Global asymptotic stability of bifurcating, positive equilibria of p-Laplacian boundary value problems with p-concave nonlinearities
aa r X i v : . [ m a t h . A P ] M a y GLOBAL ASYMPTOTIC STABILITY OF BIFURCATING, POSITIVE EQUILIBRIAOF p -LAPLACIAN BOUNDARY VALUE PROBLEMS WITH p -CONCAVENONLINEARITIES BRYAN P. RYNNE
Abstract.
We consider the parabolic, initial value problem v t = ∆ p ( v ) + λg ( x, v ) φ p ( v ) , in Ω × (0 , ∞ ) ,v = 0 , in ∂ Ω × (0 , ∞ ) , (IVP) v = v > , in Ω × { } , where Ω is a bounded domain in R N , for some integer N > , with smooth boundary ∂ Ω , φ p ( s ) := | s | p − sgn s , s ∈ R , ∆ p denotes the p -Laplacian, with p > max { , N } , v ∈ C (Ω) , and λ > . Thefunction g : Ω × [0 , ∞ ) → (0 , ∞ ) is C and, for each x ∈ Ω , the function g ( x, · ) : [0 , ∞ ) → (0 , ∞ ) isLipschitz continuous and strictly decreasing.Clearly, (IVP) has the trivial solution v ≡ , for all λ > . In addition, there exists < λ min ( g ) <λ max ( g ) ( λ max ( g ) may be ∞ ) such that: • if λ ( λ min ( g ) , λ max ( g )) then (IVP) has no non-trivial, positive equilibrium; • if λ ∈ ( λ min ( g ) , λ max ( g )) then (IVP) has a unique, non-trivial, positive equilibrium e λ ∈ W ,p (Ω) .We prove the following results on the positive solutions of (IVP): • if < λ < λ min ( g ) then the trivial solution is globally asymptotically stable; • if λ min ( g ) < λ < λ max ( g ) then e λ is globally asymptotically stable; • if λ max ( g ) < λ then any non-trivial solution blows up in finite time. Introduction
We consider the parabolic, initial-boundary value problem v t = ∆ p ( v ) + λg ( x, v ) φ p ( v ) , in Ω × (0 , ∞ ) ,v = 0 , in ∂ Ω × (0 , ∞ ) ,v = v > , in Ω × { } , (1.1)where Ω is a bounded domain in R N , for some integer N > , with smooth boundary ∂ Ω , φ p ( s ) := | s | p − sgn s , s ∈ R , and ∆ p denotes the p -Laplacian, with p > max { , N } , v ∈ C (Ω) ,and λ > .We suppose that g : Ω × [0 , ∞ ) → (0 , ∞ ) is C and, for each x ∈ Ω ,g ( x, · ) : [0 , ∞ ) → (0 , ∞ ) is strictly decreasing, (1.2) g ∞ ( x ) := lim ξ →∞ g ( x, ξ ) < g ( x ) := g ( x, . (1.3)We also suppose that g is Lipschitz with respect to ξ , in the following sense: for any K > thereexists L K such that | g ( x, ξ ) − g ( x, ξ ) | L K | ξ − ξ | , x ∈ Ω , ξ , ξ K. (1.4)We are interested in positive solutions of (1.1), so we introduce the following notation: C (Ω) (respectively W ,p , + (Ω) ) denotes the set of ω ∈ C (Ω) (respectively ω ∈ W ,p (Ω) ) with ω > on Ω . It is known that for any v ∈ C (Ω) the problem (1.1) has a unique, positive solution t → v λg,v ( t ) ∈ W ,p , + (Ω) , on some maximal interval (0 , T ) , where we may have T < ∞ or T = ∞ (what we mean by a solution will be made precise in Theorem 3.1 below). We are interestedin the asymptotic behaviour of these solutions. This asymptotic behaviour is determined by thestructure of the set of positive equilibria of (1.1), so we first describe this.For a given λ > , a positive equilibrium is a time-independent solution u ∈ W ,p , + (Ω) of (1.1),that is, u satisfies ∆ p ( u ) + λg ( u ) φ p ( u ) = 0 (this will be made precise in Section 2.5 below).For any λ > the function v ≡ (or ( λ, v ) = ( λ, ) is a ( trivial ) equilibrium. In addition,the complete structure of the set of non-trivial, positive equilibria of (1.1) is as follows (seeTheorem 2.3 below). There exists < λ min ( g ) < λ max ( g ) (we may have λ max ( g ) = ∞ ) suchthat: • if λ ( λ min ( g ) , λ max ( g )) then (1.1) has no non-trivial equilibrium in W ,p , + (Ω) ; • if λ ∈ ( λ min ( g ) , λ max ( g )) then (1.1) has a unique, non-trivial equilibrium e λ ∈ W ,p , + (Ω) .We will prove the following results on the asymptotic behaviour of the positive solutions of(1.1). For any = v ∈ C (Ω) : • < λ < λ min ( g ) = ⇒ lim t →∞ k v λg,v ( t ) k ,p = 0 • λ min ( g ) < λ < λ max ( g ) = ⇒ lim t →∞ k v λg,v ( t ) − e λ k ,p = 0 • λ max ( g ) < λ = ⇒ there exists T < ∞ such that lim t ր T | v λg,v ( · ) | = ∞ Regarding (1.1) as a bifurcation problem, these results can be interpreted as saying that: • when λ < λ min ( g ) , the trivial equilibrium is globally stable; • as λ increases through λ min ( g ) , the solution ( λ, loses stability, and a continuum, E + , ofglobally stable, positive equilibrium solutions bifurcates from the point ( λ min ( g ) , (in a sense, there is a supercritical, transcritical bifurcation at λ min ( g ) , with exchange ofstability between the equilibria); • as λ increases through λ max ( g ) , the continuum E + ‘meets infinity’ and then disappears,after which all non-trivial, positive solutions blow up in finite time.These results are consistent with a bifurcation analysis of the corresponding semilinear ( p = 2 )problem, using the ‘principle of linearised stability’ to obtain local stability. Such problems havebeen extensively investigated, see [12] and the references therein for a summary of the mainresults. However, we do not use bifurcation theory to obtain our results, which usually yields localstability results. Instead, we use a mixture of comparison and compactness arguments to obtainthe above results.For the quasilinear problem involving the p -Laplacian with p > considered here, these resultsare consistent with the results on ‘linearised stability’ in the ‘ p -concave’ case in [13] (condition(1.2) is termed ‘ p -concavity’ in [13]; this terminology has been used in other publication for verysimilar, but slightly different, conditions). However, the term ‘linearised stability’ in [13] refers tothe sign of the principal eigenvalue of the linearisation of the problem at an equilibrium solution e λ , not to the dynamic (time-dependent) stability that we consider. In the quasilinear case it isnot clear that ‘linearised stability’, in this sense, implies stability in the usual dynamic sense. Evenif such a result could be proved, it would give local rather than global stability. LOBAL ASYMPTOTIC STABILITY OF POSITIVE SOLUTIONS 3
The convergence results that we obtain say nothing about the rate of convergence. In particular,we do not obtain the exponential convergence that would be obtained from any sort of ‘linearisedstability’ analysis, if such were possible. Convergence rates for quasilinear problems are discussedin [4], together with a broad survey of the literature relating to this. It is also noted in [4] thatthe known results are limited, and difficult to apply. In particular, the results discussed in [4] saynothing about the problem considered here.2.
Preliminaries
Notation.
We let C (Ω) denote the standard space of real valued, continuous functionsdefined on Ω , with the standard sup-norm on | · | (throughout, all function spaces will be real); L q (Ω) , q > , denotes the standard space of functions on Ω whose q th power is integrable, withnorm k · k q ; W ,p (Ω) denotes the standard, first order Sobolev space of functions on Ω whichare zero on ∂ Ω , with norm k · k ,p . By our assumption that p > N , the space W ,p (Ω) iscompactly embedded into C (Ω) . We also define the set of positive functions in W ,p (Ω) to be W ,p , + (Ω) := { ω ∈ W ,p (Ω) : ω > on Ω } . The dual space of W ,p (Ω) is denoted by W − ,p ′ (Ω) ,where p ′ := p/ ( p − is the conjugate exponent of p .If h : Ω × [0 , ∞ ) → R is continuous then, for any ω ∈ C (Ω) , we define h ( ω ) ∈ C (Ω) by h ( ω )( x ) := h ( x, ω ( x )) , x ∈ Ω . Clearly, the ‘Nemitskii’ mapping ω → h ( ω ) : C (Ω) → C (Ω) is continuous. In particular, werepeatedly use the Nemitskii mapping φ p : ω → φ p ( ω ) : C (Ω) → C (Ω) .2.2. The p -Laplacian. Formally, the p -Laplacian is defined by ∆ p ω := ∇ · ( |∇ ω | p − ∇ ω ) , for suitable ω , where | v | := ( v + · · · + v N ) / for v ∈ R N . More precisely, for any ω ∈ W ,p (Ω) ,we define ∆ p ( ω ) ∈ W − ,p ′ (Ω) by Z Ω ∆ p ( ω ) ϕ := − Z Ω |∇ ω | p − ∇ ω · ∇ ϕ, ∀ ϕ ∈ W ,p (Ω) . (2.1)A precise definition of what is meant by a solution of (1.1) will be given in Section 3 below.2.3. Principal eigenvalues of the p -Laplacian. We briefly consider the weighted, nonlineareigenvalue problem − ∆ p ( ψ ) = µρφ p ( ψ ) , ψ ∈ W ,p (Ω) , (2.2)where µ ∈ R and the weight function ρ ∈ L (Ω) . We say that µ is an eigenvalue of (2.2), with eigenfunction ψ ∈ W ,p (Ω) \ { } , if the following weak formulation of (2.2) holds Z Ω |∇ ψ | p − ∇ ψ · ∇ ϕ = µ Z Ω ρφ p ( ψ ) ϕ, ∀ ϕ ∈ W ,p (Ω) . (2.3)A principal eigenvalue of (2.2) is an eigenvalue µ which has a positive eigenfunction ψ ∈ W ,p (Ω) (which we will normalise by, say, | ψ | = 1 ). The following result is well known — see,for example, [5, Sections 3-4]. BRYAN P. RYNNE
Lemma 2.1.
Suppose that the weight function ρ satisfies: ρ > on Ω , with ρ > on a set ofpositive Lebesgue measure. Then the eigenvalue problem (2.2) has a unique principal eigenvalue µ ( ρ ) . This eigenvalue has the properties, µ ( ρ ) > , ψ ( ρ ) > on Ω , and Z Ω |∇ ω | p > µ ( ρ ) Z Ω ρ | ω | p , ∀ ω ∈ W ,p (Ω) . (2.4) In addition, if ρ , ρ are two such weight functions, then ρ ρ on Ω and ρ < ρ on a set of positive Lebesgue measure = ⇒ µ ( ρ ) > µ ( ρ ) . Hence, by (1.3) and Lemma 2.1, we may define < λ min ( g ) := µ ( g ) < λ max ( g ) := ( µ ( g ∞ ) < ∞ , if g ∞ = 0 (in L ∞ (Ω) ), ∞ , if g ∞ = 0 (in L ∞ (Ω) ).and we denote the corresponding normalised principal eigenfunctions by ψ min ( g ) , ψ max ( g ) .2.4. An energy functional.
We now define an ‘energy’ functional for (1.1) on W ,p , + (Ω) . Let F ( x, ξ ) := Z ξ g ( x, s ) s p − ds, ( x, ξ ) ∈ Ω × [0 , ∞ ) ,E λg ( ω ) := 1 p Z Ω |∇ ω | p − λ Z Ω F ( ω ) , ω ∈ W ,p , + (Ω) . By the continuity of the embedding W ,p , + (Ω) ֒ → C (Ω) , the energy functional E λg : W ,p , + (Ω) → R is continuous. Lemma 2.2. If λ < µ ( g ∞ ) then there exists an increasing function M λ : R → (0 , ∞ ) suchthat, | ω | + k ω k ,p < M λ ( E λg ( ω )) , ω ∈ W ,p , + (Ω) . Proof.
Suppose the contrary, so there exists R ∈ R and = ω n ∈ W ,p , + (Ω) , n = 1 , , . . . , such that E λg ( ω n ) R and lim n →∞ k ω n k ,p = ∞ (since p > N , | ω | C k ω k ,p , for someconstant C ). Let e ω n := ω n / k ω n k ,p , n = 1 , , . . . . By the compactness of the embedding W ,p (Ω) ֒ → C (Ω) , we may assume that e ω n → e ω ∞ in C (Ω) , for some e ω ∞ ∈ C (Ω) , and itsuffices to show that this leads to a contradiction.By definition, E λg ( ω n ) = 1 p k ω n k p ,p (cid:26)Z Ω |∇ e ω n | p − λp Z Ω F ( ω n ) k ω n k p ,p (cid:27) , n > . (2.5)We now show that, as n → ∞ , p Z Ω F ( ω n ) k ω n k p ,p → Z Ω g ∞ e ω p ∞ . (2.6)By (1.2) and (1.3) there exists C > such that, for any n > , p | F ( ω n ) | k ω n k p ,p | g | | ω n | p k ω n k p ,p C, (2.7) LOBAL ASYMPTOTIC STABILITY OF POSITIVE SOLUTIONS 5 and similarly, using (1.3), for any x ∈ Ω and ǫ > , there exists C ( x, ǫ ) > such that, for any n > , p F ( ω n )( x ) k ω n k p ,p C ( x, ǫ ) + ( g ∞ ( x ) + ǫ ) ω n ( x ) p k ω n k p ,p → ( g ∞ ( x ) + ǫ ) e ω ∞ ( x ) p . Combining this with a similar lower bound shows that p F ( ω n )( x ) k ω n k p ,p → g ∞ ( x ) e ω ∞ ( x ) p , x ∈ Ω , (2.8)so (2.6) follows from (2.7), (2.8) and the dominated convergence theorem.Now suppose that R Ω g ∞ e ω p ∞ > . Then, by Lemma 2.1, for n > , Z Ω |∇ e ω n | p > µ ( g ∞ ) Z Ω g ∞ e ω pn → µ ( g ∞ ) Z Ω g ∞ e ω p ∞ > , (2.9)and combining (2.5), (2.6) and (2.9) shows that E λg ( ω n ) → ∞ (since λ < µ ( g ∞ ) ). However,this contradicts the initial assumption that E λg ( ω n ) R for all n > . Next, suppose that R Ω g ∞ e ω p ∞ = 0 , with k e ω ∞ k p > . Then, by Lemma 2.1, for n > , Z Ω |∇ e ω n | p > µ ( ) k e ω n k pp → µ ( ) k e ω ∞ k pp > (2.10)(where denotes the weight function that is identically 1 on Ω ), and combining (2.5), (2.6) and(2.10) again yields the contradiction E λg ( ω n ) → ∞ .Finally, suppose that k e ω ∞ k p = 0 . Since k e ω n k ,p = 1 , n > , this implies that R Ω |∇ e ω n | p → , so combining (2.5) and (2.6) again yields a contradiction, and so completes the proof ofLemma 2.2. (cid:3) Existence and uniqueness of non-trivial, positive equilibria. A positive equilibrium of(1.1) is a solution of the problem − ∆ p ( u ) = λg ( u ) φ p ( u ) , u ∈ W ,p , + (Ω) . (2.11)More precisely, a solution of (2.11) is defined to be a function u ∈ W ,p , + (Ω) which satisfies thefollowing weak formulation of (2.11), Z Ω |∇ u | p − ∇ u · ∇ ϕ = λ Z Ω g ( u ) φ p ( u ) ϕ, ∀ ϕ ∈ W ,p (Ω) . (2.12)Clearly, for any λ ∈ R , the function u = 0 is a ( trivial ) positive solution of (1.1) and (2.11).We now describe the structure of the set of non-trivial, positive equilibria. Let Λ := ( λ min ( g ) , λ max ( g )) . Theorem 2.3. ( a ) If λ Λ then (2.11) has no non-trivial solution u ∈ W ,p , + (Ω) . ( b ) If λ ∈ Λ then (2.11) has a unique, non-trivial solution e λ ∈ W ,p , + (Ω) , and e λ > on Ω . BRYAN P. RYNNE ( c ) The mapping λ → e λ : Λ → W ,p , + (Ω) is continuous, and lim λ ց λ min ( g ) k e λ k ,p = 0 , λ max ( g ) < ∞ = ⇒ lim λ ր λ max ( g ) k e λ k ,p = ∞ . (2.13) Proof.
Parts ( a ) and ( b ) are proved in [8, Theorems 1, 2]. We observe that: ( i ) the strict positivity of e λ on Ω is not stated in [8, Theorem 2], but is derived in its proof; italso follows from Lemma 2.1; ( ii ) part ( a ) also follows from Lemma 2.1 and the definitions of λ min ( g ) , λ max ( g ) .To prove part ( c ) , suppose firstly that λ n ∈ Λ , n = 1 , , . . . , is such that lim n →∞ λ n = λ ∞ < ∞ , lim n →∞ k e λ n k ,p = ∞ . (2.14)Defining w n := e λ n / k e λ n k ,p , n = 1 , , . . . , it follows from the compactness and continuityproperties described on p. 229 of [7] that we may suppose that w n → w ∞ ∈ W ,p , + (Ω) , with w ∞ = 0 and − ∆ p ( w ∞ ) = λ ∞ gφ p ( w ∞ ) ,g ( x ) = lim n →∞ g ( e λ n ( x )) , x ∈ Ω . (2.15)By (1.2) and (1.3), g g on Ω , so it follows from (2.15) and the invertibility of the operator ∆ p (see [7]) that we must have g > on a set of positive measure. Hence, by Lemma 2.1 and(2.15), w ∞ ( x ) > for each x ∈ Ω , so that e λ n ( x ) → ∞ , and g ( x ) = g ∞ ( x ) . Thus, by thedefinition of λ max ( g ) and (2.15), λ ∞ = λ max ( g ) . We conclude that the mapping λ → e λ :Λ → W ,p , + (Ω) is bounded on any closed, bounded subinterval of Λ , and hence, again using thecontinuity properties in [7], this mapping is continuous on Λ .Next, by similar arguments, it can be shown that if λ n → λ min ( g ) then k e λ n k ,p cannot bebounded away from 0, and if λ n → λ max ( g ) then k e λ n k ,p cannot be bounded, which proves(2.13), and so completes the proof of the theorem. (cid:3) Remark 2.4. ( a ) Theorem 2.3 shows that the set of non-trivial, positive equilibria, which we willdenote by E + , is a Rabinowitz-type continuum in Λ × W ,p , + (Ω) , which bifurcates from ( λ min ( g ) , and ‘meets infinity’ at λ max ( g ) . ( b ) It is shown in [9], [10] that if Ω is a ball, then E + is in fact a smooth curve of radiallysymmetric solutions. 3. Time-dependent solutions
In Section 2.5 we discussed equilibrium (time-independent) solutions of equation (1.1). In thissection we will discuss time-dependent solutions of (1.1). We first describe an existence anduniqueness result, and then a comparison result, which will be used to determine the long-timebehaviour of the solutions.3.1.
Existence and uniqueness of positive solutions.
In this section we will discuss the ex-istence, uniqueness and properties of solutions of the time-dependent problem (1.1). To stateprecisely what we mean by a solution of (1.1) we define the spaces Σ( T ) := C ([0 , T ) , L (Ω)) ∩ C ((0 , T ) , W ,p (Ω)) ∩ W , ((0 , T ) , L (Ω)) , T > LOBAL ASYMPTOTIC STABILITY OF POSITIVE SOLUTIONS 7 (we allow T = ∞ here, and likewise for other such numbers below). The space W , ((0 , T ) , L (Ω)) is defined in [15, Example 10.2], using the notation H ((0 , T ) , L (Ω)) ; the space W , ((0 , T ) , L (Ω)) can be defined by a simple adaptation of the definition in [15]. We will search for a solution of(1.1) in Σ( T ) , for some T > . Thus, in this setting, a solution v will be regarded as a time-dependent mapping t → v ( t ) : (0 , T ) → W ,p (Ω) , with ∆ p ( v ( t )) ∈ W − ,p ′ (Ω) defined by (2.1),for each t ∈ (0 , T ) , and satisfying the initial condition at t = 0 as a limit in L (Ω) . More(or less) regularity at t = 0 can be attained, depending on the regularity of v (for example if v ∈ W ,p , + (Ω) then the solution will belong to C ([0 , T ) , W ,p (Ω)) ), but the above setting willsuffice here.In view of this, we will rewrite (1.1) in the form dvdt = ∆ p ( v ) + λg ( v ) φ p ( v ) , v (0) = v ∈ C (Ω) . (3.1)The following theorem describes the existence and uniqueness of solutions of (3.1), and variousadditional properties which will be required below. This theorem does not require g to be positive,nor to satisfy the conditions (1.2), (1.3). It does, however, assume that g is defined on Ω × R rather than on Ω × [0 , ∞ ) (which we have assumed so far, since we are mainly interested in positivesolutions). Once we have established the general existence of solutions, we will then prove theirpositivity, and thereafter the values of g ( x, ξ ) , ξ < , will be irrelevant. If g is only defined on Ω × [0 , ∞ ) ab initio, then we may simply extend it to Ω × R by setting g ( x, − ξ ) = g ( x, , ξ > . Theorem 3.1.
Suppose that g : Ω × R → R satisfies the Lipschitz condition (1.4) on Ω × R , and λ > , v ∈ C (Ω) . Then (3.1) has a unique solution v λg,v ∈ Σ( T λg,v ) , defined on a maximalinterval [0 , T λg,v ) , for some T λg,v > , having the following properties. ( a ) v λg,v (0) = v . ( b ) The function v λg,v : [0 , T λg,v ) → L (Ω) is differentiable at almost all t ∈ [0 , T λg,v ) ,and at such t , d v λg,v dt ( t ) , ∆ p ( v λg,v ( t )) ∈ L (Ω) , and d v λg,v dt ( t ) = ∆ p ( v λg,v ( t )) + λg ( v λg,v ( t )) φ p ( v λg,v ( t )) , in L (Ω) . ( c ) The function E λg ( v λg,v ( · )) : (0 , T λg,v ) → R is absolutely continuous, decreasing and ddt E λg ( v λg,v ( t )) = − (cid:13)(cid:13)(cid:13) ddt v λg,v ( t ) (cid:13)(cid:13)(cid:13) , a.e. t ∈ (0 , T λg,v ) . (3.2) ( d ) The interval [0 , T λg,v ) on which the solution exists is maximal, in the sense that T λg,v < ∞ = ⇒ lim sup t ր T λg,v | v λg,v ( t ) | = ∞ . (3.3) If the set of equilibria of (3.1) is bounded in C (Ω) then (3.3) holds with lim rather than lim sup . BRYAN P. RYNNE
Proof.
Let θ ∈ C ∞ ( R , R ) be a decreasing function with θ ( s ) = ( , s , , s > , and for any integer n > , define ˆ f n : Ω × R → (0 , ∞ ) by ˆ f n ( x, ξ ) := θ ( | ξ | /n ) g ( x, ξ ) φ p ( ξ ) , ( x, ξ ) ∈ Ω × R . Since ˆ f n is bounded and Lipschitz, the results of [1] and [4] show that the problem ˆ v t = ∆ p (ˆ v ) + λ ˆ f n (ˆ v ) , ˆ v (0)= v , (3.4)has a unique solution ˆ v n ∈ Σ( ∞ ) having the properties ( a ) - ( c ) (we discuss this further in Re-mark 3.2 below). Clearly, ˆ v n is a solution of (3.1) on the time interval [0 , T n ) , where T n := sup { T : | ˆ v n ( t ) | n : t ∈ [0 , T ) } , n > , and letting T λg,v := lim n →∞ T n , we see that (3.1) has a unique solution v λg,v ∈ Σ( T λg,v ) , having the properties ( a ) - ( c ) , and T λg,v < ∞ = ⇒ lim t n ր T λg,v | v λg,v ( t n ) | = ∞ , (3.5)for some sequence ( t n ) in (0 , T λg,v ) . That is, (3.3) holds.Now suppose that there exists M > such that | e | < M for any equilibrium solution e of(3.1), and that the overall limit in (3.3) does not exist. Then there exists another sequence ( s n ) in (0 , T λg,v ) such that s n ր T λg,v and the sequence ( | v λg,v ( s n ) | ) is bounded. Combining thiswith (3.5) and the continuity of the mapping t → | v λg,v ( t ) | on (0 , T λg,v ) , we may supposefurther that | v λg,v ( s n ) | = K , n = 1 , , . . . , for some K > M . It now follows from property ( c ) and the definition of E that the sequence ( v λg,v ( s n )) is bounded in W ,p (Ω) so, by the argumentin Section 4.1 below (after taking a subsequence if necessary) there exists v ∞ ∈ W ,p (Ω) suchthat | v λg,v ( s n ) − v ∞ | → and v ∞ is an equilibrium of (3.1) (it is assumed in Section 4.1that < λ < λ max ( g ) , but this assumption is only used to obtain such a bounded sequencein W ,p (Ω) ). However, this implies that | v ∞ | = K > M , which contradicts the choice of M above, so we conclude that the limit in (3.3) must in fact exist, that is, property ( d ) must hold.This completes the proof of Theorem 3.1. (cid:3) Remark 3.2. ( a ) The existence and uniqueness of a solution ˆ v n of (3.4), and the fact that ˆ v n hasproperties ( a ) - ( b ) , as asserted in the proof of Theorem 3.1, follows by combining various standardresults on maximal monotone operators. Specifically, [2, Theorem 3.2], [1, Theorems 3.4, 3.11]and [1, Remark 3.6(5)]. How these results combine to give a solution with the desired propertiesis discussed in detail in [4, Remark 2.2]. It should be noted that, with the sign of f used here,the functions ˆ f n are not monotone but, by assumption (1.4), they satisfy the Lipschitz conditionimposed on the function f in assumption (2 . in [4]. Thus, to apply the discussion in [4] to theproblem (3.4) above, we set (in the notation in [4]) f = 0 and f = ˆ f n . LOBAL ASYMPTOTIC STABILITY OF POSITIVE SOLUTIONS 9
The fact that ˆ v n also has property ( c ) follows from [2, Lemma 3.3], and the argument in theproof of [4, Lemma 3.1]. ( b ) Existence and uniqueness of a local (in time) solution of (3.1), with weaker properties thanthose stated in Theorem 3.1, is proved in [11, Theorem 2.1], and global existence and uniquenessof such solutions of (3.4) (under similar Lipschitz conditions) is proved in [11, Theorem 3.1].Hence, the solution v λg,v given by Theorem 2.3 is unique in a considerably broader solutionspace than Σ p .3.2. Comparison results.
We now consider the auxiliary problem dwdt = ∆ p ( w ) + λγφ p ( w ) , w (0) = w ∈ C (Ω) , (3.6)where γ ∈ L ∞ (Ω) is independent of v , and γ > on Ω . This is a special case of (3.1) (with g ( x, v ) having the form γ ( x ) ) so, by Theorem 3.1, the problem (3.6) has a unique solution w λγ,w defined on a maximal interval [0 , T λγ,w ) . Remark 3.3.
Theorem 3.1 was stated, and proved, for continuous functions g depending on ( x, ξ ) (and Lipschitz with respect to ξ ), but the results quoted from [4] (see Remark 3.2) in theproof of Theorem 3.1 apply equally to the problem (3.6), containing an x -dependent function γ ∈ L ∞ (Ω) .We now describe a ‘comparison’ result for solutions of (3.1) and (3.6). For any T > andfunctions ω , ω ∈ Σ( T ) , we write ω > ω on [0 , T ) if ω ( t ) > ω ( t ) , on Ω , for each t ∈ [0 , T ) .From now on we suppose that g satisfies our basic hypotheses, that is, g is positive and satisfies(1.2) and (1.3). Lemma 3.4. If g ∞ > γ > and v > w > on Ω , then T λg,v T λγ,w and v λg,v > w λγ,w on [0 , T λg,v ) .Proof. The proof follows, with minor modifications, the proof of [14, Theorem 2.5], usingour assumptions on g (in particular, the assumption g ∞ > γ implies that g ( v ) > γ for any v ∈ W ,p , + (Ω) ). We omit the details. However, we note that [14, Theorem 2.5] considers equationsof the form v t = ∆ p ( v ) + λφ p ( v ) , but the proof can be adapted to give the above result; theargument in [14] is based on the proof of [6, Lemma 3.1, Ch. VI], which considered the equation v t = ∆ p ( v ) . (cid:3) If γ = 0 and w = 0 , then clearly w , ≡ , and since g ∞ > , Lemma 3.4 now yields thefollowing positivity result for the solution v λg,v of (3.1) found in Theorem 3.1. Corollary 3.5. If v ∈ C (Ω) then v λg,v ( t ) ∈ W ,p , + (Ω) for all t ∈ (0 , T λg,v ) . In the next section we will use the comparison result Lemma 3.4 to describe the behaviour ofsolutions of (3.1). The following criterion for finite time blow-up of solutions of (3.6) will beuseful.
Lemma 3.6. If λ > µ ( γ ) and = w ∈ C (Ω) , then T λγ,w < ∞ . Proof.
This can be proved by following, almost verbatim, the proof of [14, Theorem 3.5], whichdeals with the case γ ≡ . (cid:3) Global stability and instability of the equilibria
For any λ > the time-dependent problem (3.1) has the trivial equilibrium solution u = 0 ,and also, by Theorem 2.3, for each λ ∈ ( λ min ( g ) , λ max ( g )) there is a unique, non-trivial, positiveequilibrium e λ ∈ W ,p , + (Ω) . We will now consider the global stability, and instability, of theseequilibria. Theorem 4.1.
Suppose that = v ∈ C (Ω) . ( a ) If < λ λ min ( g ) then T λg,v = ∞ and lim t →∞ k v λg,v ( t ) k ,p = 0 . ( b ) If λ min ( g ) < λ < λ max ( g ) then T λg,v = ∞ and lim t →∞ k v λg,v ( t ) − e λ k ,p = 0 . ( c ) If λ max ( g ) < λ then T λg,v < ∞ , that is, the solution v λg,v blows up in finite time. Proof of Theorem 4.1 ( a ) , ( b ) . Suppose that < λ < λ max ( g ) = µ ( g ∞ ) . Let v = v λg,v ( T λg,v / ∈ W ,p , + (Ω) . Then E λg ( v ) is defined and, by Lemma 2.2 and Theorem 3.1 ( d ) - ( e ) , E λg ( v λg,v ( t )) E λg ( v ) = ⇒ k v λg,v ( t ) k ,p M λ ( E λg ( v )) , T λg,v t < T λg,v = ⇒ T λg,v = ∞ and E λg ( v λg,v ( · )) is bounded on (0 , ∞ )= ⇒ lim t →∞ E λg ( v λg,v ( t )) exists. (4.1)From now on, ( t n ) will denote an increasing sequence in (0 , ∞ ) such that t n → ∞ ; we willchoose various such sequences below. By (4.1), the sequence ( v λg,v ( t n )) is bounded in W ,p (Ω) ,so we may also suppose (after taking a subsequence if necessary) that v λg,v ( t n ) ⇀ v ∞ in W ,p (Ω) , | v λg,v ( t n ) − v ∞ | → , (4.2)for some v ∞ ∈ W ,p , + (Ω) (where ⇀ denotes weak convergence in W ,p (Ω) ). The argument in theproof of [4, Lemma 3.1] now shows that v ∞ is an equilibrium solution of (3.1), that is, v ∞ is asolution of (2.11). Hence, by Theorem 2.3, we have the following cases. ( a ) If < λ λ min ( g ) then v ∞ = 0 . ( b ) If λ min ( g ) < λ < λ max ( g ) then either v ∞ = 0 or v ∞ = e λ .In case ( a ) , a simple contradiction argument (using the preceding results) now shows that wemust have lim t →∞ | v λg,v ( t ) − σ λ | = 0 . In case ( b ) , suppose that there exists sequences ( t n ) , ( t n ) , such that | v λg,v ( t n ) | → , | v λg,v ( t n ) − e λ | → . Then, by continuity of the mapping t → | v λg,v ( t ) | , there exists a sequence ( t n ) such that | v λg,v ( t n ) | = | e λ | / , n > . But this contradicts the preceding results, so we must have lim t →∞ | v λg,v ( t ) − v ∞ | = 0 , foreither v ∞ = 0 or v ∞ = e λ . The following lemma shows that, in fact, the latter holds – the proofwill use some results from Section 4.2, so will be given in Section 4.3 below. Lemma 4.2.
For λ min ( g ) < λ < λ max ( g ) and = v ∈ C (Ω) , lim t →∞ | v λg,v ( t ) − e λ | = 0 . LOBAL ASYMPTOTIC STABILITY OF POSITIVE SOLUTIONS 11
Now, to simplify the notation, and to combine the two cases ( a ) and ( b ) , we define σ λ ∈ W ,p , + (Ω) by σ λ := ( , if < λ λ min ( g ) , e λ , if λ min ( g ) < λ < λ max ( g ) , and the preceding results show that lim t →∞ | v λg,v ( t ) − σ λ | = 0 , < λ < λ max ( g ) . (4.3)Thus, it only remains to prove the convergence with respect to the W ,p (Ω) norm.By integrating (3.2) with respect to t and using the existence of the limit lim t →∞ E λg ( v λg,v ( t )) (by (4.1)), we see that the function on the right hand side of (3.2) lies in L (0 , ∞ ) , so we maychoose a sequence ( t n ) such that k ∆ p ( v λg,v ( t n )) + λg ( v λg,v ( t n )) φ p ( v λg,v ( t n )) k → . (4.4)We may also suppose that (4.2) holds, with v ∞ = σ λ . Hence, by (2.1), (2.12) and (4.4), Z Ω (cid:0) ∆ p ( v λg,v ( t n )) + λg ( v λg,v ( t n )) φ p ( v λg,v ( t n )) (cid:1) v λg,v ( t n ) → , (by (4.1) and (4.4)) = ⇒ Z Ω |∇ v λg,v ( t n ) | p → λ Z Ω g ( σ λ ) σ pλ = Z Ω |∇ σ λ | p , (by (2.1) and (2.12))and combining this with (4.2) yields v λg,v ( t n ) ⇀ σ λ , in W ,p (Ω) , k v λg,v ( t n ) k ,p → k σ λ k ,p . (4.5)Hence, by the uniform convexity of W ,p (Ω) and [3, Proposition 3.32], k v λg,v ( t n ) − σ λ k ,p → , which implies that E λg ( v λg,v ( t n )) → E λg ( σ λ ) , and so lim t →∞ E λg ( v λg,v ( t )) = E λg ( σ λ ) (4.6)(since this limit exists, by (4.1)).Now suppose that there exists a sequence ( t n ) and ǫ > such that k v λg,v ( t n ) − σ λ k ,p > ǫ ,and also that (4.2) holds. Combining this with (4.6) (and the form of E λg ) shows that (4.5)again holds, and so (by uniform convexity) k v λg,v ( t n ) − σ λ k ,p → , which contradicts this choiceof sequence ( t n ) . Hence, we must have k v λg,v ( t ) − σ λ k ,p → , which completes the proof ofparts ( a ) and ( b ) of Theorem 4.1.4.2. Proof of Theorem 4.1 ( c ) . By hypothesis, λ > λ max ( g ) = µ ( g ∞ ) , so by Lemmas 3.4and 3.6 (with γ = g ∞ and w = v ), T λg,v T λg ∞ ,v < ∞ , which proves part ( c ) of Theo-rem 4.1. (cid:3) Proof of Lemma 4.2.
By the arguments preceding Lemma 4.2 in Section 4.1, it sufficesto show that if we suppose that lim t →∞ | v λg,v ( t ) | = 0 , (4.7) then we can obtain a contradiction.For any δ > , define g δ ∈ C (Ω) by g δ := g ( x, δ ) , x ∈ Ω . By the properties of g , and the principal eigenvalue function µ ( · ) (see Lemma 2.1 and [5]), wehave g δ g and lim δ ց | g δ − g | = 0 = ⇒ µ ( g δ ) > µ ( g ) and lim δ ց µ ( g δ ) = µ ( g ) (the final limiting result is not explicitly stated in [5], but it can readily be proved using theminimisation characterisation of µ ( ρ ) in (1 . of [5]; the argument is similar to the proof of [5,Proposition 4.3]). Hence, since λ > λ min ( g ) = µ ( g ) , we may choose δ sufficiently small that λ > µ ( g δ ) . Also, by (4.7), we may choose t δ > such that | v λg,v ( t ) | δ/ , t > t δ . (4.8)Now, by regarding t δ as the initial time, and v δ := v λg,v ( t δ ) as the initial value, we can follow theargument in Section 4.2 to show that T λg,v δ T λg δ ,v δ < ∞ , that is, v λg,v δ blows up in finite time(the inequality λ > µ ( g δ ) provides the analogue here of the inequality λ > λ max ( g ) = µ ( g ∞ ) used in Section 4.2). This clearly contradicts (4.8), and so proves Lemma 4.2. (cid:3) References [1]
G. Akagi, M. Otani , Evolution inclusions governed by subdifferentials in reflexive Banach spaces,
J. Evolution Equations (2004), 519–541.[2] H. Brezis,
Op´erateurs Maximaux Monotones et Semi-groupes de Contraction dans les Espace deHilbert,
North-Holland, Vol. 5, North-Holland, Amsterdam (1973).[3]
H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations,
Springer, NewYork (2011).[4]
R. Chill, A. Fiorenza , Convergence and decay rate to equilibrium of bounded solutions ofquasilinear parabolic equations,
J. Differential Equations (2006), 611–632.[5]
M. Cuesta,
Eigenvalue problems for the p -Laplacian with indefinite weights, Electron. J. Differ-ential Equations
No. 33.[6]
E. DiBenedetto,
Degenerate Parabolic Equations,
Springer, New York (1993).[7]
M. A. del Pino, R. F. Man´asevich,
Global bifurcation from the eigenvalues of the p -Laplacian, J. Differential Equations (1991), 226–251.[8] J. I. D´ıaz, J. E. Sa´a,
Existence et unicit´e de solutions positives pour certaines ´equations ellip-tiques quasilin´eaires,
C. R. Acad. Sci. Paris S´er. I Math. (1987), 521–524.[9]
J. Garcia-Melian, J. Sabina de Lis,
A local bifurcation theorem for degenerate elliptic equa-tions with radial symmetry,
J. Differential Equations (2002), 27–43.[10]
F. Genoud,
Bifurcation along curves for the p -Laplacian with radial symmetry, Electron. J. Dif-ferential Equations (2012).[11]
Z. Jun Ning , Existence and nonexistence of solutions for u t = div ( |∇ u | p − ∇ u ) + f ( ∇ u, u, x, t ) , J. Math. Anal. Appl. (1993), 130–146.[12]
J. Karatson, P. L. Simon,
On the stability properties of nonnegative solutions of semilinearproblems with convex or concave nonlinearity,
J. Comput. Appl. Math. (2001), 497–501.[13]
J. Karatson, P. L. Simon,
On the linearised stability of positive solutions of quasilinear problemswith p -convex or p -concave nonlinearity, Nonlinear Analysis (2001), 4513–4520. LOBAL ASYMPTOTIC STABILITY OF POSITIVE SOLUTIONS 13 [14]
Y. Li, C. Xie,
Blow-up for p -Laplacian parabolic equations, Electron. J. Differential Equations (2003).[15] M. Renardy, R. C. Rogers , An Introduction to Partial Differential Equations , Springer, 1993.
Department of Mathematics and the Maxwell Institute for Mathematical Sci-ences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland.
E-mail address ::