{d}}\mu=\sqrt{\det{(g_{ij})}}\,\textup{{d}}x$, and

[TABLE]

1.6. Plan of the paper

We prove in Section 2 several necessary auxiliary results. In Subsection 2.2 we present some embeddings which are not used in the following, but which may be of independent interest. In Section 3 we prove the results concerning the subcritical case, in Section 4 we prove Theorem 1.8 dealing with the case of the Euclidean metric, and finally in Section 5 Theorem 1.9 about interface blow up.

2. Embeddings

Let us note that, since $g$ is nondecreasing,

[TABLE]

This property will be used without further notice. We also employ throughout the notation

[TABLE]

for all measurable $I\subset M$.

2.1. Embeddings involving $\omega$

We begin with one of our main tools; actually an analogous embedding was proved in [20] in the Euclidean setting. A proof in our setting may follow [3] (where again the setting was different); we sketch here the proof of the case we need, for the reader’s convenience.

Lemma 2.1**.**

Let $u\in W^{1,p}(M)$, $0<r<q\leq Np/(N-p)$. Then

[TABLE]

where

[TABLE]

Proof.

We confine ourselves to the case $q\leq p$, which is the one of our interest here.

Introduce the standard rearrangement function

[TABLE]

Then write for convenience of notation

[TABLE]

We have for a $k>0$ to be selected presently

[TABLE]

Next we invoke Chebychev inequality

[TABLE]

to bound

[TABLE]

The last equality in (2.6) is our choice of $k$, which amounts to $\mu_{k}=\gamma E$. Note that we may assume $\mu_{0}$ as large as necessary, by approximating $u$ while keeping all the involved integral quantities stable. Thus we can safely assume that such a value of $k$ exists. Hence we absorb $I_{2}+I_{3}$ on the left hand side of (2.5). We then reason as in [19] to obtain

[TABLE]

We have exploited here the fact that $t\mapsto tg(t)^{-1}$ is increasing as it follows from our assumption that $\omega$ is nondecreasing.

Finally (2.3) follows from (2.7) and from our choice $\mu_{k}=\gamma E$. ∎

Corollary 2.2**.**

Let $u\in W^{1,p}(M)$ and $0<r<p$. Then

[TABLE]

where

[TABLE]

Proof.

We select $q=p$ in Lemma 2.1. The statement follows from an elementary computation, when we also bound by means of Hölder’s inequality

[TABLE]

∎

Corollary 2.3**.**

Assume $\psi(s)=s^{p}\rho(s)$ satisfies (1.18), and that $u\in W^{1,p}(M)$ has support of finite measure. Then for all $R>0$,

[TABLE]

Proof.

Let $R>0$ be fixed, and let $\zeta$ be a cutoff function in $B_{2R}$, with

[TABLE]

Then for $u$ as in the statement,

[TABLE]

Then we invoke (1.18) to infer

[TABLE]

where we have used Hardy’s inequality (2.20). Next we bound for $v=\mu(\operatorname{supp}u)$

[TABLE]

where we have used embedding (2.3) with $q=p$ as well as (2.9).

Note that, on appealing once more to Hardy’s inequality (2.20), we prove

[TABLE]

On using (2.14) in (2.12), (2.13) we finally get (2.10). ∎

Next Theorem is not used in the following, but it may be of independent interest.

Theorem 2.4**.**

Let the metric in $M$ be Euclidean, i.e., $\omega$ be constant. Then for all $u\in W^{1,p}(M)$ with support $\operatorname{supp}u$ of finite measure we have for all $R>0$

[TABLE]

Proof.

We may assume $u\geq 0$ and split

[TABLE]

Next by the standard Euclidean Sobolev embedding, for $p^{*}=Np/(N-p)$,

[TABLE]

Next by the same token

[TABLE]

Collecting the estimates above we obtain (2.15). ∎

Theorem 2.5**.**

Let the metric in $M$ be Euclidean, i.e., $\omega$ be constant. Assume that that $\psi_{\alpha}(s)=s^{\alpha}\rho(s)$ is nonincreasing for $s>s_{0}$, for some given $s_{0}>0$, $N\geq\alpha\geq p$. Then for $u\in W^{1,p}(M)$ and $p(N-\alpha)/(N-p)<p_{1}<Np/(N-p)$ we have

[TABLE]

Proof.

First we remark that owing to our assumption on $\psi$

[TABLE]

Indeed for $d(x)\geq s_{0}$ we have

[TABLE]

and for $p_{1}$ as in the statement

[TABLE]

as one can immediately check. Next we apply Hölder inequality

[TABLE]

Finally we apply Sobolev embedding to prove (2.18). ∎

We conclude by proving Hardy inequality, which has been used above. Its proof of course does not rely on the previous results.

Theorem 2.6** (Hardy inequality).**

For any $u\in W^{1,p}(M)$ we have

[TABLE]

Proof.

We may assume $u\geq 0$. With the notation of Lemma 2.1, we have

[TABLE]

On the other hand

[TABLE]

Therefore (2.21) gives on integrating by parts

[TABLE]

Next we apply our assumption (1.7) in (2.22) and after applying Hölder inequality we arrive at

[TABLE]

This immediately yields when we invoke (1.6)

[TABLE]

that is (2.20), by Polya-Szego principle. ∎

2.2. A general embedding

The results of this Subsection seem to us to be of independent interest. They follow from a more direct and sharper approach based on (2.25). However, they lead to formal complications which in practice make their use in our approach prohibitive, though they may be applicable in some special cases.

We start assuming

[TABLE]

Here $G:[0,+\infty)\to[0,+\infty)$ is a convex and increasing function, with $G(0)=0$. We remark that formally $G$ is the inverse function of the function $g$ introduced in (1.4), and that (2.25) could be actually proved by arguments relying on isoperimetric properties, under extra assumptions.

We assume in the following that $p>1$ is such that the Cauchy problem

[TABLE]

has a maximal solution $\mathcal{A}$ with $\mathcal{A}(s)>0$, $\mathcal{A}^{\prime}(s)>0$ for $s>0$. Then we define

[TABLE]

We also extend for notational simplicity $\mathcal{A}$ to $\boldsymbol{R}$ as an even function, so that $\mathcal{A}(s)=\mathcal{A}(\lvert s\rvert)$ for $s\in\boldsymbol{R}$.

We assume also that for some $C>1$

[TABLE]

The first inequality in (2.27) follows from the convexity of $\mathcal{A}$, which is in turn a simple consequence of its definition; we remark that the second inequality is satisfied e.g., if $s\mapsto\mathcal{A}^{\prime}(s)s^{-\alpha}$ is nonincreasing for some $\alpha>0$, with $C=\alpha+1$.

We also assume that $\mathcal{B}$ is convex.

Remark 2.7*.*

For example, in the standard Euclidean case of $\boldsymbol{R}^{N}$ we have that the admissible $p$ are those in $(1,N)$ and

[TABLE]

For notational brevity we introduce the function

[TABLE]

Lemma 2.8**.**

Let $p>1$ be as above; then for $u\in W^{1,p}(M)$

[TABLE]

Proof.

Choose in (2.25) $f(x)=\mathcal{A}(u(x))$ and obtain

[TABLE]

Then we use (2.26) and apply $G^{(-1)}$ to get (2.28). ∎

Note that according to the definitions above

[TABLE]

whence we get, on invoking (2.27),

[TABLE]

Our next result should be considered as a Faber-Krahn inequality.

Corollary 2.9**.**

If $u\in W^{1,p}(M)$ has bounded support then

[TABLE]

where $v=\lvert\operatorname{supp}u\rvert$.

Proof.

Let $v=\lvert\operatorname{supp}u\rvert$; we may assume $u\geq 0$. We start with

[TABLE]

where we used (2.28).

Thus we obtain, also employing Jensen inequality and (2.29),

[TABLE]

∎

Finally we prove a weighted version of our previous result.

Corollary 2.10**.**

Let $\psi$ be nondecreasing. Under the assumptions of Corollary 2.9, we have for all $R>0$, setting $A^{R}=\operatorname{supp}u\setminus B_{R}$,

[TABLE]

Proof.

Fix $R>0$ and assume without loss of generality that $u\geq 0$. Let us begin with estimating

[TABLE]

Here we have exploited the monotonicity of $\psi$ and Hardy’s inequality (2.20).

Next from Jensen inequality and the definition of $\mathcal{B}$, as well as from its assumed convexity, we find

[TABLE]

Since $\rho$ is nonincreasing we can bound by means of (2.28)

[TABLE]

Finally (2.31) follows from (2.32)–(2.34), and from applying $\mathcal{B}^{(-1)}$ as in the proof of Corollary 2.9, as well as from (2.29). ∎

2.3. Caccioppoli inequality

We’ll use the following inequalities.

Lemma 2.11**.**

Let $u$ be a solution of (1.1)–(1.2), and let $\theta>0$, with $\theta>2-m$ if $m<1$, $k>h>0$ be given. Let $\zeta\in C^{1}_{0}((0,+\infty))$, $0\leq\zeta\leq 1$. Then

[TABLE]

provided the right hand side in (2.35) is finite. Here $H(h,k)=(k/(k-h))^{(m-1)_{-}}$.

Lemma 2.12**.**

Let $u$ be a solution of (1.1)–(1.2), and let $\theta\geq p-1$. Let $\zeta\in C^{1}_{0}(M)$, $0\leq\zeta\leq 1$. Then

[TABLE]

provided the right hand side in (2.36) is finite.

The proofs of lemmas 2.11 and 2.12 are standard and we omit them.

2.4. Proof of Theorem 1.2

Take in (1.1) as testing function a standard cut off function $\zeta=\zeta(x)$ such that $\zeta\in C^{1}_{0}(\boldsymbol{R}^{N})$, with $\zeta=1$ in $B_{R}$, $R>\bar{R}$. Thus

[TABLE]

since $\operatorname{\nabla}\zeta=0$ on the support of $u$.

3. The subcritical case

Proof of Theorem 1.4.

For $R>0$ to be chosen, we introduce the sequence of increasing annuli

[TABLE]

We assume that $\operatorname{supp}u_{0}\subset B_{R/4}$ and $0<\eta,\sigma\leq 1/4$. Thus $u_{0}=0$ on all $A_{n}$.

Let us also set for a fixed $\theta>0$ as in Lemma 2.12

[TABLE]

for a sequence of cutoff functions $\zeta_{n}\in C^{1}_{0}(A_{n+1})$ such that

[TABLE]

As a consequence of Lemma 2.12 we have for $n\geq 0$

[TABLE]

Next we bound the right hand side of (3.3) by means of the embedding in Corollary 2.2, of (1.5) and of Young’s inequality, to obtain

[TABLE]

Here $b=2^{p+N(p-r)/r}$, and $\delta>0$ is to be chosen presently. Indeed, exploiting recursively (3.3), (3.4) we find for $j\geq 1$

[TABLE]

where we have set

[TABLE]

Thus on choosing $\delta<b^{-1}$, we infer as $j\to+\infty$, switching back to the notation $u^{1+\theta}=v^{s}$, $u=v^{r}$,

[TABLE]

We have used here (1.8) to bound

[TABLE]

Define next a decreasing sequence of annuli

[TABLE]

We apply (3.5) recursively with $A_{0}=D_{n+1}$, $A_{\infty}=D_{n}$, $\sigma=\eta=2^{-n-2}$, $n\geq 0$, to obtain, when recalling our definitions of $r$ and of $\beta$ in (2.2),

[TABLE]

where $b$ is as above. From (3.6) we get after an application of Hölder’s inequality

[TABLE]

on invoking assumption (1.8). Next we collect (3.6) and (3.7) (written for $n+1$) to get

[TABLE]

for $n\geq 0$. It follows from [12, Lemma 5.6 Ch. II] that $Y_{n}\to 0$ provided

[TABLE]

In turn, in view of the bound (1.12) and of our assumption (1.6), (3.9) is implied by

[TABLE]

Note that according to the definition of $Y_{n}$ in practice we have proved that $u(x,t)=0$ for $x\in M\setminus B_{R}$, if $R$ satisfies (3.10), and of course the condition $\operatorname{supp}u_{0}\subset B_{R/4}$ stated at the beginning of the proof; the sought after result follows immediately. ∎

Proof of Theorem 1.6.

For a $k>0$ to be selected, and a fixed $\theta>0$ as in Lemma 2.11, define for $n\geq 0$

[TABLE]

Here $\sigma\in(0,1/4]$. We also define for a fixed $t>0$ the decreasing sequence

[TABLE]

We introduce the notation $G_{n}(\tau)=\operatorname{supp}v_{n}(\tau)$, $0<\tau<t$. Note that according to our assumptions, we have $G_{n}(\tau)\subset B_{Z(\tau)}$.

We begin by an application of Hölder’s inequality and then of embedding (2.10), obtaining for $0<\tau<t$ the bound

[TABLE]

Next we select $R=L_{n+1}(\tau)$ according to

[TABLE]

where the inequality follows from (1.6) and from $G_{n+1}(\tau)\subset B_{Z(\tau)}\subset B_{Z(t)}$. Then, according to the definition of $\psi$, both the terms in brackets in (3.14) can be bounded in the same way leading us to

[TABLE]

where we have used assumption (1.19). In turn by definition of $L_{n+1}(\tau)$ and by $\alpha<p$ we have in (3.16)

[TABLE]

where we have estimated, appealing again to $G_{n+1}(\tau)\subset B_{Z(t)}$,

[TABLE]

Thus, collecting (3.14)–(3.18), we get, integrating also over $\tau_{n+1}<\tau<t$,

[TABLE]

with

[TABLE]

We use the above estimate together with a standard application of Caccioppoli inequality in Lemma 2.11, and Young inequality arriving at

[TABLE]

Here $\delta>0$ is to be chosen, $\ell=1+(m-1)_{-}$ and $b=2^{\ell p/(p-s)}$; we have set

[TABLE]

We can iterate (3.20) obtaining for $j\geq 1$

[TABLE]

We select $\delta<b^{-1}$ and let $j\to+\infty$, arriving at the basic estimate needed to start our second and last iterative process:

[TABLE]

The iteration makes use of the following definitions

[TABLE]

We apply Chebychev inequality as well as (3.24) with $\sigma=2^{-2n-2}$, to get, recalling the definitions of $v_{n}$ and of $s$,

[TABLE]

for $b=4^{1+\theta+\ell p/(p-s)}$. Invoking next [12, Lemma 5.6 Ch. II] we get that $Y_{n}\to 0$ as $n\to+\infty$ provided

[TABLE]

We remark that this amounts to $u(x,t)\leq k$, $x\in M$.

Next we note that since $H_{0}(\tau)\subset B_{Z(t)}$, $0<\tau<t$, we have

[TABLE]

according to assumption (1.16). Using (3.29) in (3.28), together with $s/(p-s)=(1+\theta)/(p+m-3)$, we see that (3.28) is implied by

[TABLE]

Finally we substitute (1.6) in (3.30) to transform it into

[TABLE]

whence (1.21). ∎

4. The case of the Euclidean metric

We use here the embedding in Theorem 2.5.

Proof of Theorem 1.8.

We use the notation introduced in (3.13), (3.21), (3.22), (3.25), (3.26). Fix $p_{1}\in(p,Np/(N-p))$; the value of $p_{1}$ will not affect the functional form of the final estimate. We have by Hölder inequality and by the embedding in (2.18)

[TABLE]

Then reasoning as in (3.20) we find

[TABLE]

Here $\delta>0$ is to be chosen and $b=2^{\ell p/(p-s)}$. We remark that straightforward arguments yield

[TABLE]

After selecting $\delta$ suitably small the same iterative procedure as in (3.23) leads us to

[TABLE]

As in (3.27) we get

[TABLE]

for $b=4^{1+\theta+\ell p/(p-s)}$; owing to [12, Lemma 5.6 Ch. II] and taking into account the definition of $s$, we get that $Y_{n}\to 0$ as $n\to+\infty$ provided

[TABLE]

In order to bound $Y_{0}$ we appeal once more to Chebychev inequality to find for $q>0$

[TABLE]

Thus, on defining

[TABLE]

we conclude for all $t>0$

[TABLE]

We are left with the task of estimating $E_{q+1}(\tau)$. We select $q>0$ large enough to have

[TABLE]

indeed the leftmost side of (4.8) is less than $p$ since in our assumptions $\alpha>p$, while $(1+q)<p+m+q-2$ since $p+m>3$. Then from (1.1) we get for $v=u^{(p+m+q-2)/p}$ the equality in

[TABLE]

where the inequality follows from an application of Theorem 2.5 with $p_{1}=p_{1}^{\prime}$ as in (4.8). A direct integration gives

[TABLE]

actually we integrate over $(t_{0},t)$ and then let $t_{0}\to 0+$, to circumvent possible problems with the local summability of the initial data.

Finally we substitute (4.10) in (4.7) and arrive at the sought after estimate. ∎

5. Interface blow up

Proof of Theorem 1.9.

We assume by contradiction that $u(t)$ is compactly supported for all $t>0$.

Let us compute by Hölder and Hardy inequalities

[TABLE]

where our assumption (1.23) implies

[TABLE]

In a similar fashion

[TABLE]

where again according to our assumption (1.24) for suitable $\theta>0$

[TABLE]

On using (5.2) and the equation (1.1), we prove that, for $v=u^{(p+m+\theta-2)/p}$,

[TABLE]

Note that here $\theta>0$ however is small enough for our assumptions (1.24) to hold true. We integrate the last differential inequality to obtain

[TABLE]

However owing to (5.1) and to an application of Hölder inequality

[TABLE]

Again integrating the equality in (5.3) we get

[TABLE]

which combined with (5.4) yields finally

[TABLE]

Indeed, since we are assuming by contradiction that the support of the solution is bounded over $(0,t+1)$, and therefore conservation of mass takes place in the same interval, according to Theorem 1.2. But (5.7) is clearly inconsistent when $t\to+\infty$, thereby proving our statement. ∎