Asymptotic properties of solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density on manifolds

We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.


Introduction
We consider the Cauchy problem ρ(x)u t − ∆ p,m (u) = 0 , x ∈ M , t > 0 , (1.1) Here M is a complete Riemannian manifold of topological dimension N, with infinite volume.In local coordinates x i , we denote The first author is member of the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM).
1 where (g ij ) denotes the Riemannian metric, (g ij ) = (g ij ) −1 , ∇ u is the gradient with respect to (g ij ), and g ij ∂u ∂x j ∂u ∂x i .
We always assume that 1 < p < N, and that either or p + m − 3 < 0 , m > 0 .
(1.4) In this paper we follow an approach ultimately based on the classical DeGiorgi estimates ( [12,24]); a new technical tool is the weighted Faber-Krahn type inequality of Lemmas 2.10 and 2.11.This inequality takes into account the asymptotic behaviors both of the density function ρ and of the volume growth of the manifold at infinity.We use the isoperimetrical properties of the manifold, which also allow us to prove new embedding results which we think are of independent interest.
We establish in the slow diffusion case (1.3), and also in the fast diffusion case (1.4) under additional assumptions, the decay rate for large times of nonnegative solutions, for initial data of finite mass.In the degenerate case we also estimate the finite speed of propagation for the support of solutions with a bounded support.These results apply in a subcritical case where, roughly speaking, the density function decays not too fast at infinity.Where we have explicit solutions, that is in the Euclidean case, our estimates reduce to the known optimal ones.
Still in the slow diffusion case, but when the density function decays fast enough, we investigate the behavior of the solutions for large t; we obtain under different assumptions a universal bound for solutions and a result of interface blow up.The universal bound is the same as in the Euclidean case, as expected; see also Subsection 1.3.
The various cases recalled above are discriminated in terms of the behavior of a universal function involving the density and the volume growth of the manifold (see Remark 1.2).
The interest of this problem appeared first in the case M = R 3 with the Euclidean metric, where [23], [35] obtained the first surprising results, in symmetric cases, on the qualitative properties of solutions to the porous media equation with inhomogeneous density.The interface blow up in the Euclidean setting was first shown in [21]; [38] extended it to a wide class of doubly degenerate parabolic equations.
Let us here explicitly recall the behavior of solutions in the Euclidean case, when ρ(x) = (1 + |x|) −α , x ∈ R N , for a given 0 < α ≤ N.Such a behavior strongly depends on the interplay between the nonlinearities appearing in the equation, at least from a two-fold point of view, that is both as far as the sup bounds for solutions are concerned, and then also if we consider the property of finite speed of propagation, see [38] for the following results; see also [14].Concerning the first issue, the limiting threshold is α = p: in the range α ≤ p the sup estimates for solutions keep the same structure as the ones known for the homogeneous doubly nonlinear equation (say with α = 0 above), although of course the exact dependence on the parameters of the problem is not the same.More explicitly, the solution is bounded above by the product of a negative power of time and a certain positive power of the initial mass.Instead in the range α > p one can prove a universal bound: a suitable negative power of time still is present, but the initial mass disappears from the estimate.Concerning the second issue of finite speed of propagation, which is connected to conservation of mass, a second explicit threshold α * ∈ (p, N) appears.We assume clearly that the initial data has compact support.Then in the subcritical range α < α * the property of finite speed of propagation is preserved for all times, i.e., the solution is compactly supported for t > 0. As a result, also the property of conservation of mass is valid for all times.In the supercritical range α > α * the evolution of the support is quite different: finite speed of propagation (that is boundedness of support) and conservation of mass can not hold true for all t > 0.
In Subsection 1.3 below we present a class of examples and a more detailed comparison with the Euclidean case.Before describing the results of this paper, we recall that parabolic problems in a Euclidean metric with inhomogeneous density were studied in [25], [26] (blow up phenomena); [34], [22] (asymptotic expansion of the solution of the porous media equation); [29], [20] (critical case).We quote for results related to ours [17,19,30] for the porous media equation and [13] for anisotropic operators.Still on the subject of porous media like equations on Riemannian manifolds, besides the seminal papers [31,33] on the properties of the support of solutions, we also quote [18,39].
The main goal of the present paper is to investigate the behavior of solutions around the threshold discriminating between the cases described above for the sup estimates, in terms of the density function ρ, the nonlinearities in the equation, and of course the Riemannian geometry of M. Our results on the subcritical L ∞ bound in fact apply when we, clearly, are in the subcritical case, but not too far from the threshold.In the Euclidean example just discussed, i.e., α should be not too small, which is not restrictive in the light of the purpose of this paper.On the other hand, we provide a unified approach to both the degenerate (1.3) and the singular (1.4) cases.The case where we are far below the threshold calls for a different approach; this will be the subject of a forthcoming paper.Such a requirement of closeness to the threshold is not needed by the other results.See also [1] for the Euclidean case; we borrow the energetic setting of [3,5,6,7]; see also [37].An important role is played by the function On the geometry of the manifold M we need the following requirements, of isoperimetrical character.We assume that for all bounded and Lipschitz domains U ⊂ M is nondecreasing in (0, +∞).
We list here all the assumptions required in the following; all of them are needed for the subcritical sup estimate, while the other results employ only a subset of such hypotheses.
1.1.1.Volume growth conditions.We require the growth conditions and its counterpart for a given 0 < c < 1.
The following condition of non-parabolicity of the manifold is needed to prove global embedding results: In some cases we need that R N /V (R) is nondecreasing, which is implied by the assumption (1.9) In addition we require 1.1.2.Density decay conditions.In order to get the subcritical sup estimate, the density function ρ is required to satisfy: where 0 < α 1 < α 2 < p are given constants.Note that (1.8), (1.12) imply that the function V ρ (R) = ρ(R)V (R) is bounded from above and below by constant multiples of the same increasing function (see Lemma 2.4 below); however we need state the more precise assumption ( We denote the inverse function of V ρ by R ρ .This hypothesis will be assumed implicitly throughout.Finally, as we remarked above, we need to be not too far from the threshold, in Theorem 1.5; exactly, this means that, on setting p * = Np/(N − p), for a suitably chosen s 0 > 0.
Remark 1.1.One can see easily that assumption (1.8) is a consequence of the following alternative assumption: R → V (R)/R q is nondecreasing for R > 0 for some q > p.
Assumption (1.8) is not merely a technical device; if it fails we do not expect decay of solutions for large times, see [23].
Remark 1.2.Consider the functions Owing to our assumptions, b 1 is increasing if and only if ψ is.In turn, this is automatically satisfied in the degenerate case (1.3), at least if (1.12) is assumed, but it is not necessarily valid in the singular case (1.4).However, when ψ is increasing we denote its inverse by Z; the latter function provides in some cases an estimate of the finite speed of propagation of the support of a solution.
The definition of weak solution to (1.1)-(1.2) is in fact standard; the problem is given the integral formulation , with bounded support.In general, the existence can be proved following the methods of [9,10] in the setting of energy solutions, i.e., assuming ) for all T < +∞.In the subcritical cases where the assumptions of Theorem 1.5 are in force, in fact we can prove existence even for initial data which are Radon measures, such that ρu 0 has finite mass in M.This follows from the estimate in Theorem 1.5 and from a standard approximation procedure via a sequence of solutions to initial-value boundary problems, with vanishing Dirichlet data, in a sequence of invading compact domains.In the latter case however the solution satisfies loc (M × (0, T )) for all T < +∞.In addition, the proof requires an estimate of the L 1 norm of u m−1 |∇ u| p−1 up to time t = 0 (sometimes called an entropy estimate); this bound can be achieved following [2,4], and again using Theorem 1.5.The uniqueness of solutions for problems of the kind we consider is well known to be a difficult problem in general; see e.g., the seminal paper [15], and more recently [32].
Thus in the following theorems, we refer to a solution u constructed by approximation as shown above.Also, in the proofs for simplicity we work sometimes with a strong almost everywhere formulation of the differential equation, to avoid the by now standard regularization arguments.We denote by γ, γ 0 , . . ., constants (varying from line to line) depending only on the parameters of the problem.
Remark 1.3.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 for the solution referred to just above Here γ depends on M and ρ, but not on u.Notice that this bound follows without assuming finite speed of propagation.However, it is easy to prove that if the support of the solution is bounded over [0, T ], mass is conserved exactly, that is Remark 1.4.It follows without difficulty from our arguments that the radial character and the assumptions on ρ can be replaced by analogous statements on a radial function ρ such that 1.2.Main results.We begin with the subcritical sup estimate.
Next we deal with the finite speed of propagation.
Theorem 1.6.Assume that supp u 0 ⊂ B R 0 and that we are in the degenerate case (1.3), with ψ increasing, ψ(0+) = 0. Assume further (1.5) and that for a suitable C > 0 for a constant γ > 0 independent of u.
The reason why we can avoid in Theorem 1.6 several of the global assumptions of Theorem 1.5 is that in the proof we work in bounded sets, shaped like annuli.Thus we donâĂŹt need to apply a weighted Sobolev inequality, which dispenses us from assuming non-parabolicity as in (1.8).By the same token, we need only a standard doubling property for ρ.
In the supercritical case (1.24) we can prove the following universal, or absolute, sup bound for the solution, which is in fact in its functional form independent of the initial data and of the geometry of M. Theorem 1.7.Assume that (1.3), (1.5)-(1.10)hold true, and that for a suitable c > 0 for some α > p.
where γ does not depend on u.
Finally, if the second threshold is exceeded, which we essentially assume in (1.26), the finite speed of propagation property fails.
Then the boundedness of the support of u(t) fails over (0, t) for a sufficiently large t > 0.
The proof and the result of Theorem 1.5 apply, for large t, in the two cases: i) p + m − 3 < 0, m > 0: we have to assume ) is automatically satisfied.We assume (1.28), (1.29).
It can be computed that the function ψ takes here the form where λ > 0 owing to our assumptions.Then Hence Theorem 1.5 applies and yields an asymptotic decay rate of t −δ 1 (ln t) δ 2 with Also the estimate of finite speed of propagation in Theorem 1.6 is in force for large times, provided p + m − 3 > 0, and (1.28) is satisfied.
Concerning the universal bound result in Theorem 1.7, the necessary assumptions in this example are fulfilled if α > p, p + m − 3 > 0.
Finally, the interface blow up phenomenon of Theorem 1.8 takes place if p + m − 3 > 0 and which guarantees (1.26).Note that α * > p.
If we take formally above β = 1, µ = ν = 0, we recover the results known in the setting of the Euclidean space ( [38]).Furthermore, the example is still admissible, under the same assumptions, if f and ρ are modified by multiplying them by a factor 1 + H(τ ) where H is a sufficiently regular and fast decaying function as τ → +∞.1.4.Plan of the paper.We prove in Section 2 several auxiliary inequalities.In Section 3 we prove Theorem 1.5 in the degenerate case, while the proof in the singular case, being a minor variant of the previous one, is dealt with in the short Section 4. The finite speed of propagation property is proved in Section 5.The supercritical universal sup bound is proved in Section 6, while finally the interface blow up is treated in Section 7.

Auxiliary results
We begin by stating the following trivial result.
Lemma 2.2.Assume (1.8); then for the same constant c as in (1.8) Assume further (1.12); then a constant c > 0 exists such that Proof.The inequality (2.4) follows from (1.8), since on taking there k = V (s) we get when we also exploit the fact that the integrand is decreasing.
Let us begin by checking that W is nondecreasing in (V ρ (1), +∞); in this case R ρ (s) > 1. Write The first factor is nondecreasing by assumption (1.12).As to the second factor, set R = R ρ (s).Then the quantity in square brackets in such a factor equals Here, the first factor is nondecreasing by assumption (1.11); the second one is nondecreasing by assumption (1.9).
Next, we note that clearly two constants It is easy to check that owing to our assumptions N > p > α 2 > α 1 .Thus for such a β and c 0 , c 1 given in (2.28) we have proved (2.1).Our claim (2.25) follows.
Our second claim (2.26) is a direct consequence of (1.11) and of (2.6).
Proof.Our assumptions match the ones made in [8], when we show that the inequality follows from our assumptions.Indeed, setting R = V (−1) (s), we have, with the notation of the proof of Lemma 2.3, and therefore by (2.20) Then (2.29) follows from (2.31) and (1.8).
The weighted Sobolev inequality in [28] was proved under the assumption that the Ricci curvature is nonnegative and the volume growth is such to guarantee the hyperbolicity of the manifold.Instead in [8] we applied, to the same end, a symmetrization technique relying on an isoperimetrical inequality, as well as an assumption of p-hyperbolicity of M.
Next we prove the following Hardy inequality, needed below; see also the survey [11] on this subject.(2.32) Proof.We may assume u ≥ 0. Introduce for λ ≥ 0 the standard rearrangement function (2.33) We have (2.34) On the other hand Therefore (2.34) gives on integrating by parts (2.36)This immediately yields when we invoke (1.10) that is (2.32), by Polya-Szego principle (see [8]).
We state first an estimate where the density function ρ does not appear.

.38)
where (2.39) Proof.We confine ourselves to the case q ≤ p, which is the one of our interest here.The case q > p can be proved reasoning as in [4].
Introduce the standard rearrangement function as in (2.33).Then write for convenience of notation We have for a k > 0 to be selected presently Next we invoke Chebychev inequality to bound The last equality in (2.41) is our choice of k, which amounts to µ k = γS q .Note that we may assume µ 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 (2.40).We then reason as in [36] to obtain We have exploited here the fact that t → th(t) −1 is increasing as it follows from our assumption that ω is nondecreasing.Finally (2.38) follows from (2.42) and from our choice µ k = γS q .
Corollary 2.9.Let u ∈ W 1,p (M) and 0 < r < p. Then Proof.We select q = p in Lemma 2.8.The statement follows from an elementary computation, when we also bound by means of Hölder's inequality (2.44) Next we state some weighted estimates where the estimated norms involve the weight ρ.We denote in the following for k > 0 (2.45) Lemma 2.10 (Faber-Krahn type estimate).Assume that 1 < p < N, and that the assumptions of Lemma 2.6 and of Theorem 2.7 hold true.We further assume (1.12), (1.14).
Then for all u ∈ W 1,p (M), k > 0 we have (2.46) Proof.We split the integral between B(R) and its complement; let A k = {u > k}.We first have from Lemma 2.6, |∇ u| p dµ =: I 1 . (2.47) We used assumption (1.14) and the estimate (2.19), at least for R > s 0 as in (1.14).For R ≤ s 0 we simply note that for a suitable 0 < c < 1, as ω(0+) > 0 due to the locally Euclidean behavior of the Riemannian metric.
Next we have by means of Hardy inequality (2.32), and also taking into account (2.3), Finally we select R so that, essentially, I 1 and I 2 contribute the same quantity, i.e., ρ(R) or as one can immediately see Proof.As in the proof of Lemma 2.10, we split the integral between B(R) and its complement; let A k = {u > k} and ν ρ (k) be defined as in (2.45).We first have from Hölder inequality, Lemma 2.6, and from assumption (1.14), (2.53) We have used in last inequality our assumption (1.10).

Next we apply again Hölder inequality to get
The second factor in (2.54) is majorized, owing to our assumption (2.52), by , (2.55) when we apply also Lemma 2.6.The first integral in (2.54) is bounded with the help of Hardy inequality (2.32), to obtain finally (2.56) In last inequality we used again (1.10).We require that I 1 and I 2 contribute essentially the same quantity, that is we define R by means of . With this choice of R it is trivial to check that, e.g., the right hand side of (2.57) equals ρ(R) s/p R s ν ρ (k) 1−s/p , proving the claim.Remark 2.12.Concerning assumption (2.52) we note that the exponent of R in it, that is N − s(N − p)/p, tends to p− as s → p+.Hence for suitable s in such a range, (2.52) is a consequence of (1.12) and of Lemma 2.1.
Let us also note for later use the following consequence of (2.52): rewrite the right hand side of (2.57), that is The last factor above is nondecreasing owing to our assumption (1.9); thus, owing to (2.52) we have for C > 0 as in (2.52).
In the following we let for the sake of notational simplicity for p > r > 0 (2.59) Lemma 2.13 (Sobolev-Gagliardo-Nirenberg).Assume that 1 < p < N, and that the assumptions of Lemma 2.10 hold true.Assume further that 0 < r < p. Then we have

.60)
Proof.We begin by splitting, for a k > 0 to be chosen, We first bound, on using (2.46) and Chebychev inequality, (2.62) Then we get obviously Lemma 2.14.Assume that 1 < p < s < N, and that the assumptions of Lemma 2.11 hold true.Assume further that 0 < r < p. Then we have ) Proof.As in the proof of Lemma 2.13, we begin by splitting for a k > 0 to be chosen, Then we bound J 1 essentially as we did in (2.62), but applying (2.51) rather than (2.46), and J 2 following the direct approach of (2.63).We get, collecting such estimates, after a further use of Chebyshev inequality, We have made use of (2.58) here.
Corollary 2.15.Under the assumptions and with the notation of Lemma 2.13 we have for the function W defined in (2.25) where Proof.We only need to multiply and divide the right hand side of (2.60) by S (p−α 2 )/(N −α 1 ) ; then (2.75) follows after some elementary algebra, when we recall the definitions of S and of W .
Again by simple algebra we infer also the following result.
Corollary 2.16.Under the assumptions and with the notation of Lemma 2.14 we have for the function W defined in (2.25) where H is as in Corollary 2.15.
We'll use the following inequalities.The set C 1 0 (M) denotes the C 1 (M) functions with compact support.Lemma 2.17.Let u be a solution of (1.1)-(1.2),and let θ > 0, with provided the right hand side in (2.78) is finite.
The proofs of lemmas 2.17 and 2.18 are standard and we omit them.
3. Proof of the sup estimate in the case p + m − 3 > 0.
For k > 0 to be selected later, and for θ > 0 as in Lemma 2.17, we define for n ≥ 0 Here σ ∈ (1, 1/4) is fixed.From Lemma 2.17 we infer sup for ℓ = 1 + (m − 1) − .From Hölder inequality it follows whence, on applying Corollary 2.15 to bound the first integral on the right hand side of (3.2), we get here S is defined as in (2.59).Note that by appealing again to Hölder inequality we obtain, with the notation introduced in (2.45) we get the recursive inequality where A standard iteration process then implies that, when we select e.g., ε = 1/(2b), sup where we have computed Next we set in (3.9) Note that by Chebyshev inequality we obtain there when we employ also Lemma 2.5 and By the same token, the left hand side of (3.9) is bounded from below by Thus we infer for According to the classical result [24, Lemma 5.6, Chapt.II], we get that Y i → 0 as i → +∞ provided for a suitably small constant γ 0 > 0 depending on the parameters in (3.12).When we recall the definition of the function W we see that (3.13) reduces to, when we use the bound for mass in (1.18), We conclude by invoking Remark 1.2.
4. Proof of the sup estimate in the case p + m − 3 < 0.
We borrow the notation from Section 3, assuming also θ > 3 − p − m is so large that r < p, and that s > p satisfies (2.52): see Remark 2.12.Let us also note explicitly that assumption (1.20), with some elementary algebra, yields which in turn implies immediately H(s) > 0 (H has been defined in Corollary 2.15).We start again from (3.1); on the right hand side there by means of (2.76), we get where Σ is then bounded as in (3.4).From now on, the proof proceeds formally unchanged as in Section 3, with the remarks below.We use Hölder inequality to bound the right hand side of (3.1) exactly with the right hand side of (3.5) (with the current values of s, r).Note that the condition (s − r)(N − α 1 ) required in order to apply Hölder (and, next, Young) inequality, is not automatically satisfied as in the case of Section 3; anyway, it is easily seen by a direct elementary calculation that (4.3) is equivalent to (4.1).
Then we apply Young inequality to arrive at (3.8).Here we specifically note that all exponents have the expected sign owing to (4.1), and that in (3.9) η is positive by assumption (1.20).
At the end of the proof, in order to define k by means of (3.14), we exploit our assumption that ψ is increasing.

Proof of the finite speed of propagation
Here we introduce the notation We also introduce a standard cutoff function Then for θ > 0 as in Section 3 we define Note that no A n intersects supp u 0 .Then from Lemma 2.18 we get Next we apply Corollary 2.9 to v n+1 and get for the present choice of r < p (remember that the support of v n+1 is contained in an annulus), and for all times, We integrate in time this estimate, and apply Young inequality, to bound above the right hand side of (5.1) with From (5.1) and ( 5.3) we infer, on substituting the definition of r, the recursive inequality where A standard iterative argument, relying on a suitable choice of ε, then yields the estimate sup Define next a sequence of shrinking annuli We apply inequality (5.5) with n ≥ 0, and obtain, using the the fact that ρ (5.6) In addition we require that one of the following holds: Proof.From Hölder inequality and Lemma 2.6 we infer where rp * p * −r dµ .(6.5) We majorize the last integral as where we have used the definition of ω and (1.9), (1.10).Finally, the right hand side of (6.6) is bounded by according to our assumptions on V and ρ.The last integral converges in the cases given in the statement, as a direct inspection shows.
Proof of Theorem.We use the notation introduced in Section 3, with the exception that while we still denote s = p(1+θ)/(p+m+θ−2) < p, θ > 0, we choose p < r < p * .We may apply Lemma 6.1, since (6.2) with β = p follows under our assumptions from (2.4), so that p < r < p * is in the admissible range for Lemma 6.   3) .(6.10) Note that the last exponent is greater than 1 owing to our choice r > p.
We are left with the task of estimating E q+1 (τ ); this will be accomplished by appealing again to Lemma 6.1, where we select 0 < r ′ := p(1 + q) p + m + q − 2 < p < p * .(6.14) We obtain from the differential equation (1.1), for w = u (p+m+q−2)/p the equality in where the inequality follows from an application of (6.3) with r replaced with the r ′ given in (6.14).On integrating (6.15) we get E q+1 (t) ≤ γt − 1+q p+m−3 , t > 0 ; (6.16) actually we integrate over (t 0 , t) and then let t 0 → 0+ in order to circumvent possible problems with the local summability of the initial data.Finally we substitute (6.16) in (6.13) to prove the claim of the Theorem.

Interface blow up
Let us reason by contradiction, and assume that the support of u(t) stays bounded for all times.
Then we get for all times and for a fixed θ > 0, by means of Hölder and Hardy inequality

1. 1 .
Assumptions.In what follows d(x) denotes the geodesic distance of x from a fixed point x 0 ∈ M, |U| is the Riemannian volume of U ⊂ M, and |∂U| N −1 the corresponding area of its boundary.

(3. 4 )
Next we integrate in time the estimate (3.3), and after a further application of Hölder inequality we bound the right hand side of(3.1)