Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds
aa r X i v : . [ m a t h . A P ] A p r BLOW-UP AND GLOBAL EXISTENCEFOR THE POROUS MEDIUM EQUATION WITH REACTIONON A CLASS OF CARTAN-HADAMARD MANIFOLDS
GABRIELE GRILLO, MATTEO MURATORI, AND FABIO PUNZO
Abstract . We consider the porous medium equation with power-type reaction terms u p on negatively curved Rie-mannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If p > m ,small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If p < m ,large data blow up at worst in infinite time, and under the stronger restriction p ∈ (1 , (1 + m ) / all data give rise tosolutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The resultsare in several aspects significantly different from the Euclidean ones, as has to be expected since negative curvatureis known to give rise to faster diffusion properties of the porous medium equation. Introduction
We consider solutions to the nonlinear evolution problem ( u t = ∆( u m ) + u p in M × (0 , T ) ,u = u in M × { } , (1.1)where M is an N -dimensional complete, simply connected Riemannian manifold with non-positive sectional curvatures (namely a Cartan-Hadamard manifold ) and ∆ is the Laplace-Beltrami operator on M , m > , p > , T ∈ (0 , ∞ ] . The assumption m > corresponds toconsider the slow diffusion case, see [41]. The initial datum u will be always assumed to benonnegative and bounded. We shall always suppose, without further comment, that N ≥ .We shall concentrate on the situation in which the curvature boundsRic o ( x ) ≤ − ( N − h or Ric o ( x ) ≥ − ( N − k hold for some h, k > , where Ric o ( x ) is the Ricci curvature at x in the radial direction ∂∂r .In particular a global negative curvature condition like the one above, involving sectionalcurvatures, implies that the spectrum of − ∆ on M is bounded away from zero, see [25],this being in sharp contrast with the Euclidean setting and resembling, to some extent,the case of a bounded Euclidean domain. In fact, the basic example we have in mind isthe hyperbolic space H nh , namely the complete, simply connected manifold with sectionalcurvatures everywhere equal to − h . It is known that, on H nh , Brownian motion, associatedto − ∆ by a standard procedure, has an expected speed which is linear for large times (seee.g. [5, Cor. 5.7.3]), hence diffusion occurs at a much faster rate than the one typical of theEuclidean situation.The behaviour of solutions to the porous medium or fast diffusion equations u t = ∆( u m ) in M × (0 , T ) . (1.2)has been the subject of recent results (see e.g. [13], [14], [15], [16], [17], [33], [35], [34],[42]). That even nonlinear diffusion gives rise to speedup phenomena can be seen at least Mathematics Subject Classification.
Primary: 35R01. Secondary: 35K65, 58J35, 35A01, 35A02,35B44.
Key words and phrases.
Porous medium equation; Cartan-Hadamard manifolds; large data; a prioriestimates; blow-up. in two different ways, see [42] for M = H N , and [17] for extensions on some more generalmanifolds. First, the L ∞ norm of a solution corresponding to a compactly supported datumobeys the law k u ( t ) k ∞ ≍ (cid:16) log tt (cid:17) / ( m − as t → + ∞ , which is faster than the correspondingEuclidean bound, the latter being given in term of a power t − α ( N,m ) with α ( N, m ) = ( m − /N )) − < / ( m − . Besides, the volume V ( t ) of the support of the solution u ( t ) satisfies V ( t ) ≍ t / ( m − while in the Euclidean situation one has V ( t ) ≍ t β ( N,m ) with β ( N, m ) = n (2 + n ( m − − = < / ( m − .The behaviour of solutions to problem (1.1) is therefore determined by competing phe-nomena: the diffusion pattern associated to − ∆ , the reaction due to the power source, andthe (slow, but faster than in the Euclidean case) diffusion properties of the porous mediumequation u t = ∆( u m ) . In fact, in the case of linear diffusion ( m = 1 ) it is known (see [2],[43], [44], [32]) that, when M = H N , for all p > and sufficiently small nonnegative datathere exists a global in time solution. The situation is different in R N : indeed, blowupoccurs for all nontrivial nonnegative data when p ≤ /N , while global existence prevailsfor p > /N (for more specific results, see e.g. [4], [6], [8], [7], [18], [22], [36], [39], [45],[47].)To understand more precisely the differences between the Euclidean results and the onesproved in the present paper, let us first summarize qualitatively some of the former ones,quoting from [38]. For subsequent, more detailed results see e.g. [9], [28], [40] and referencesquoted therein. The case M = R N . We suppose that the initial datum is nonnegative, nontrivial andcompactly supported . In this case we have: • ([38, Th. 1, p. 216]) For any p > , all sufficiently large data give rise to solutionsblowing up in finite time; • ([38, Th. 2, p. 217]) if p ∈ (cid:0) , m + N (cid:1) , all data give rise to solutions blowing up infinite time; • ([38, Th. 3, p. 220]) if p > m + N , all sufficiently small data give rise to solutionsexisting globally in time.Let us mention that further nonexistence results for quasilinear parabolic equations, alsoinvolving p -Laplace type operators, have been obtained in [26], [27], [31] (see also [24] forthe case of Riemannian manifolds). Moreover, in [37] problem u t = ∆( u m ) + λu p in Ω × (0 , T ) ,u = 0 on ∂ Ω × (0 , T ) ,u = u in Ω × { } , where Ω is a bounded domain of R N and λ is a positive parameter, has been studied. Let λ (Ω) be the first eigenvalue of − ∆ in Ω , completed with homogeneous Dirichlet boundaryconditions. It is shown that (see [37, Theorem 1.3]) there exists a global solution for any u ∈ L q (Ω) , q > , and for any λ ≤ λ (Ω) . In addition, when p > m , or p = m and λ > λ (Ω) (see [37, Section 4]), then depending on the initial datum u , solutions may or may not existfor all times. Analogous results can also be found in [38, Chapter VII].Existence of global solutions and blow-up in finite time for problem (1.1) have been studiedin [48], if the volume of geodesic balls grows as a power of the radius R , namely as R α with α ≥ ; this occurs, in particular, when Ricci curvature is nonnegative. However, we shouldnote that such a condition tipically is not satisfied in our setting, in which the volume ofgeodesic balls can grow exponentially or faster with respect to the radius. In particular, in [48] it is proved that if m < p ≤ m + 2 /α , then problem (1.1) does not have global(nontrivial) solutions. Instead, if α = N, p > m + 2 /N , under a suitable assumption on themetric tensor, there exists a global solution of (1.1), for some u . Such results extend someof those in [38] to general Riemannian manifolds.The situation on negatively curved manifolds is significantly different, as we now brieflysummarize by singling out qualitatively some of our results.
The case of a Cartan-Hadamard manifold M . We suppose that the initial datum is nonnegative, nontrivial and compactly supported . In this case we have: • (see Theorems 3.1, 3.6) If p > m and upper curvature bounds (see (2.7)) hold, allsufficiently small data give rise to solutions existing globally in time. Besides, a classof sufficiently small data shows propagation properties identical to the ones valid forthe unforced porous medium equation (1.2). Moreover, small data non necessarilywith compact support and possibly with arbitrarily large L p norms ( p ≥ ) give riseto solutions existing globally in time if p > m , and also if p = m and a suitablecurvature bound holds; • (see Theorem 3.2) If p > m and lower curvature bounds (see (2.6)) hold, all suffi-ciently large data give rise to solutions blowing up in finite time; • (see Theorem 3.3). If p ∈ (cid:0) , m (cid:3) and upper curvature bounds (see (2.7)) hold, alldata exist globally in time; • (see Theorem 3.4). If p ∈ (1 , m ) and lower curvature bounds (see (2.6)) hold, allsufficiently large data give rise to solutions blowing up at worst in infinite time.Thus the overall picture is considerably different from the Euclidean one, on the one handsince the main critical exponent turns out to be p = m , on the other hand since a completelynew phenomenon, namely existence of solutions blowing up in infinite time, appears when p ≤ (1 + m ) / . We do not know, and leave these as challenging open problems, whether for p ∈ (cid:0) m , m (cid:1) solutions corresponding to small initial data exist for all time and if solutionscorresponding to large data blow up in finite or infinite time.The paper is organized as follow. Section 2 contains some geometric preliminaries, therelevant notation, a concise discussion of Laplacian comparison in Riemannian geometry,and, finally, a brief discussion of local existence of solution to (1.1) and comparison princi-ples. Section 3 contains the statements of our main results. Section 4 contains two generalauxiliary lemmas, that will be repeatedly used in the construction of the barriers we needin the proofs of our main results. Such proofs are contained in Section 5.2. Preliminaries
Notations from Riemannian geometry.
Let M be a complete noncompact Rie-mannian manifold. Let ∆ denote the Laplace-Beltrami operator, ∇ the Riemannian gradientand d V the Riemannian volume element on M .We consider Cartan-Hadamard manifolds, i.e. complete, noncompact, simply connectedRiemannian manifolds with nonpositive sectional curvatures everywhere. Observe that onCartan-Hadamard manifolds the cut locus of any point o is empty [11, 12]. Hence, for any x ∈ M \ { o } one can define its polar coordinates with pole at o , namely r ( x ) := d ( x, o ) and θ ∈ S N − . If we denote by B R the Riemannian ball of radius R centred at o and S R := ∂B R ,there holds meas ( S R ) = Z S N − A ( R, θ ) dθ dθ . . . dθ N − , GABRIELE GRILLO, MATTEO MURATORI, AND FABIO PUNZO for a specific positive function A which is related to the metric tensor, [11, Sect. 3]. Moreover,it is direct to see that the Laplace-Beltrami operator in polar coordinates has the form ∆ = ∂ ∂r + m ( r, θ ) ∂∂r + ∆ S r , (2.1)where m ( r, θ ) := ∂∂r (log A ) and ∆ S r is the Laplace-Beltrami operator on S r . Thanks to(2.1), we have that m ( r, θ ) = ∆ r ( x ) for every x ≡ ( r, θ ) ∈ M \ { o } . Let A := (cid:8) f ∈ C ∞ ((0 , ∞ )) ∩ C ([0 , ∞ )) : f ′ (0) = 1 , f (0) = 0 , f > in (0 , ∞ ) (cid:9) . We say that M is a spherically symmetric manifold or a model manifold if the Riemannianmetric is given by ds = dr + ψ ( r ) dθ , where dθ is the standard metric on S N − and ψ ∈ A . In this case, we shall write M ≡ M ψ ;furthermore, we have A ( r, θ ) = ψ ( r ) N − η ( θ ) for a suitable angular function η , so that ∆ = ∂ ∂r + ( N − ψ ′ ψ ∂∂r + 1 ψ ∆ S N − . Note that ψ ( r ) = r corresponds to M = R N , while ψ ( r ) = sinh r corresponds to M = H N ,namely the N -dimensional hyperbolic space.For any x ∈ M \ { o } , we denote by Ric o ( x ) the Ricci curvature at x in the radial direction ∂∂r . Let ω be any pair of tangent vectors from T x M having the form (cid:0) ∂∂r , V (cid:1) , where V is aunit vector orthogonal to ∂∂r . We denote by K ω ( x ) the sectional curvature at x ∈ M of the -section determined by ω .2.2. Laplacian comparison.
Let us recall some crucial Laplacian comparison results. Itis by now classical (see e.g. [10] and [11, Section 15]) that ifRic o ( x ) ≥ − ( N − k for all x ≡ ( r, θ ) ∈ M \ { o } for some k > , then m ( r, θ ) ≤ ( N − k coth( kr ) for all r > , θ ∈ S N − . So, in particular, m ( r, θ ) ≤ ( N − k coth( k ) for all r ≥ , θ ∈ S N − , (2.2)since r coth r is decreasing. On the other hand, ifRic o ( x ) ≤ − ( N − h for all x ≡ ( r, θ ) ∈ M \ { o } (2.3)for some h > , then (see [46, Theorem 2.15]) m ( r, θ ) ≥ ( N − h coth( hr ) ≥ ( N − h for all r > , θ ∈ S N − (2.4)(the second inequality is merely due to the fact that coth( r ) ≥ for all r > ). Letus observe that the latter implication is based upon the assumption that M is a Cartan-Hadamard manifold. Indeed, on general Riemannian manifolds with a pole o , namely witha point o ∈ M having empty cut locus, inequality (2.4) is valid, provided thatK ω ( x ) ≤ − h for all x ≡ ( r, θ ) ∈ M \ { o } . Clearly, (2.3) is a weaker condition than the previous one concerning the sectional curvature.
In the special case of a model manifold M ψ , for any x ≡ ( r, θ ) ∈ M ψ \ { o } we haveK ω ( x ) = − ψ ′′ ( r ) ψ ( r ) , Ric o ( x ) = − ( N − ψ ′′ ( r ) ψ ( r ) . In particular, as ψ ∈ A , the condition ψ ′′ ≥ in (0 , ∞ ) is necessary and sufficient for M ψ to be a Cartan-Hadamard manifold. Finally, note that for any Cartan-Hadamard manifoldwe have K ω ( x ) ≤ , therefore the Laplace comparison theorem easily gives that m ( r, θ ) ≥ N − r for any x ≡ ( r, θ ) ∈ M \ { o } . Main assumptions and consequences.
Throughout the paper we shall work underthe following assumption: M is a Cartan-Hadamard manifold of dimension N ≥ . (2.5)Besides, one or both the following curvature bounds will be required:Ric o ( x ) ≥ − ( N − k for some k > (2.6)Ric o ( x ) ≤ − ( N − h for some h > . (2.7)2.4. Local existence and comparison.
In this brief section we first give a precise meaningto the concept of solution to (1.1) we shall deal with, and then establish some elementaryexistence results and comparison principles, which are essential to be able to exploit all ofthe barrier functions we provide below.
Definition 2.1.
Let u ∈ L ∞ ( M ) , with u ≥ . Let T > and p, m > . We say thata nonnegative function u ∈ L ∞ ( M × (0 , S )) (for all S < T ) is a (very weak) solution toproblem (1.1) if it satisfies − Z M Z T u ϕ t dt d V = Z M u ( x ) ϕ ( x, d V ( x ) + Z M Z T ( u m ∆ ϕ + u p ϕ ) dt d V (2.8) for all nonnegative ϕ ∈ C ∞ c ( M × [0 , T )) .Similarly, we say that a nonnegative function u ∈ L ∞ ( M × (0 , S )) (for all S < T ) is a(very weak) subsolution [supersolution] to problem (1.1) if it satisfies (2.8) with “ = ” replacedby “ ≤ ” [“ ≥ ”]. In order to give a (rather standard) local existence result, let us briefly discuss about minimal solutions . To this end, we first need to introduce the auxiliary problems (let
R > ) u t = ∆( u m ) + u p in B R × (0 , T ) ,u = 0 on ∂B R × (0 , T ) ,u ( · ,
0) = u in B R . (2.9) Definition 2.2.
Let u ∈ L ∞ ( B R ) , with u ≥ . Let T > and p, m > . We say thata nonnegative function u ∈ L ∞ ( B R × (0 , S )) (for all S < T ) is a (very weak) solution toproblem (2.9) if it satisfies − Z B R Z T u ϕ t dt d V = Z B R u ( x ) ϕ ( x, d V ( x ) + Z B R Z T ( u m ∆ ϕ + u p ϕ ) dt d V (2.10) for all nonnegative ϕ ∈ C ∞ ( B R × [0 , T ]) with ϕ = 0 on ∂B R for all t ∈ [0 , T ] and at t = T . GABRIELE GRILLO, MATTEO MURATORI, AND FABIO PUNZO
Similarly, we say that a nonnegative function u ∈ L ∞ ( B R × (0 , S )) (for all S < T ) is a(very weak) subsolution [supersolution] to problem (2.9) if it satisfies (2.10) with “ = ” replacedby “ ≤ ” [“ ≥ ”]. Note that problem (2.9) admits a nonnegative solution u R ∈ L ∞ ( B R × (0 , S )) , for all S < T R , where T R is the maximal existence time, i.e. k u R ( t ) k ∞ → + ∞ as t → T − R ;moreover, for problem (2.9) the comparison principle between sub– and supersolutions holds(see [1]). Observe that T R ≥ T for any R > , where T can be quantified depending on theinitial datum u by simple comparison with the solution of the associated ODE: ( x ′ = x p ,x (0) = k u k ∞ , that is u R ( x, t ) ≤ k u k ∞ (cid:16) − ( p − k u k p − ∞ t (cid:17) p − = ⇒ T R ≥ T := 1( p − k u k p − ∞ . (2.11)Moreover, such a solution is unique. In particular, u R ≤ u R +1 , so that the family u R ismonotone increasing and, thanks to the upper bound (2.11), it converges as R → ∞ tosome solution u to (1.1). Such a solution is necessary smaller than any other solution, dueagain to comparison on balls (see also Proposition 2.4 below). In this sense it is referredto as minimal . We can define the maximal existence time T of u as the supremum over all S > for which lim R →∞ u R ∈ L ∞ ( M × (0 , S )) : note that u does solve (1.1), at least in thesense of Definition 2.1, up to such time.As a reference for these results, we quote e.g. [1], where in fact the authors mainly dis-cuss the one-dimensional Euclidean case, for more general nonlinearities. However, theirarguments are easily adaptable to our framework as well. We omit details. See also [29, 30]for similar techniques applied to a related (but substantially different) problem in generalEuclidean space.In agreement with the above discussion, we can state the following existence result. Proposition 2.3 (Existence) . Let u ∈ L ∞ ( M ) , with u ≥ . Then there exists a solutionto problem (1.1) , in the sense of Definition 2.1, with T = 1( p − k u k p − ∞ , which is obtained as a monotone limit of the solutions to the approximate problems (2.9) .Moreover, such a solution is minimal , in the sense that any other solution is larger. By taking advantage of the construction of the minimal solution, we can readily prove afundamental comparison theorem.
Proposition 2.4 (Comparison with supersolutions) . Let u ∈ L ∞ ( M ) , with u ≥ . Let u be a supersolution to (1.1) (for some T > ), according to Definition 2.1. Then, if u is theminimal solution provided by Proposition 2.3, there holds u ≤ u in M × (0 , T ) . (2.12) In particular, if the supersolution exists at least up to time T , then also u does, i.e. themaximal existence time for u is at least T . Proof.
It is enough to apply the above-mentioned comparison results in B R : since u is clearlyalso a supersolution to (2.9), for each R > , we have u R ≤ u in B R × (0 , T ) . (2.13)By passing to the limit as R → ∞ in (2.13) we obtain (2.12), which trivially ensures that u does exist at least up to T , by the definition of maximal existence time. (cid:3) We also have a similar result for subsolutions.
Proposition 2.5 (Comparison with subsolutions) . Let u ∈ L ∞ ( M ) , with u ≥ . Let u bea solution (for some T ≡ T > ) and u be a subsolution (for some T ≡ T > ) to (1.1) ,according to Definition 2.1. Suppose that u has the following additional property: supp u | M × [0 ,S ] is compact for all S < T . Then u ≥ u in M × (0 , T ∧ T ) . (2.14) Proof.
Again, it is sufficient to apply comparison on balls: we fix any
S < T ∧ T andobserve that, if R is so large that supp u ⌋ M × [0 ,S ] ⊂ B R , then u and u are a supersolutionand a subsolution, respectively, to (2.9), whence u ≥ u in B R × (0 , S ) . Inequality (2.14) then just follows by letting R → ∞ and using the arbitrariness of S . (cid:3) In the sequel, by “solution” to (1.1) we shall tacitly mean the minimal one, according toProposition 2.3, to which therefore the crucial comparison results provided by Propositions2.4–2.5 are directly applicable.
Remark 2.6.
It can be shown that if, for some
C > , Ric o ( x ) ≥ − C (1 + r ( x )) for all x ∈ M , then the comparison principle between any bounded sub– and supersolution holds for prob-lem (1.1); consequently, problem (1.1) admits a unique solution in L ∞ ( M × (0 , T )) . Theseresults follow by combining the arguments of [15] (where the Cauchy problem for (1.2) hasbeen dealt with) and those in [1], in order to consider the source term u p . In this direction,let us mention that such a hypothesis on the Ricci curvature to get uniqueness is quitenatural; see, e.g., [19, 20, 21] for the linear case, m = 1 , without source terms. We omit thedetails, since a general comparison principle for problem (1.1) is not the main concern of thepresent paper; in fact, in order to prove our results, we do not need it, but it is sufficient touse the comparison principle in the form of Propositions 2.4, 2.5.3. Statements of the main results
Our first main result concerns global existence of solutions for sufficiently small, compactlysupported data, in the case p > m . Besides, such small data show propagation properties identical to the ones valid for the unforced porous medium equation (1.2). Hereafter, givena compactly supported datum u , we define R ( t ) to be the radius of the smallest ball thatcontains the support of the solution at time t . GABRIELE GRILLO, MATTEO MURATORI, AND FABIO PUNZO
Theorem 3.1.
Let assumptions (2.5) , (2.7) be satisfied. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊂ B R for some R > . Suppose that p > m and that k u k ∞ is sufficiently smallin a sense to be made more precise below. Then problem (1.1) (with T = ∞ ) has a global intime solution u ( t ) . Besides, the bound u ( x, t ) ≤ Cζ ( t ) h − ra η ( t ) i m − + ∀ t ≥ , ∀ x ∈ M holds for the following choices of functions η, ζ , of the constants C > , a > , and of theinitial data u :(i) ζ ( t ) := ( τ + t ) − m − [log( τ + t )] βm − , η ( t ) := [log( τ + t )] − β for any given β ≥ . The constants a = a ( h, N, m, R ) , τ = τ ( m, β, p, a ) must belarge enough and C m − = c a for a suitable c = c ( β, m ) . Finally, one requires that k u k ∞ ≤ C ( h, N, m, β, p, R ) is sufficiently small.As a consequence one has, for the class of data considered and all t ≥ : R ( t ) ≤ a [log( τ + t )] β , k u ( t ) k ∞ ≤ C ( τ + t ) − m − [log( τ + t )] βm − ; (3.1) (ii) ζ ( t ) := ( τ + t ) − α , η ( t ) := ( τ + t ) − β with p − < α < m − , β = 1 − α ( m − . The constants a = a ( h, N, m, R ) , τ = τ ( m, α, p, a ) must be large enough and C m − = c a for a suitable c = c ( α, m ) . Finally, one requires that k u k ∞ ≤ C ( h, N, m, β, p, R ) is sufficiently small, with C > C given in item (i).As a consequence one has, for the class of data considered and all t ≥ : R ( t ) ≤ a ( τ + t ) β , k u ( t ) k ∞ ≤ C ( τ + t ) − α . The above theorem provides upper bounds on solutions, and hence on the corresponding L ∞ norm and free boundary radius R ( t ) , that depend on the parameters involved and henceon the class of data considered. Of course, rougher bounds correspond to a wider set ofinitial data. It is important to stress that both bounds appearing in (3.1), in the case β = 1 ,correspond exactly to those valid for the free porous medium equation on H N proved in [42]and developed upon in [17], which are known to be sharp. We do not know whether therougher upper bounds stated above are optimal for some class of data, but we complementthe results by showing that, under lower curvature bounds, blow-up can occur for sufficientlybig, although compactly supported, data.In our next result we show in fact that if p > m and lower curvature bounds (see (2.6))hold, all sufficiently large data give rise to solutions blowing up in finite time. Theorem 3.2.
Let assumptions (2.5) , (2.6) be satisfied. Suppose that p > m . For any T > there exist compactly supported initial data u ∈ L ∞ ( M ) , u ≥ such that the correspondingsolution u ( t ) of problem (1.1) blows up at a time S ≤ T in the sense that k u ( t ) k ∞ → + ∞ as t → S − . More precisely, the bound u ( x, t ) ≥ Cζ ( t ) h − ra η ( t ) i m − + ∀ t ∈ (0 , T ∧ S ) , ∀ x ∈ M holds for the following choices of functions η, ζ , of the constants C, a, T > , and of the classof initial data u :(i) ζ ( t ) := ( T − t ) − α [ − log( T − t )] βm − , η ( t ) := [ − log( T − t )] − β for every t ∈ [0 , T ) , with T ∈ (0 , and α > m − , β > or α = 1 m − , < β ≤ . The constant C = C ( a, α, β, m, k, N, C , p ) must be large enough ( C is as in (4.16) )and that T = T ( a, C, p, m, α, β, N, k, C ) ∈ (0 , is small enough. Finally one re-quires that supp u ⊃ B R with R = R ( a, T, β ) large enough and inf B R u ≥ K ( C, T, m, α, β ) large enough;(ii) ζ ( t ) := ( T − t ) − α , η ( t ) := ( T − t ) β for every t ∈ [0 , T ) , with α > m − , < β ≤ α ( m − − . The constant C = C ( a, α, β, m, k, N, C , p ) must be large enough and that T = T ( a, C,p, m, α, β, N, k, C ) ∈ (0 , is small enough. Finally one requires that supp u ⊃ B R with R = R ( a, T, β ) large enough and inf B R u ≥ K ( C, T, α ) large enough. We comment that the above result will be shown by constructing appropriate subsolutionsthat blow up everywhere in M at time T , with support becoming the whole M exactly attime T . This does not rule out the possibility that u blows up locally in the L ∞ norm atsome earlier time S .We now discuss the case p < m . As concerns global existence results, we are forced torestrict ourselves to the range < p ≤ (1 + m ) / . In that range, and under an upper boundon curvature (see (2.7)), all compactly supported data give rise to a global in time solution. Theorem 3.3.
Let assumptions (2.5) , (2.7) be satisfied. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊂ B R for some R > . Suppose that < p ≤ m + 12 . Then problem (1.1) (with T = ∞ ) has a global in time solution u ( t ) . More precisely thebound u ( x, t ) ≤ Cζ ( t ) h − ra η ( t ) i m − + ∀ t ≥ , ∀ x ∈ M holds for the following choices of functions η, ζ , of the constants C, a > :(i) If < p < m +12 one chooses ζ ( t ) := ( τ + t ) α , η ( t ) := ( τ + t ) − β , with α ≥ m − p + 1 , β = 1 + α ( m − , τ ≥ , supposing in addition that a ≥ R ∨ H with H = H ( m, N, h, β ) sufficiently large,and that C = C ( m, N, h, a, p ) satisfies the (compatible) bounds c a / ( m − p ) ≤ C ≤ c a / ( m − , where c , c depend on m, N, h, p and τ = τ ( C, α, m, u ) is sufficientlylarge.(ii) If p = ( m + 1) / one chooses ζ ( t ) := exp { α ( τ + t ) } , η ( t ) := exp {− β ( τ + t ) } , with α ≥ α ( N, m, p, h ) > , β = α ( m − , τ ≥ , supposing in addition that the conditions on a, C, τ given in item i) above hold. In the whole range < p < m we can prove qualitatively similar lower bounds. In fact,if p ∈ (1 , m ) and lower curvature bounds (see (2.6)) hold, all sufficiently large data give riseto solutions blowing up at worst in infinite time. Theorem 3.4.
Let assumptions (2.5) , (2.6) be satisfied. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊃ B R for some R > . Suppose that < p < m and that < α < m − , β = α ( m −
1) + 12 . Then the bound u ( x, t ) ≥ Cζ ( t ) h − ra η ( t ) i m − + ∀ t ∈ (0 , S ) , ∀ x ∈ M holds, S ≤ + ∞ being the maximal existence time, for the following choices of functions η, ζ ,of the constants C, a > , and of the class of initial data u : ζ ( t ) := ( τ + t ) α , η ( t ) := ( τ + t ) − β for every t ∈ [0 , ∞ ) , where C = C ( a, m, k, N, α, β, C ) ( C is as in (4.16) ) and must be sufficiently large, τ = τ ( a, C, p, m, α, β, N, k, C ) ≥ is sufficiently large, and finally one requires that R = R ( a, τ, β ) is large enough and that inf B R u ≥ K ( C, τ, α ) is large enough. Remark 3.5.
We stress that, in the range < p ≤ (1+ m ) / , the combination of the resultsgiven in Theorems 3.3, 3.4 shows that large data give rise to solutions existing for all timesbut blowing up pointwise everywhere as t → + ∞ .Let M = H Nh be the simply connected manifold with sectional curvatures everywhereequal to − h . Then λ ( H Nh ) = ( N − h . Let v > be a positive, bounded solution of theequation − ∆ H Nh v = λ v . It is known that v is radial w.r.t. a given pole and monotonicallydecreasing as a function of the geodesic distance. Notice that v can be chosen so that k v k ∞ ≤ .As a final result, we show that data which are below a suitable profile related either tothe equation − ∆ u = u q , or to a ground state of − ∆ , both equations being in principleconsidered on H nh , and the corresponding solutions being transplanted on M , give rise toglobal in time solutions when p ≥ m . In fact, we remind the reader that the equation − ∆ u = u q , on H nh (3.2)admits strictly positive solutions for all q > , see [3]. Stationary solutions to (1.1) correspondto solutions of (3.2) with q = p/m , which is larger than one iff p > m . Notice that positive,bounded, energy solutions to (3.2) do exist (and are unique up to hyperbolic translations) when q ∈ (cid:16) , N +2 N − (cid:17) due to the results of [23]. We shall show that small data not necessarilywith compact support and possibly with large L p norms ( p ≥ ) give rise to solutions existingglobally in time. Theorem 3.6.
Let assumptions (2.5) , (2.7) be satisfied. Suppose that p ≥ m and, in thecase p = m only, that radial sectional curvatures K ω satisfy K ω ( x ) ≤ − h for all x ∈ M \{ o } ,with h ≥ / ( N − .Let v be a ground state of the Laplacian on H nh and, for p > m , let V be a strictly positivesolution to (3.2) with q = p/m , and transplant such functions on M . Suppose that, in case p = m , u ≤ v m and that, in case p > m , u ≤ V m . Then problem (1.1) (with T = ∞ ) hasa global in time solution u ( t ) that satisfies ≤ u ( t ) ≤ v m or, respectively, ≤ u ( t ) ≤ V m ,for all t ≥ . Remark 3.7.
By the results of [3] one knows that there exist, for all q > , infinitely manystrictly positive solutions to (3.2). All of them have polynomial decay at infinity, exceptthe unique (up to translations) energy solution. In particular, the L p norm ( p ≥ ) of datacomplying with the assumptions of Theorem 3.6 can be arbitrarily large. The same commentapplies when p = m since any ground state of − ∆ can be chosen. Notice that data might nothave compact support provided they are positive and below the suitable stationary profile.4. A family of supersolutions and subsolutions
We recall that, throughout this section, m > and p > .In order to construct a family of supersolutions and of subsolutions of equation u t = ∆( u m ) + u p in M × (0 , T ) , (4.1)consider two functions η, ζ ∈ C ([0 , T ]; R + ) and two constants C > , a > . Define u ( x, t ) ≡ u ( r ( x ) , t ) := Cζ ( t ) h − ra η ( t ) i m − + . (4.2)For further references, we compute u t − ∆( u m ) − u p . To this aim, set F ( r, t ) := 1 − ra η ( t ) , D := { ( x, t ) ∈ ( M \ { o } ) × (0 , T ) | < F ( r, t ) < } . For any ( x, t ) ∈ D we have u t ( r, t ) = Cζ ′ ( t ) F m − − Cm − ζ ( t ) F m − − η ′ ( t ) η ( t ) ra η ( t )= Cζ ′ ( t ) F m − − Cm − ζ ( t ) η ′ ( t ) η ( t ) F m − − + Cm − ζ ( t ) η ′ ( t ) η ( t ) F m − ; (4.3) u mr ( r, t ) = − C m ma ( m − ζ m ( t ) η ( t ) F m − ; (4.4) u mrr ( r, t ) = C m ma ( m − ζ m ( t ) η ( t ) F m − − . (4.5) By (2.1), (4.3)–(4.5), u t − ∆( u m ) − u p = CF m − − n F (cid:20) ζ ′ ( t ) + C m − ma ( m − ζ m ( t ) η ( t ) m ( r, θ ) + ζ ( t ) m − η ′ ( t ) η ( t ) (cid:21) − ζ ( t ) m − η ′ ( t ) η ( t ) − C m − ma ( m − ζ m ( t ) η ( t ) − C p − ζ p ( t ) F p − mm − o in D . (4.6) Proposition 4.1 (Supersolution conditions) . Let assumptions (2.5) , (2.7) be satisfied. Let T ∈ (0 , ∞ ] , ζ, η ∈ C ([0 , T ); R + ) . If, for all t ∈ (0 , T ) , − η ′ ( t ) η ( t ) ≥ C m − ma ( m − ζ m − ( t ) (4.7) and ζ ′ ( t ) + C m − ma ( m − ζ m ( t ) η ( t ) (cid:20) ( N − h − η ( t ) a ( m − (cid:21) ≥ C p − ζ p ( t ) , (4.8) then u as defined in (4.2) is a weak supersolution of equation (4.1) .Proof. In view of (2.5), (2.7) (2.4), (4.6) and the fact that u is radially decreasing, for any ( x, t ) ∈ D we get u t − ∆( u m ) − u p ≥ CF m − − n ξ ( t ) F − δ ( t ) − γ ( t ) F p − mm − o , (4.9)where ξ ( t ) := ζ ′ ( t ) + C m − ma ( m −
1) ( N − hζ m ( t ) η ( t ) + ζ ( t ) m − η ′ ( t ) η ( t ) ,δ ( t ) := ζ ( t ) m − η ′ ( t ) η ( t ) + C m − ma ( m − ζ m ( t ) η ( t ) , (4.10) γ ( t ) := C p − ζ p ( t ) . (4.11)For every t ∈ (0 , T ) and F ∈ [0 , , let us define ϕ ( F, t ) := ξ ( t ) F − δ ( t ) − γ ( t ) F p − mm − . Note that (4.7) implies ϕ (0 , t ) ≥ for every t ∈ (0 , T ) , whereas (4.8) implies ϕ (1 , t ) ≥ for every t ∈ (0 , T ) . Therefore, since F ϕ ( F, t ) is concave (recall that p, m > ), ϕ ( F, t ) ≥ for every ≤ F ≤ and t ∈ (0 , T ) . (4.12)Thus, because for each ( x, t ) ∈ D there holds < F ( x, t ) < , due to (4.12) and (4.9) wededuce that u t − ∆( u m ) − u p ≥ in D . Now observe that u ∈ C ( M × [0 , T )) , u m ∈ C (( M \ { o } ) × [0 , T )) (recall (4.4)) and, by thedefinition of u , u ≡ in M \ D \ [ { o } × (0 , T )] . Hence, u t − ∆( u m ) − u p ≥ in ( M \ { o } ) × (0 , T ) in the weak sense. On the other hand, thanks to a standard Kato-type inequality (note that u mr (0 , t ) ≤ ), we can easily infer that u t − ∆( u m ) − u p ≥ weakly in M × (0 , T ) . (cid:3) In order to construct subsolutions, we need to introduce some preliminary materials. Let σ ( t ) := ζ ′ ( t ) + C m − ma ( m −
1) ( N − k coth( k ) ζ m ( t ) η ( t ) + ζ ( t ) m − η ′ ( t ) η ( t ) , (4.13) δ ( t ) := ζ ( t ) m − η ′ ( t ) η ( t ) , (4.14)and σ ( t ) := ζ ′ ( t ) + C m − ma ( m −
1) ( N − C ζ m ( t ) η ( t ) + ζ ( t ) m − η ′ ( t ) η ( t ) , (4.15)where ( N − C ≥ ( r,θ ) ∈ [0 , × S N − m ( r, θ ) r . (4.16)Note that such a C > does exist since M is locally Euclidean, i.e. m ( r, θ ) ∼ N − r as r → .Let us set w ( x, t ) ≡ w ( r ( x ) , t ) := ( u ( x, t ) in ( M \ B ) × (0 , T ) ,v ( x, t ) in B × (0 , T ) , (4.17)where v ( x, t ) ≡ v ( r ( x ) , t ) := Cζ ( t ) (cid:20) − η ( t )2 a ( r + 1) (cid:21) m − + , ( x, t ) ∈ B × [0 , T ) . (4.18)Notice that w m is of class C . Proposition 4.2 (Subsolution conditions) . Let assumptions (2.5) , (2.6) be satisfied. Let T ∈ (0 , ∞ ] , ζ, η ∈ C ([0 , T ); R + ) with < η ( t ) ≤ a for all t ∈ (0 , T ) . (4.19) Let σ, δ, γ, σ , δ be defined by (4.13) , (4.10) , (4.11) , (4.15) and, respectively, (4.14) . Assumethat, for all t ∈ (0 , T ) , "(cid:18) m − p − m (cid:19) m − p − − (cid:18) m − p − m (cid:19) p − mp − σ p − mp − + ( t ) ≤ δ ( t ) γ m − p − ( t ) , (4.20) ( m − σ ( t ) ≤ ( p − m ) γ ( t ) , (4.21) p − mm − [ σ ( t ) − δ ( t )] ≤ γ ( t ) . (4.22) Then w as defined in (4.17) is a weak subsolution of equation (4.1) .Proof. Let u be as in (4.2), and set E := { ( x, t ) ∈ ( M \ B ) × (0 , T ) : 0 < F ( r, t ) < } . In view of (2.5), (2.6), (2.1), (2.2) and again the fact that u is radially decreasing, we deducethat u t − ∆( u m ) − u p ≤ CF m − − n σ ( t ) F − δ ( t ) − γ ( t ) F p − mm − o , (4.23) Given (4.23), we can suppose with no loss of generality that σ ( t ) ≥ for all t ∈ (0 , T ) . Let ϕ ( F, t ) := σ ( t ) F − δ ( t ) − γ ( t ) F p − mm − for all F ∈ [0 , and t ∈ (0 , T ) . Observe that (for better readability from now on we omit time dependence) ∂ϕ ∂F ( F, t ) = σ − p − mm − γ F p − m − ; as a consequence, ∂ϕ ∂F ( F, t ) = 0 if and only if F = F := (cid:18) m − p − m σγ (cid:19) m − p − , and F is the maximum point of the (concave) function F ϕ ( F, t ) . Thanks to (4.21), ≤ F ≤ . Moreover, an explicit computation shows that ϕ ( F , t ) = σ p − mp − γ m − p − "(cid:18) m − p − m (cid:19) m − p − − (cid:18) m − p − m (cid:19) p − mp − − δ . (4.24)From (4.20) and (4.24) we obtain ϕ ( F ) ≤ which, combined with (4.23), yields u t − ∆( u m ) − u p ≤ in E . Since u ∈ C ( M × [0 , T )) , u m ∈ C (( M \ { o } ) × [0 , T )) and, by the definition of u , u ≡ in M \ B \ E , there holds u t − ∆( u m ) − u p ≤ weakly in ( M \ B ) × (0 , T ) . (4.25)Now let v be as in (4.18). Set P := { x ∈ B × (0 , T ) : 0 < G ( r, t ) < } , where the function G is defined as G ( r, t ) := 1 − η ( t )2 a ( r + 1) . For any ( x, t ) ∈ P , we have: v t ( r, t ) = Cζ ′ ( t ) G m − − Cm − ζ ( t ) η ′ ( t ) η ( t ) G m − − + Cm − ζ ( t ) η ′ ( t ) η ( t ) G m − ; (4.26) v mr ( r, t ) = − C m ma ( m − ζ m ( t ) η ( t ) rG m − ; v mrr ( r, t ) = − C m ma ( m − η ( t ) ζ m ( t ) G m − + C m ma ( m − ζ m ( t ) η ( t ) r G m − − ≥ − C m ma ( m − η ( t ) ζ m ( t ) G m − . (4.27)In view of (4.16), (2.1) and (4.26)–(4.27), we deduce that v t − ∆( u m ) − v p ≤ C G m − − n σ ( t ) G − δ ( t ) − γ ( t ) G p − mm − o . (4.28) Due to (4.19), for each ( x, t ) ∈ P there holds ≤ G ( r, t ) ≤ . So, (4.28) and (4.22) yield v t − ∆( v m ) − v p ≤ in P ≡ B × (0 , T ) , (4.29)in the classical sense. Because w ∈ C ( M × [0 , T )) and w m ∈ C ( M × [0 , T )) (note that byconstruction u = v and u mr = v mr on ∂B × (0 , T ) ), from (4.25) and (4.29) the thesis easilyfollows. (cid:3) Proofs of the main results
We provide here complete proofs of our main results, by using explicit barrier argumentsbased on the results of Section 4 and on the comparison results given in Section 2.4.5.1.
Supersolutions.
We now provide some explicit supersolutions from which the resultsof Theorems 3.1, 3.3, 3.6 will follow.
Lemma 5.1.
Let assumptions (2.5) , (2.7) be satisfied. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊂ B R for some R > . Suppose that p > m . Let ζ ( t ) := ( τ + t ) − α [log( τ + t )] βm − , η ( t ) := [log( τ + t )] − β with α = m − , β ≥ . Suppose that C m − a ≤ m − m β , (5.1) a ≥ h ( N − m − , (5.2) α ≤ ( N − h C m − ma ( m − , (5.3) and that τ = τ ( m, β, p, a ) ≥ e is large enough. Then the function u defined in (4.2) is aweak supersolution of equation (4.1) with T = ∞ . Moreover, if a ≥ R , k u k ∞ ≤ C m − τ − α (log τ ) βm − , (5.4) then u is also a supersolution of problem (1.1) with T = ∞ .Proof. Condition (4.7) with T = ∞ reads β [log( τ + t )] β − ≥ mC m − a ( m −
1) ( τ + t ) − α ( m − = mC m − a ( m − for all t > , which holds due to (5.1) and the fact that τ ≥ e . Moreover, condition (4.8) with T = ∞ reads − α ( τ + t ) − α − [log( τ + t )] βm − + βm − τ + t ) − α − [log( τ + t )] βm − − + C m − ma ( m −
1) ( τ + t ) − αm [log( τ + t )] βm − (cid:20) ( N − h − [log( τ + t )] − β a ( m − (cid:21) ≥ C p − ( τ + t ) − αp [log( τ + t )] βpm − for all t > , which is fulfilled, in view of (5.2) and (5.3), provided τ = τ ( m, β, p, a ) ≥ e is so large that(note that α + 1 = αm ) α ( τ + t ) − α − [log ( t + τ )] βm − ≥ C p − ( t + τ ) − αp [log( τ + t )] βpm − for all t > , (5.5)where in the r.h.s. one can replace C with the upper bound given in (5.1). We point outthat in this last inequality the existence of such a τ is ensured since p > m . Hence, in viewof Proposition 4.1, we obtain that u is a weak supersolution of equation (4.1). In addition,(5.4) implies that (recall the explicit expression (4.2)) u ( x ) ≤ u ( x, for all x ∈ M . (5.6)Hence u is also a supersolution of problem (1.1).Finally, let us briefly explain how the above conditions can be made compatible: firstone picks C so as to satisfy (5.1) as equality, which means that C m − ∼ a , then plugs thischoice in (5.3) and selects a so large that both (5.3) and (5.2) are met. Lastly, τ ≥ e istaken so large that (5.5) holds upon the previous choices. (cid:3) Lemma 5.2.
Let assumptions (2.5) , (2.7) hold and suppose that p > m . Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊂ B R for some R > . Let ζ ( t ) := ( τ + t ) − α , η ( t ) := ( τ + t ) − β with p − < α < m − , β = 1 − α ( m − . (5.7) Suppose that (5.1) , (5.2) , (5.3) hold and that τ = τ ( m, α, p, a ) ≥ is large enough. Then thefunction u defined in (4.2) is a weak supersolution of equation (4.1) with T = ∞ . Moreover,if a ≥ R , k u k ∞ ≤ C m − τ − α , (5.8) then u is also a supersolution of problem (1.1) .Proof. Condition (4.7) with T = ∞ reads β ( τ + t ) β − α ( m − ≥ mC m − a ( m − for all t > , which holds for all τ ≥ , in view of (5.1), providing that β − α ( m − ≥ , the latter inequality being trivially guaranteed by (5.7). Furthermore, condition (4.8) with T = ∞ reads − α ( τ + t ) − α − + mC m − a ( m −
1) ( τ + t ) − αm − β (cid:20) ( N − h − ( τ + t ) − β a ( m − (cid:21) ≥ C p − ( τ + t ) − αp for all t > , which is fulfilled, thanks to (5.2) and (5.3), if β − α ( m − ≤ , α ( p − m ) ≥ β > (recall (5.7))and τ = τ ( m, α, p, a ) ≥ is so large that for all t > α ( τ + t ) − α − ≥ C p − ( τ + t ) − αp ; this is always possible thanks to the first (lower) inequality in (5.7). Hence, in view ofProposition 4.1, u is a weak supersolution of equation (4.1). The fact that C , a and τ can be chosen so as to satisfy the above conditions can be justified similarly to the end of theproof of Lemma 5.1.Finally, (5.8) yields (5.6), thus u is also a supersolution of problem (1.1). (cid:3) Proof of Theorem 3.1 . We use comparison with the barriers constructed in Lemmas 5.1,5.2 for solutions to approximating problems that involve homogeneous Dirichlet boundaryconditions on balls of radius R with R → + ∞ , see Proposition 2.4. The bounds still holdin such limit and yield part i ) of the claim by using Lemma 5.1 and part ii ) of the claim byusing Lemma 5.2. It is standard although tedious to check that the conditions on the initialdata considered in item ii ) give rise to a larger class than the one singled out in item i ). (cid:3) Lemma 5.3.
Let assumptions (2.5) , (2.7) be satisfied. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊂ B R for some R > . Suppose that < p < m + 12 . (5.9) Let ζ ( t ) := ( τ + t ) α , η ( t ) := ( τ + t ) − β , with α ≥ m − p + 1 , β = 1 + α ( m − , τ ≥ . (5.10) Suppose that (5.1) – (5.2) hold and that C m − p a ≥ m − m ( N − h . (5.11) Then the function u defined in (4.2) is a weak supersolution of equation (4.1) . Moreover, if a ≥ R ∨ H , Cτ α ≥ m − k u k ∞ , (5.12) with H = H ( m, N, h, β ) sufficiently large, then u is also a supersolution of problem (1.1) .Proof. Condition (4.7) with T = ∞ reads β ( τ + t ) β − − α ( m − ≥ mC m − ( m − a for all t > , which is satisfied, due to (5.1) and the fact that τ ≥ , whenever β − − α ( m − ≥ ⇐⇒ β ≥ α ( m − . (5.13)Furthermore, condition (4.8) with T = ∞ becomes α ( τ + t ) α − + mC m − a ( m −
1) ( τ + t ) αm − β (cid:20) ( N − h − ( τ + t ) − β a ( m − (cid:21) ≥ C p − ( τ + t ) αp for all t > , which is fulfilled, thanks to (5.2), (5.11) and τ ≥ , providing that and α ( m − p ) ≥ β . (5.14)It is straightforwardly checked that (5.10) (and (5.9)) ensures that both (5.13) and (5.14)hold. Hence, from Proposition 4.1 we get that u is a supersolution of equation (4.1). As concerns the compatibility of the conditions involving C and a , we just point out that(5.9) is crucial in order to guarantee that one can pick a so large that also (5.11) (in additionto (5.1)–(5.2)) holds.Finally, (5.12) implies (5.6), so u is also a supersolution of problem (1.1). (cid:3) Lemma 5.4.
Let assumptions (2.5) , (2.7) be satisfied. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊂ B R for some R > . Suppose that < p ≤ m + 12 . (5.15) Let ζ ( t ) := exp { α ( τ + t ) } , η ( t ) := exp {− β ( τ + t ) } , τ ≥ , with α ≥ α ( N, m, p, h ) > , β = α ( m − . (5.16) Suppose that (5.1) , (5.2) and (5.11) hold. Then the function u defined in (4.2) is a weaksupersolution of equation (4.1) . Moreover, if a ≥ R , C exp { ατ } ≥ m − k u k ∞ , (5.17) then u is also a supersolution of problem (1.1) .Proof. Condition (4.7) with T = ∞ reads β exp { [2 β − α ( m − τ + t ) } ≥ mC m − ( m − a for all t > , which is satisfied, in view of (5.1) and the fact that τ ≥ , as long as β − α ( m − ≥ ⇐⇒ β ≥ α ( m − . (5.18)Furthermore, condition (4.8) with T = ∞ reads α exp { α ( τ + t ) } + mC m − a ( m −
1) exp { ( αm − β )( τ + t ) } (cid:20) ( N − h − exp {− β ( τ + t ) } a ( m − (cid:21) ≥ C p − exp { αp ( τ + t ) } for all t > , which is fulfilled, due to (5.2), (5.11) and the fact that τ ≥ , providing that α ( m − p ) ≥ β . (5.19)Observe that (5.15) and (5.16) guarantee the validity of (5.18) and (5.19). Hence, Proposi-tion 4.1 ensures that u is a supersolution of equation (4.1).As concerns the compatibility of the conditions involving C and a , we point out thatif (5.15) holds with strict inequalities then the same comments as in the end of the proofof Lemma 5.3 apply. Otherwise, in the critical case p = m +12 , by substituting C with ther.h.s. of (5.1) in condition (5.11) one sees that the only degree of freedom left to make theinequality hold is the one given by α (through β ), which should be taken sufficiently largedepending on N, m, p, h (i.e. larger than a value that we labeled α ).Finally, from (5.17) there follows (5.6), so u is also a supersolution of problem (1.1). (cid:3) Proof of Theorem 3.3 . We use comparison with the barriers constructed in Lemmas 5.3,5.4 for solutions to approximating problems that involve homogeneous Dirichlet boundaryconditions on balls of radius R with R → + ∞ , see Proposition 2.4. The bounds still holdin such limit and yield part i ) of the claim by using Lemma 5.3 and part ii ) of the claim byusing Lemma 5.4. (cid:3) We now turn to the proof of Theorem 3.6. Its statement will follow from the next result.
Lemma 5.5.
Let assumptions (2.5) , (2.7) be satisfied. Suppose that p ≥ m and that, incase p = m only, radial sectional curvatures K ω satisfy K ω ( x ) ≤ − h for all x ∈ M \ { o } ,with h ≥ / ( N − .Consider a ground state v of the Laplacian on H nh and, for p > m , a strictly positivesolution V to (3.2) with q = p/m , and transplant such functions on M . Then u = v m when p = m , u = V m when p > m , are supersolutions of equation (4.1) . Moreover, if u ≤ v m , or u ≤ V m respectively, then u is also a supersolution of problem (1.1) .Proof. By the properties of v recalled above and Laplacian comparison (2.4) we compute: − ∆ v = − v ′′ − m ( r, θ ) v ′ ≥ − v ′′ − ( N − h coth( hr ) v ′ = − ∆ H nh v = λ ( H nh ) v ≥ v in M , where we have used the known bound (see [25]) λ ≥ ( N − h and the running curvature assumption.Since p = m , the function u := v m is a positive stationary supersolution of equation (4.1).In fact − ∆ u m = − ∆ v ≥ v = u p . Clearly, u is also a supersolution of problem (1.1), provided that u ≤ u = v m in M . Anessentially identical proof works for the case p > m by replacing v with V . (cid:3) Proof of Theorem 3.6 . We use comparison with the barrier constructed in Lemma5.5 for solutions to approximating problems that involve homogeneous Dirichlet boundaryconditions on balls of radius R with R → + ∞ , see Proposition 2.4. The bounds still holdin such limit. (cid:3) Subsolutions.
We now provide some explicit supersolutions from which the results ofTheorems 3.2, 3.4 will follow.
Lemma 5.6.
Let assumptions (2.5) , (2.6) hold and assume that p > m . hold. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊃ B R for some R > . Let ζ ( t ) := ( T − t ) − α [ − log( T − t )] βm − , η ( t ) := [ − log( T − t )] − β for every t ∈ [0 , T ) , with T ∈ (0 , and α > m − , β > or α = 1 m − , < β ≤ . (5.20) Suppose that C m − a ≥ max (cid:26) α ( m − mk coth( k )( N − , α ( m −
1) + βmC ( N − , ˜ C (cid:27) , (5.21) C m − a ≥ β ( m − m , (5.22) for a suitable constant ˜ C = ˜ C ( p, m, N, k ) > , and that T = T ( a, C, p, m, α, β, N, k, C ) ∈ (0 , is small enough ( C is as in (4.16) ). Then the function u defined in (4.2) is a weaksubsolution of equation (4.1) . Moreover, if R ≥ a ( − log T ) β , u ≥ CT − α ( − log T ) βm − in B R , (5.23) then u is also a subsolution of problem (1.1) .Proof. We take T = T ( a, β ) > so small that (4.19) is fulfilled. With the above choices of ζ and η , in view of (5.20) (for the moment we only use α ≥ m − ), the first inequality of(5.21) and the fact that T ≤ , we have that (recall the definition of σ given by (4.13)) σ ( t ) = α ( T − t ) − α − [ − log( T − t )] βm − + C m − m ( N − k coth( k ) a ( m −
1) ( T − t ) − αm [ − log( T − t )] βm − ≤ C m − m ( N − k coth( k ) a ( m −
1) ( T − t ) − αm [ − log( T − t )] βm − . (5.24)Furthermore, upon recalling the definition of δ given by (4.10), thanks to (5.22) we obtainthe estimate δ ( t ) = − βm − T − t ) − α − [ − log( T − t )] βm − − + C m − ma ( m − ( T − t ) − αm [ − log( T − t )] β (2 − m ) m − ≥ C m − m a ( m − ( T − t ) − αm [ − log( T − t )] β (2 − m ) m − (5.25)as long as T ∈ (0 , is so small that ( T − t ) − α − [ − log( T − t )] βm − − ≤ ( T − t ) − αm [ − log( T − t )] β (2 − m ) m − ∀ t ∈ (0 , T ) . Note that such a choice of T is always feasible thanks to (5.20). Now set K := (cid:18) m − p − m (cid:19) m − p − − (cid:18) m − p − m (cid:19) p − mp − > . (5.26)Due to (5.24) and (5.25), condition (4.20) is implied by K p − p − m C m − m ( N − k coth( k ) a ( m −
1) ( T − t ) − αm [ − log( T − t )] βm − ≤ C ( p − m − p − m (cid:18) C m − m a ( m − (cid:19) p − p − m ( T − t ) − αp ( m − − αm ( p − p − m [ − log( T − t )] βm − ∀ t ∈ (0 , T ) . (5.27)Note that αm ≤ αp m − p − m + αm p − p − m if and only if ( p − m )( m − ≥ , which trivially holds since p > m . Hence, in view of the third inequality in (5.21), we havethat (5.27) (and so (4.20)) is fulfilled: we point out that, for this purpose, the hypothesis p > m is essential (at p = m the dependence on C and a vanishes and there is no moredegree of freedom to make (5.27) hold). Moreover, from (5.24) we deduce that (4.21) issatisfied whenever mk coth( k )( N − p − m ≤ a C p − m ( T − t ) − α ( p − m ) [ − log( T − t )] β ( p − m − ∀ t ∈ (0 , T ) , (5.28)and to this aim it is enough to choose T = T ( a, C, p, m, α, β, N, k ) > small enough.Finally, thanks to the middle inequality in (5.21), we have that (recall that σ and δ aredefined by (4.15) and (4.14), respectively) σ ( t ) − δ ( t ) ≤ C m − m ( N − C a ( m −
1) ( T − t ) − αm [ − log( T − t )] βm − . We therefore deduce that inequality (4.22) is satisfied provided p − mm − +1 m ( N − C ( m − ≤ a C p − m ( T − t ) − α ( p − m ) [ − log( T − t )] β ( p − m − ∀ t ∈ (0 , T ) . (5.29)Similarly to (5.28), it is plain that (5.29) holds if T = T ( a, C, p, m, α, β, N, C ) > is smallenough. Since we have established that (4.20), (4.21) and (4.22) hold, from Proposition 4.2we get that u is a subsolution of equation (4.1). Furthemore, (5.23) implies that u ( x, ≤ u ( x ) for all x ∈ M , (5.30)so that u is also a subsolution of problem (1.1). (cid:3) Lemma 5.7.
Let assumptions (2.5) , (2.6) hold and suppose that p > m . Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊃ B R for some R > . Let ζ ( t ) := ( T − t ) − α , η ( t ) := ( T − t ) β for every t ∈ [0 , T ) , with α > m − , < β ≤ α ( m − − . (5.31) Suppose that (5.21) – (5.22) hold, and that T = T ( a, C, p, m, α, β, N, k, C ) ∈ (0 , is smallenough ( C is as in (4.16) ). Then the function u defined in (4.2) is a weak subsolution ofequation (4.1) . Moreover, if R ≥ aT − β , u ≥ CT − α in B R , (5.32) then u is also a subsolution of problem (1.1) .Proof. We take T = T ( a, β ) ∈ (0 , so small that (4.19) is fulfilled. In view of the firstinequality in (5.21) and (5.31) (for the moment we only use the fact that β ≤ α ( m − − ),we have (recall that σ is defined in (4.13)) σ ( t ) = α ( T − t ) − α − + C m − m ( N − k coth( k ) a ( m −
1) ( T − t ) − αm + β − βm − T − t ) − α − ≤ C m − m ( N − k coth( k ) a ( m −
1) ( T − t ) − αm + β . (5.33) Moreover, thanks to (5.22) and (5.31) (recall that δ is defined in (4.10)), δ ( t ) = − βm − T − t ) − α − + C m − ma ( m − ( T − t ) − αm +2 β ≥ C m − m a ( m − ( T − t ) − αm +2 β ; (5.34)note that here we do need that β ≤ α ( m − − . Now let K be defined as in (5.26). By virtueof (5.33) and (5.34), condition (4.20) is implied by K p − p − m C m − m ( N − k coth( k ) a ( m −
1) ( T − t ) − αm + β ≤ C ( p − m − p − m (cid:18) C m − m a ( m − (cid:19) p − p − m ( T − t ) − αp ( m − β − αm )( p − p − m ∀ t ∈ (0 , T ) . (5.35)Observe that αm − β ≤ αp m − p − m + ( αm − β ) p − p − m holds if and only if ( p − m )[ α ( m − − β ] ≥ , which is guaranteed since p > m and (5.31) holds. Therefore, from the third inequality in(5.21) we infer that (5.35) is fulfilled: we point out, once again, that here it is essential that p > m , for the same reasons as in the proof of Lemma 5.6. On the other hand, from (5.33)we deduce that (4.21) is satisfied provided mk coth( k )( N − p − m ≤ a C p − m ( T − t ) − α ( p − m ) − β ∀ t ∈ (0 , T ) ; to this end, it suffices to pick T = T ( a, C, p, m, α, β, N, k ) > small enough.Furthermore (recall that σ and δ are defined by (4.15) and (4.14), respectively), thanksto the central inequality in (5.21) (actually here one can replace β with in such inequality),we deduce that σ ( t ) − δ ( t ) ≤ C m − m ( N − C a ( m −
1) ( T − t ) − αm + β . We therefore infer that inequality (4.22) is met provided p − mm − +1 m ( N − C ( m − ≤ a C p − m ( T − t ) − α ( p − m ) − β ∀ t ∈ (0 , T ) . (5.36)It is apparent that (5.36) is satisfied if T = T ( a, C, p, m, α, β, N, C ) > is small enough.Since (4.20), (4.21) and (4.22) hold, from Proposition 4.2 we get that u is a subsolution ofequation (4.1). Finally, (5.32) yields (5.30), so u is also a subsolution of problem (1.1). (cid:3) Proof of Theorem 3.2 . We use comparison with the barriers constructed in Lemmas5.6, 5.7, see Proposition 2.5. This yield part i ) of the claim by using Lemma 5.6 and part ii ) of the claim by using Lemma 5.7. (cid:3) Lemma 5.8.
Let assumptions (2.5) , (2.6) be satisfied. Let u ∈ L ∞ ( M ) , u ≥ with supp u ⊃ B R for some R > . Suppose that < p < m . (5.37) Let ζ ( t ) := ( τ + t ) α , η ( t ) := ( τ + t ) − β for every t ∈ [0 , ∞ ) . Suppose that (5.22) holds, < α < m − , β = α ( m −
1) + 12 , (5.38) C m − a ≥ max (cid:26) α ( m − mk coth( k )( N − , α ( m − mC ( N − (cid:27) (5.39) and that τ = τ ( a, C, p, m, α, β, N, k, C ) ≥ is sufficiently large. Then the function u definedin (4.2) is a weak subsolution of equation (4.1) . Moreover, if R ≥ aτ β , u ≥ Cτ α in B R , (5.40) then u is also a subsolution of problem (1.1) .Proof. We take τ β ≥ a , so (4.19) is fulfilled. In view of (5.38), τ ≥ and the first inequalityin (5.39), we have that (recall that σ is defined in (4.13)) σ ( t ) = α ( τ + t ) α − + C m − m ( N − k coth( k ) a ( m −
1) ( τ + t ) αm − β − βm − τ + t ) α − ≤ C m − m ( N − k coth( k ) a ( m −
1) ( τ + t ) αm − β . (5.41)Here we are only using the fact that β ≤ α ( m −
1) + 1 . Moreover, thanks to (5.22), (5.38),(5.39) and τ ≥ (recall that δ is defined in (4.10)), δ ( t ) = − βm − τ + t ) α − + C m − ma ( m − ( τ + t ) αm − β ≥ C m − m a ( m − ( τ + t ) αm − β . (5.42)Let K be defined as in (5.26). Due to (5.41) and (5.42), condition (4.20) is implied by K p − p − m C m − m ( N − k coth( k ) a ( m −
1) ( τ + t ) αm − β ≤ C ( p − m − p − m (cid:18) C m − m a ( m − (cid:19) p − p − m ( τ + t ) αp ( m − αm − β )( p − p − m ∀ t > . (5.43)Observe that αm − β < αp m − p − m + ( αm − β ) p − p − m holds if and only if ( p − m )[ α ( m − − β ] > , which is valid thanks to (5.37) and (5.38)It is therefore apparent that one can choose τ ( a, C, p, m, α, β, N, k ) ≥ sufficiently largeto make (5.43) (and so (4.20)) hold. Note that here the extrema of the inequality ( p = m or β = α ( m − ) have to be excluded, otherwise one does not have enough degrees of freedomon C, a to make (5.43) hold).Moreover, from (5.41) we deduce that (4.21) is fulfilled whenever mk coth( k )( N − p − m ≤ a C p − m ( τ + t ) α ( p − m )+ β ∀ t ≥ . (5.44) From (5.38) it follows that α ( p − m ) + β > . So, (5.44) is satisfied, providing again that τ ( a, C, p, m, α, β, N, k ) ≥ is sufficiently large. Furthermore, thanks to the last inequalityin (5.39) (recall that σ and δ are defined by (4.15) and (4.14), respectively), we have that σ ( t ) − δ ( t ) ≤ C m − m ( N − C a ( m −
1) ( τ + t ) αm − β . Hence, from (5.38) we infer that inequality (4.22) is satisfied provided p − mm − +1 m ( N − C ( m − ≤ a C p − m ( τ + t ) α ( p − m )+ β ∀ t ≥ . (5.45)Clearly, (5.45) holds as long as τ ( a, C, p, m, α, β, N, C ) ≥ is sufficiently large. Since (4.20),(4.21) and (4.22) hold, from Proposition 4.2 we get that u is a subsolution of equation (4.1).Finally, (5.40) yields (5.30), so that u is also a subsolution of problem (1.1). (cid:3) Proof of Theorem 3.4 . We use comparison with the barriers constructed in Lemma5.8, see Proposition 2.5. This yields immediately the claim. (cid:3)
Acknowledgements.
G.G. was partially supported by the PRIN Project “Equazioni allederivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate,e applicazioni” (Italy). M.M. and F.P. were partially supported by the GNAMPA Project“Equazioni diffusive non-lineari in contesti non-Euclidei e disuguaglianze funzionali asso-ciate” (Italy). All authors have also been supported by the Gruppo Nazionale per l’AnalisiMatematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale diAlta Matematica (INdAM, Italy).
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