On global existence and stationary solutions for two classes of semilinear parabolic problems

On global existence and stationary solutions for two classes of semilinear parabolic problems

Pavol Quittner

Abstract. We investigate stationary solutions and asymptotic behaviour of solutions of two boundary value problems for semilinear parabolic equations. These equations involve both blow up and damping terms and they were studied by several authors. Our main goal is to fill some gaps in these studies.

Keywords: global existence, blow up, semilinear parabolic equation, stationary solution Classification: 35K60, 35J65, 35B40

1. Introduction.

Consider the following two problems

(NBC)











ut=△u−au^p in (0,∞)×Ω,

∂u

∂n =u^q on (0,∞)×∂Ω, u(0, x) =u0(x) x∈Ω,

(DGT)







u_t=△u− |∇u|^q+λu^p in (0,∞)×Ω,

u= 0 on (0,∞)×∂Ω,

u(0, x) =u0(x) x∈Ω,

where Ω is a smoothly bounded domain inR^N,p, q >1,a, λ >0 andu₀∈W^1,∞(Ω) is a non-negative function. These problems were studied by many authors (see e.g. [CFQ], [E], [FQ1], [LGMW] in the case of (NBC) and [AW], [C], [CW1], [CW2], [F], [KP], [Q1] in the case of (DGT)). In both problems there is a blow-up term (u^q andλu^p) and a damping term (−au^pand−|∇u|^q). These terms cause that the corresponding solutions admit an interesting asymptotic behaviour which strongly depends on the parameters p, q, a, λ. The main purpose of this paper is to fill some gaps in the studies of these problems; i.e. to investigate the behaviour of the

This research was done while the author was visiting CEREMADE at the University Paris Dauphine which was possible due to a fellowship from the French Government. The author thanks Profs. J.P. Aubin and H. Frankowska for their hospitality and Prof. M. Struwe for the discussions concerning Theorem 4.2

solutions for those parameters p, q, a, λ or N for which the results in the above mentioned papers are not satisfactory.

In the case of (NBC) (the problem withNonlinear BoundaryConditions), the study was almost completely done in [CFQ] forN = 1. Particularly, it was shown that the exponentp= 2q−1 is critical for the blow up in the following sense:

(i) ifp <2q−1 (orp= 2q−1 anda < q) then there exist solutions of (NBC) which blow up in finite time,

(ii) if p >2q−1 (or p= 2q−1 and a > q) then all solutions of (NBC) exist globally and are globally bounded,

(iii) if p= 2q−1 and a= q then all nontrivial solutions of (NBC) exist glob- ally but they are unbounded; they tend pointwise to a singular stationary solution.

The assertions (i) and (ii) were shown also for Ω being a ball in R^N, N > 1.

However, if Ω is a general bounded domain inR^N,N >1, then [CFQ] or [E] imply blow up of suitable solutions of (NBC) only forp≤qand the global existence and boundedness is shown in [CFQ] only forp > c(q),q < ^N+1_N−1, where

c(q) := N−q(N−2)

N+ 1−q(N−1)(q+ 1)−1>2q−1.

The main result of this paper concerning the global existence for (NBC) in the case of a general domain Ω is the following:

(a) if p > 2q−1 then all solutions of (NBC) exist globally and are globally bounded,

(b) if p < 2q−1 (or p = 2q−1 and a is sufficiently small) then there exist initial functionsu₀ such that the corresponding solutions of (NBC) blow up in L^∞(Ω)-norm.

It has to be mentioned that in the case (b) we do not know whether the blow up occurs in finite or infinite time. We find only a subsolutionu⁺such that any positive stationary solution has to intersectu⁺ so that the solution of (NBC) starting at u⁺ cannot be bounded. In the case (a) we show that a simple substitution leads to the casep > c(q) which was already solved in [CFQ]. Hence, forp >2q−1 we obtain global existence, boundedness and also the existence of a positive stationary solution of (NBC).

Considering the (positive) stationary solutions of (NBC) we are mainly interested in the caseq < p < 2q−1, N >1. The results of [CFQ] imply that in this case there existsa₀≥0 such that the stationary problem corresponding to (NBC) has

(j) no positive solutions fora < a₀,

(jj) at least one positive solution fora > a₀,

(jjj) in the subcritical case (q < _N−2^N ) at least two positive solutions for a ∈ {a₁, a₂,· · · }, where a_k→ ∞.

Moreover, if Ω is a ball thena0>0 and (NBC) has at least one positive stationary symmetric solution fora=a₀ and at least two positive stationary symmetric solu- tions fora > a₀ (see [CFQ] for a more precise information forN = 1). The main

difficulty in proving this additional property for a general domain is the absence of apriori estimates for stationary solutions. In this paper we show that for a general domain Ω

(α) a₀>0,

(β) in the subcritical case, (NBC) has at least two positive stationary solutions for almost alla > a₀.

Moreover, for Ω being a ball in R² we find apriori estimates for all positive sta- tionary solutions (note that the apriori estimates in [CFQ] forN >1 concern only symmetric solutions).

The proof of (α) is based on the apriori estimate of min{u(x) ;x∈∂Ω}, whereu is any positive stationary solution. The proof of (β) is based on a trick of M. Struwe [S1].

Concerning the problem (DGT) (the problem withDampingGradientTerm), it is known that forp > q >1 blow-up of solutions inL^∞-norm in finite time can occur (see [CW1], [KP], [F], [Q1]) while forp≤qany solutionuis bounded in [0, T)×Ω, whereT is the maximal existence time foru(see [F]). In this paper we show that if the existence timeTof a solutionuof (DGT) is finite then lim_t→T−ku(t,·)kL^∞(Ω)= +∞. Consequently, the solution exists globally ifp≤q.

Our main results concerning the stationary solutions of (DGT) are the following:

(k) if q ≥p then there exists λ₀ >0 such that the stationary problem corre- sponding to (DGT) has

(k1) no positive solutions forλ < λ0,

(k2) at least one positive solution forλ=λ₀, (k3) at least two positive solutions forλ > λ₀,

(kk) ifp <(N+ 2)/(N−2) (in the caseN >2) andq <min{2,(N+ 2)/N}then there exists λ₀ ≥ 0 such that (DGT) has at least one positive stationary solution for anyλ > λ₀ (see Theorem 6.2 and Remark 6.2).

The stationary problem for (DGT) was studied also in [AW], [C], [CW1], [CW2], [FQ2], [SZ]. However, these studies concern mostly the case where Ω is a ball inR^N or Ω =R^N when one can make use of the symmetry of the solution and apply time map technique (shooting method). Let us also emphasize that in the case (k) we do not need any subcriticality condition for por q since we work with dynamical methods in this case. The proof of (kk) is based on the use of Leray-Schauder degree and the apriori estimates from [BT] and [FLN]. Finally, let us mention that in the case (k), i.e. q≥p,

(kα) ifp=qthen λ₀ ≥pdiam (Ω)^−p (see [C, Theorem 4]),

(kβ) ifq= 2 and N= 1 then there are exactly two positive solutions forλ > λ₀ (see [S, Example 3.2.2])

and in the casep <(N+ 2)/(N−2) and Ω being a ball in R^N

(kkα) ifq <2p/(p+ 1) ( ⇒ q <min(2,(N+ 2)/N)) then there exists a positive solution for anyλ >0. Moreover, this solution is unique if N = 1 (see [C, Theorem 3]).

(kkβ) If q = 2p/(p+ 1) and λ ≤ (2p)^p/(p+ 1)^2p+1 then (DGT) does not have

positive stationary solutions. The estimate on λ is precise if and only if N = 1 (see [C, Theorem 3], [FQ2]).

(kkγ) If q > 2p/(p+ 1) and λ >0 is small then (DGT) does not have positive stationary solutions (see the proof of Theorem 3 (iii) in [C]).

2. Global existence for (NBC).

In this section we show that the assumptionp >2q−1 implies the global existence and boundedness of solutions of (NBC). Our results also imply the existence of a positive stationary solution for (NBC), since the zero solution is unstable.

Due to the results of [CFQ], it is sufficient to consider the caseN >1. As shown in [CFQ], (NBC) generates a local semiflow in{u∈W^1,r(Ω) ;u≥0}for anyr > N. Hence, we shall supposeu₀∈W^1,r(Ω) for somer > N,u₀≥0.

In [CFQ], the estimates from [FK] were used to get the global existence and boundedness results for (NBC) under the assumption

(2.1) q < N+ 1

N−1, p > c(q) := N−q(N−2)

N+ 1−q(N−1)(q+ 1)−1.

We use this information and a simple substitution to get the desired result.

Theorem 2.1. Ifp > 2q−1 >1then any solution of (NBC)exists globally and stays uniformly bounded.

Proof: Letube a maximal solution of (NBC),m≥1. Then v :=u^m solves the problem

(2.2)















v_t=△v−m−1 m

1

v|∇v|²−mav^p∗ in (0, T)×Ω,

∂v

∂n=mv^q∗ on (0, T)×∂Ω,

v(0, x) =u^m₀(x) x∈Ω,

where p^∗ = (p+m−1)/m, q^∗ = (q+m−1)/mand T is the maximal existence time foru. Using the comparison principle one simply getsv ≤w, wherewsolves the problem

(2.3)











w_t=△w−maw^p∗ in (0, T)×Ω,

∂w

∂n =mw^q∗ on (0, T)×∂Ω, w(0, x) =u^m₀ (x) x∈Ω.

Now it is sufficient to verify that the couple (p^∗, q^∗) fulfils the condition (2.1) iff 2m >(q−1)(N−1) and 2m(p+ 1−2q)>(q−1) (p−1)(N−1)−(q−1)(N−2)

, which is clearly true if m is sufficiently large. Hence, for m large one can apply the results of [CFQ] for the solutionwto get its global existence and boundedness and, consequently, also the global existence and boundedness for v and u (it is obvious that the linear factormin (2.3) does not play any significant role in [CFQ,

Theorem 4.6]).

Corollary 2.1. Ifp >2q−1then(NBC) has a positive stationary solution.

Proof: Put u₀ ≡ε, whereε >0 is small enough. Since (NBC) possesses a Lya- punov function Φ (see [CFQ]) and Φ(u₀)<0 = Φ(0), theω-limit set of the solution starting atu₀ consists of (nonnegative) equilibria which are different from 0. Due to the maximum principle, these equilibria are positive.

3. Blow up for (NBC).

In this section we shall suppose thatp≤2q−1 (andais small enough ifp= 2q−1) and we shall show that there exists a solution of (NBC) which blows up (in finite or infinite time). As a by-product of our considerations we obtain also an apriori bound for min

x∈∂Ωu(x), whereuis any positive stationary solution of (NBC).

Lemma 3.1. Let α > 2 be fixed and u_δ(x) := ₁

ε δ−dist(x, ∂Ω)+α

, where δ >0,αε^α(q−1)=δ^α(q−1)+1 andv⁺:= max(v,0). Ifδ is sufficiently small thenu_δ is a subsolution for(NBC) and any positive stationary solutionuof (NBC) fulfils min∂Ω(u−u_δ)<0.

Proof: One can easily verify that u_δ fulfils the boundary condition in (NBC) for any δ > 0. Further suppose that dist (x, ∂Ω) ≤ δ and δ is sufficiently small.

Denotingd(x) := dist (x, ∂Ω) and ϕ(d) :=₁

ε(δ−d)⁺α one hasu_δ(x) =ϕ d(x) and

△u_δ= (ϕ^′′◦d)|∇d|²+ (ϕ^′◦d)△d.

Let y =y(x) ∈∂Ω be the closest point to x in ∂Ω and let n=n(x) be the unit (outward) normal to∂Ω aty(x). Then we have

|∇d|²=∂d

∂n 2

= 1, ∂²d

∂n² = 0, |△d| ≤C,

where C is some constant depending only on the curvature of ∂Ω (cf. [GT, Lem- mas 14.16 and 14.17]). Using these estimates and the inequality p≤2q−1 (and a≪1 ifp= 2q−1) one can easily check that

△u_δ≥ 1

2ϕ^′′◦d≥aϕ^p◦d=au^p_δ

for δ sufficiently small, where the inequalities are strict if d(x) < δ. Henceu_δ is a (strict) subsolution forδ≤δ₀ andu_δ

∂Ω = ^δ_εα

→+∞asδ→0+.

Now suppose that u is a positive stationary solution, u ≥ u_δ0 on ∂Ω. Put Ω⁻ :={x∈Ω ; u(x)< u_δ0(x)}. If Ω⁻ 6=∅ then the functionw :=u−u_δ0 fulfils w= 0 on∂Ω⁻ and△w=△u− △u_δ0 ≤au^p−au^p_δ

0 <0 in Ω⁻, i.e. w >0 in Ω⁻ which is a contradiction. Hence Ω⁻=∅andu≥u_δ0 in Ω.

Choose δ≤δ₀ such that u≥ u_δ in Ω andu(x₀) = u_δ(x₀) at some x₀ ∈ Ω. This choice leads to a contradiction with the maximum principle:

ifx₀∈Ω thenu_δ(x₀)6= 0 and△u(x₀)≥ △u_δ(x₀)> au^p_δ(x₀) =au^p(x₀);

ifx0∈∂Ω then△u_δ(x)≥au^p_δ(x) +η > au^p(x) =△u(x) for someη >0 and all x∈Ω close tox₀which gives a contradiction with^∂(u_∂n^δ−u)(x₀) = 0, (u_δ−u)(x₀) = 0,

u_δ≤u.

Corollary 3.1. The solution of(NBC)starting atu_δblows up.

Proof: Letu be the solution starting atu_δ. Then u_t ≥0 due to the maximum principle. Ifuis bounded, thenu(t,·) has to converge to a stationary solutionw≥ u_δ since the orbit{u(t,·) ;t≥0} is relatively compact in the appropriate Sobolev space (see [CFQ]). However, this gives us a contradiction with min

∂Ω(w−u_δ)<0.

4. Stationary solutions for (NBC).

Supposeq < p <2q−1,N >1 and put

a₀:= inf{a >0 ; there exists a positive stationary solution of (NBC)}. It follows from [CFQ] that a0 < ∞. First we prove the assertion (α) from the introduction.

Theorem 4.1. If a > 0 is small enough then (NBC) does not have positive sta- tionary solutions.

Proof: By contradiction. Suppose that fora_m↓0 there exist positive stationary solutionsu_m. By Lemma 3.1 we have min_∂Ωu_m =u_m(x_m)≤K for some x_m ∈

∂Ω and a positive constant K. Let Ω_m be the component of the set {x ∈ Ω ; u_m(x) < 2u_m(x_m)} containing x_m in its closure. Let v_m be the solution of the problem△vm= 0 in Ωm,vm =um on∂Ωm. Then ^∂v_∂n^m(xm)≤0 since vm attains its minimum at xm (and ∂Ωm∩Um =∂Ω∩Um for some neighbourhoodUm of xm). On the other hand, putting wm := um −vm we have wm = 0 on ∂Ωm, 0 ≤ △wm =amu^p_m ≤am2^pu^p_m(xm) in Ωm ⊂ Ω. The standard regularity theory implies now

(4.1) u^q_m(x_m) =∂u_m

∂n (x_m)≤∂w_m

∂n (x_m)≤Ca_m2^pu^p_m(x_m)≤Ca_m2^pK^p for suitable C > 0, hence um(xm) → 0 as m → ∞. Using (4.1) again, we get 1≤Cam2^pu^p−qm (xm)→0, which is a contradiction.

Now supposeq < p <2q−1,q < _N−2^N ifN >2, and a > a₀. Then it follows from [CFQ] that there exists a positive stationary solution u of (NBC) which is a local minimizer of the corresponding functional

Φ(u) = Φ_a(u) = 1 2

Z

Ω|∇u|²dx+ a p+ 1

Z

Ω|u|^p+1dx− 1 q+ 1

Z

∂Ω|u|^q+1dS in the Sobolev spaceW^1,2(Ω).

Puttingw_ε(x) :=ε^−q+δq−1 ε−dist (x, ∂Ω)+

, where 0< δ < ^2q−1−p_p−q , one can straight- forwardly check that Φ(w_ε+u)→ −∞asε→0+. Hence to obtain a second critical point of Φ (lying aboveu) one can use the mountain pass theorem for Φ with respect to the convex set{w∈W^1,2(Ω) ;w≥u}similarly as in the proof of Theorem 2.1 (i) in [CFQ]. The difficulty consists in verifying the corresponding Palais-Smale condi- tion (cf. also Remark 2.4 in [CFQ]). Using a trick of M. Struwe we are able to do this only for almost alla≥a₀.

Theorem 4.2. Letq < p <2q−1, q < _N−2^N . Then for a.a. a≥a₀, the problem (NBC)has at least two positive stationary solutions.

Proof: Fixa₂> a₁> a₀and letua1 be a positive solution corresponding toa₁. As shown in [CFQ], choosingu_a2 a global minimizer of Φ_a2 inK₁ :={u∈W^1,2(Ω) ; 0 ≤ u ≤ ua1} we get a stationary solution of (NBC) with 0 < ua2 < ua1 in Ω, Φa2(ua2)<Φa2(0) = 0. PutS :={u∈K1; Φa2(u) = Φa2(ua2)}. Then Φ^′_a2(u) = 0 for anyu∈S and the setS is compact since Φ^′_a2 has the form identity +F, where F mapsK₁ into a compact set. Moreover,ν₀:= ¹₂dist (S,{u;u≥ua1})>0. Next we show by contradiction that there existsν >0 (ν≤ν₀) such that

δ:=δ(ν) := inf{Φa2(u) ; dist(u, S) =ν} −Φa2(ua2)>0.

Hence assume thatδ(νn)≤0 for some νn ↓0. Let nbe fixed andν :=νn. Then there existum such that dist (um, S) = ν and lim sup_m→∞Φa2(um)≤Φa2(ua2).

Consequently, um = u^S_m+vm, where u^S_m ∈ S and kvmk = ν. We may suppose u^S_m→u^S ∈S andv_m⇀ v,kvk ≤ν.

Ifv_m→v then dist (u^S+v, S) =ν, Φ_a2(u^S+v)≤Φ_a2(u_a2).

Ifv_m 6→v then Φ_a2(u^S+v)<lim sup_m→∞Φ_a2(u^S_m+v_m)≤Φ_a2(u_a2) so that u^S+v /∈S, 0<dist (u^S+v, S)≤ kvk ≤ν.

Let w^S be a local minimizer of Φa2 in {u;ku−u^Sk ≤ dist (u^S+v, S)} such thatw^S ∈/ S. By the definition ofS we havew^S ∈/K₁. By the same way as in the end of the proof of [CFQ, Lemma 2.4] one getsw^S ∈C¹(Ω), kw^S−u^Sk_C¹_(Ω)→0 forν=ν_n→0. Since dist_C1(Ω)(S, C¹(Ω)\K₁)>0 by the maximum principle and w^S∈/S, we get a contradiction.

Now choose ν and δ= δ(ν) with the properties above and fix ε > 0 such that Φa2(ua1+wε)<Φa2(ua2). Further fixα∈(0, a2−a1) such that_p+1^α R

Ωu^p+1dx≤ ^δ₃ for any u ∈ {v; dist (v, S) ≤ ν} ∪ {ua1 +wε} and let ua2+α be a fixed positive stationary solution fora=a2+αlying belowua2. Put

K₂:={u∈W^1,2(Ω) ;u≥ua2+α}

P :={p˜∈C([0,1], K₂) ; ˜p(0) =ua2,p(1) =˜ ua1+wε} γ_a:= inf

˜

p∈P sup

u∈˜p([0,1])

Φ_a(u) for |a−a₂|< α.

Then, obviously,γ: (a₂−α, a₂+α)→R is a nondecreasing function so thatγ is differentiable almost everywhere. Choosea∈(a₂−α, a₂+α) such that there exists γ^′_a. We shall show that there exists a positive stationary solutionuof (NBC) with Φa(u) =γa. Since any global minimizeruaof Φa inK1 fulfils

Φ_a(u_a) ≤Φ_a(u_a2)≤Φ_a2(u_a2) +δ

3 = inf{Φ_a2(u) ; dist (u, S) =ν} −2δ 3

<inf{Φa(u) ; dist (u, S) =ν} ≤γa,

we find two positive solutions for (NBC) and we are done.

We shall proceed similarly as in [S1, Lemma 6.3]. Leta_m∈(a₂−α, a),a_m↑a, and letp_m∈Pbe such that sup_u∈pmΦ_a(u)≤γ_a+(a−a_m) (wherep_m=p_m([0,1])).

The definition ofγamimplies now thatSm:={u∈pm; Φam(u)≥γam−(a−am)} 6=

∅. Since Φa(u)≥Φam(u) we get also that

W_m0 :={u∈K₂;γ_am−(a−a_m)≤Φ_am(u)≤Φ_a(u)≤γ_a+ (a−a_m) for somem≥m₀}

is nonempty,W_m+1⊂W_m. It is easy to see that foru∈W_m0 we have 1

p+ 1 Z

Ω

u^p+1dx≤ γa−γam

a−a_m + 2 for suitablem≥m₀, so thatW_m0 is bounded inL^p+1(Ω).

Foru∈K₂, put g(u) := sup

v∈K2

ku−vk≤1

hΦ^′_a(u), u−vi, g_m(u) := sup

v∈K2

ku−vk≤1

hΦ^′_am(u), u−vi.

Let K(u) := u−Φ^′_a(u) and let P₂ be the orthogonal projection in W^1,2(Ω) onto K₂. ThenK is a compact map and

hu−K(u), u−P₂K(u)i ≤g(u) max(1,ku−P₂K(u)k).

Using the characterization of the projectionP₂ we get

hK(u)−P₂K(u), u−P₂K(u)i ≤0 for anyu∈K₂ and adding the last two inequalities we obtain

(4.2) ku−P2K(u)k ≤max(g(u),p

g(u)) for anyu∈K2.

Suppose that there existum ∈Wm such thatg(um)→0. Choosingv =um+

um

kumk in the definition ofg(um) we get−hΦ^′_a(um), umi ≤g(um)kumk. Adding this inequality to the inequality (q+ 1)Φ_a(u_m)≤C and using the boundedness ofW_m inL^p+1(Ω) we get

Z

Ω|∇u_m|²dx≤ 1

q−1g(u_m)ku_mk+ ˜C,

which gives the boundedness of {um} in W^1,2(Ω). Hence we may suppose that (a subsequence of){um} converges weakly to someu∈K2. Now the compactness of K and (4.2) give us u_m →u = P₂K(u), Φ_a(u) = γ_a. Since u_a2+α is a strict subsolution for (NBC) we get Φ^′_a(u) = 0 (cf. the proof of Lemma 2.4 in [CFQ]).

Now assume that the sequence{u_m} above does not exist, i.e. g(u) ≥ 4κ for someκ > 0 and anyu∈ W_m1. We may suppose that u_a2, u_a1+w_ε ∈/ W_m1 and that g(u)≥3κ,g(u)−g(u_m)≤κfor some neighbourhood ˜W of W_m1 in K₂ such thatua2, ua1 +wε∈/W˜ and ˜W is bounded in L^p+1(Ω). By [S2, Lemma 1.6], there exists a Lipschitz continuous vector field ˜e: ˜W →W^1,2(Ω) such that

˜

e(u) +u ∈K₂, ke(u)˜ k <1,

hΦ^′_a(u),˜e(u)i <−minng(u)² C ,1o

for anyu∈W˜, whereC >0 is a fixed constant. Consequently, ifm is sufficiently large thenhΦ^′_am(u),˜e(u)i<−β for someβ >0 and anyu∈W˜.

Now letη :W^1,2(Ω)→[0,1] be a Lipschitz function such thatη = 1 onW_m1 and η= 0 outside ˜W. Extend ˜etoK2 by lettinge(u) :=η(u)˜e(u) foru∈W˜,e(u) := 0 foru /∈W˜. The functioneis Lipschitz and

hΦ^′_am(u), e(u)i







<−β foru∈Wm1,

≤0 foru∈K₂,

= 0 foru /∈W .˜

Letψ: [0,∞)×K₂ →K₂be the solution of the initial value problem







∂

∂tψ(t, u) =e ψ(t, u) , ψ(0, u) =u.

Letp^t_m:=ψ(t, pm),q_m^t :={u∈p^t_m; Φam(u)≥γam−(a−am)}. Since_dt^dΦam ψ(t, u)

≤0 for anyuand _dt^dΦ_am ψ(t, u)

t=0≤ −β foru∈q_m^t , we get inf_u∈p^t

mΦ_am(u)<

γam fort large enough which gives us a contradiction with the definition ofγam. In the rest of this section suppose thatN = 2,q < p <2q−1.

Lemma 4.1. Letunbe positive stationary solutions of(NBC)witha=an≤A <

∞such thatUn:= max

Ω un→+∞asn→ ∞. PutVn:= max

Ω |∇un|and letε >0.

Then

n→∞lim Un^q

Vn = lim

n→∞

Vn

Un^q+ε

= 0.

Proof: Ifuis a positive stationary solution of (NBC) thenw:=|∇u|² fulfils

△w= 2pu^p−1w+ 2X

i,j

∂²u

∂x_i∂x_j 2

>0 in Ω,

hencewattains its maximum on the boundary∂Ω. Consequently,V_n=|∇u_n(˜x_n)| and U_n=u_n(x_n) for some x_n,x˜_n∈∂Ω. Putα:= q−1

q−1 +ε/2 and choose a unit vectorν_nsuch thatν_nis not tangential to∂Ω at ˜x_nand

∂un

∂νn(˜x_n) ≥ 1

2V_n. We may suppose that ˜x_n+tν_n∈Ω fort >0 small (t <4U_n^1−q). The estimate 0< u_n≤U_n implies that there existtn∈

0,4Un

V_n

i such that

∂u

∂ν_n(˜xn+tnνn) ≤ 1

4Vn so that theC^1,α(Ω)-norm ofu_n can be estimated below by

(4.3) ku_nkC^1,α ≥ 1

4^α+1V_n^1+αU_n^−α. On the other hand, theL^r-estimates (withr > _1−α^N ) imply

(4.4) ku_nkC^1,α ≤C₁ku_nkW^2,r≤C₂ ka_nu^p_nkL^r+ku^q_nkW^1,r

≤C₃(U_n^p+U_n^q−1V_n)≤C₄U_n^q−1V_n,

sinceVn≥Un^qand p <2q−1. Using (4.3) and (4.4) we getVn≤C Un^{(q−1+α)/α}= C U_n^q+ε/2, so that lim

n→∞

V_n U_n^q+ε = 0.

To showU_n^q/Vn →0, suppose the contrary, i.e. Vn ≤C U_n^q for suitable C >0 (and a suitable subsequence of{V_n}). Puty:=A_n(x−x_n)U_n^q−1 andv_n=v_n(y) :=

u_n(x)/U_n, where A_n is an orthogonal 2×2 matrix such that the transformation x7→y maps the tangent to∂Ω atxn to the line {y= (y1, y2)∈R²;y2 = 0} and the pointx_n−ν_n (whereν_n is the unit outward normal to∂Ω atx_n) to the point (0,1). Thenv_nfulfils

△vn = an

Un^2q−p−1

v_n^p in Ωⁿ,

∂v_n

∂y₂ =−v^q_n on∂Ωⁿ,

where Ωⁿ := {y =y(x) ;x ∈ Ω}. Moreover,v_n >0, max_Ωnv_n = v_n(0) = 1 and

|∇vn| ≤C. Passing to the limit we getvn→v, wherevis a nonnegative harmonic function in the halfspace [y2>0] fulfilling the boundary condition ∂v/∂y2=−v^q. Moreover,v(0) = 1,v≤1 and|∇v| ≤C. Hence,w:=−∂v

∂y2 is harmonic, bounded byC andw=v^q on [y₂= 0]. The Poisson’s formula ([SW, Theorem II.2.1]) gives us

w(0, λ) =c Z

[y2=0]

v^q(y)λ

λ²+|y|²dy≥c Z

[y2=0]∩[|y1|≤1/(2C)]

1 2

q λ

λ²+|y|²dy≥˜c/λ, since v(y) ≥ ¹₂ for |y| ≤ 1/(2C). This estimate gives us a contradiction, since ϕ(·) :=v(0,·) :R⁺→[0,1] fulfilsϕ^′(λ) =−w(0, λ)≤ −˜c/λ.

Theorem 4.3. LetΩ ={x∈R²;|x|<1}and q < p <2q−1. Then all positive stationary solutions of(NBC) are uniformly bounded for a varying in a bounded subset ofR⁺.

Proof: Suppose the contrary and let u=u_n be as in Lemma 4.1 (we shall fix n and omit the index n). Let (r, ϕ) be the polar coordinates inR² and let ˜ube the solution of the problem

△u˜ = 0 in Ω,

˜

u =u on∂Ω.

Then ˜u≥u, hence ˜ur:= ∂˜u

∂r ≤ ∂u

∂r =u^q on∂Ω.

Putw:=ru˜_r. Thenwis a harmonic function in Ω,w≤u^q≤U^q on∂Ω (where U := max_Ωu). Hencew≤U^q in Ω and ˜ur=w/r≤2U^qin{x∈R²; ¹₂ ≤ |x| ≤1}. Since ˜uis harmonic in Ω, we have|∇u(x)˜ | ≤U/dist (x, ∂Ω)≤2U ≤2U^qfor|x| ≤ ¹₂. Hence,

(4.5) u˜_r≤2U^q in Ω.

Chooseα∈(0,1) andε >0 such that

(4.6) p <2q−1−(1−α)(q−1)−αε.

Since

△(u−u) =˜ au^p≤AU^p in Ω, u−u˜ = 0 on∂Ω, theL^r-estimates imply

(4.7) ku−u˜kC^1,α≤Cku−˜ukW^2,r≤C U˜ ^p for anyr >2/(1−α). Using (4.5)–(4.7) we obtain the estimate

∂u

∂r(x)≤C₁U^q if|x|>1−U^1−q+ε.

Now our assumptions and Lemma 4.1 imply U_n^q/V_n → 0, hence V = |∇u(˜x)| = (K+ 1)U^q for some ˜x = ˜x_n ∈ ∂Ω and K = K_n → ∞. Consequently, denoting uϕ := _∂ϕ^∂u we have|uϕ(˜x)| ≥KU^q and we may suppose uϕ(˜x)≥KU^q. Let (1,ϕ)˜ be the polar coordinates of ˜xand choose ˆϕ:= sup{ϕ <ϕ˜;uϕ(1, ϕ)≤ ^K4U^q}(using obvious identification 0≡2π). Then uϕ(1,ϕ) =ˆ ^K₄U^q andω:=|ϕ˜−ϕˆ|< _K⁴U^1−q sinceuis bounded byU. Now the Schauder estimates imply

ku−u˜kC^2,µ ≤Ckau^pkC^0,µ≤CaU^p+Ca(2U^p)^1−µ pU^p−1(K+ 1)U^qµ

≤q(K+ 1)U^2q−1

for µ sufficiently small and U large. Since |u_ϕr| = |qu^q−1u_ϕ| ≤ q(K+ 1)U^2q−1 on ∂Ω, we have also|u˜_ϕr| ≤ 2q(K+ 1)U^2q−1 on ∂Ω. Now ˜u_ϕ is harmonic and, similarly as in the case of ˜u, the last estimate implies |u˜ϕr| ≤4q(K+ 1)U^2q−1 in Ω. Consequently,|uϕr| ≤5q(K+ 1)U^2q−1 in Ω.

Put S :={(r, ϕ)∈Ω ; ˆϕ < ϕ <ϕ,˜ 1−κ < r <1}, whereκ:= _20q(K+1)^K U^1−q. Then

uϕ(r,ϕ)˜ ≥KU^q−(1−r)5q(K+ 1)U^2q−1≥3

4KU^q forr≥1−κ, u_ϕ(r,ϕ)ˆ ≤K

4 U^q+ (1−r)5q(K+ 1)U^2q−1 ≤1

2KU^q forr≥1−κ.

Hence, (4.8)

Z

S

1

r²u_ϕϕdϕ dr= Z 1

1−κ

1

r² u_ϕ(r,ϕ)˜ −u_ϕ(r,ϕ)ˆ

dr≥ κKU^q 4(1−κ)² ≥4U ifK (orU) is sufficiently large. On the other hand, we know thatur≤C₁U^qin S, hence

(4.9)

Z

Surrdr dϕ≥ − Z ϕ˜

ˆ

ϕ C1U^qdϕ=−ωC1U^q≥ −4C1

K U ≥ −U

for K sufficiently large. By Lemma 4.1 we have |ur| ≤ |∇u| ≤ U^2q−1 for nsuffi- ciently large so that

(4.10)

Z

S

1

ru_rdr dϕ ≤ 1

1−κU^2q−1ωκ= 1

2q(1−κ)K²U ≤U forKlarge enough. Using (4.8)–(4.10) we get R

S△u dx≥2U. However, Z

S△u dx=a Z

Su^pdx≤aU^pκω≤U

forU and/orK large enough, which gives a contradiction.

5. Global existence for (DGT).

In this section we shall suppose that Ω is a smoothly bounded domain in R^N, N ≥1,p, q >1,r > Nmax(1, q−1) and

u₀∈W₀^1,r(Ω)⁺:={u∈W^1,r(Ω) ;u≥0 in Ω andu= 0 on∂Ω}.

It is known (see e.g. [A1]) that (DGT) generates a local semiflow on W₀^1,r(Ω)⁺ and that for any u₀ ∈ W₀^1,r(Ω)⁺ there exists a unique maximal solution u ∈ C [0, T), W₀^1,r(Ω)⁺

, whereT =T(u0) is the maximal existence time for u. More- over, this semiflow is order-preserving.

Byk · k∞we shall denote the norm inL^∞(Ω). The main result of this section is the following

Theorem 5.1. (i) IfT <∞thenlim sup_t→T−ku(t,·)k∞= +∞. (ii) Ifq≥pthenT = +∞andsup_t≥0ku(t,·)k∞<∞.

(iii) Ifq≥pandut≥0 thensup_t≥t0k∇u(t,·)k∞<∞for anyt0 >0.

Proof: To prove (i) it is sufficient to show that an L^∞-estimate for u implies also an L^∞-estimate for ∇u. More precisely, let 0 < t₀ < T₀ < T <∞, C₁ :=

max_t≤T0ku(t,·)k∞<∞andC0:=k∇u(t0,·)k∞. Then we shall show thatC0<∞ and that there exists a constant C2 =C2(C0, C1, T) such thatk∇u(t,·)k∞ ≤C2

for anyt∈[t0, T0].

By [A1, Theorem 14.6] we have u∈C (0, t₀], W^1,rq(Ω)

hence|∇u|^q ∈ C (0, t₀], L^r(Ω)

. Since W^1,r(Ω) ֒→ C(Ω), we have also u^p ∈ C [0, t₀], L^r(Ω)

and the variation of constants formula for u on the interval [t0/2, t0] gives us u(t0,·) ∈ W^2−ε,r(Ω) for any ε >0. Since W^2−ε,r(Ω)֒→C¹(Ω) forε >0 small enough, we haveC₀<∞.

Now putf(y) :=y^q,g(y) :=λy^p and chooseC^∞-functions f_k, g_k (k= 1,2, . . .) such that

• f_k=f andg_k=g on [1,∞),

• f_k≥f andg_k≤g on [0,1],f_k^′(0) = 0,

• f_k→f andg_k→g inC¹ [0,∞)

as k→ ∞. Letu_k be the solution of the problem

(DGT)_k







vt=△v−f_k(|∇v|) +g_k(v) in (t₀,∞)×Ω,

v= 0 on (t₀,∞)×∂Ω,

v(t₀, x) =u(t₀, x) x∈Ω.

Recall that u(t₀,·) ∈ W^2−ε,r(Ω) for any ε > 0. By [A2, Theorem 7.3 and Corollary 9.4], the problem (DGT)_k generates a local semiflow inW₀^1+δ,r(Ω)⁺ for 0 < δ <min(¹_r,1−^N_r,1−(q−1)^N_r) and, denoting byT_k the maximal existence time of u_k in this space, we have u_k ∈ C^∞ (t₀, T_k)×Ω

. We shall show that T_k > T₀ and k∇u_k(t,·)k^∞ ≤C₂ for any t ∈[t₀, T₀] where C₂ =C₂(C₀, C₁, T) is independent of k. Then the variation of constants formula for z_k := u−u_k, the Gronwall’s inequality forkz_k(t,·)kW^2−ε,r(Ω)and a pass to the limit fork→ ∞gives us|∇u| ≤C2.

First notice thatu_k≤uby the maximum principle and that it is sufficient to find the estimatek∇u_k(t,·)k∞≤C2for anyt∈[t0,min(T_k, T0)) since then the variation of constants formula gives an apriori bound also inW^1+δ,r(Ω), henceT_k> T₀.

Fixkand let ˜T < T_k, ˜T ≤T0. The functionw:=¹₂|∇u_k|² fulfils the equation (5.1) wt=△w−X

i,j

(u_k)²_xixj−X

j

f_k^′(∇u_k)

|∇u_k| (u_k)xjwxj+ 2g^′_k(u_k)w Since sup_t∈[t

0,T˜]2g^′_k(u_k)≤2λpmax(2, C₁^p−1) =: ˆC if kis large enough, the maxi- mum principle implies that the functionz:=we^{−C(t−tˆ 0)}attains its maximumZin Q:= [t₀,T˜]×Ω on the parabolic boundary ({t₀} ×Ω)∪([t₀,T˜]×∂Ω).

IfZ≤C₀²/2 then ¹₂|∇u_k|²=w≤ ¹2C₀²e^CTˆ in Qand we are done.

IfZ > C₀²/2 then Z=z(t, x0) for somet∈(t0,T] and˜ x0 ∈∂Ω. Consequently,

∂u_k

∂n(t, x₀)

=|∇u_k(t, x₀)|= max

x∈Ω|∇u_k(t, x)|=√

2Ze^{C(t−tˆ 0)/2}. Sinceu_kis smooth at (t, x0), we have

(5.2) 0 = (u_k)t(t, x0) =△u_k(t, x0)− |∇u_k(t, x0)|^q. Ifν is any unit tangential vector to∂Ω atx0 then, obviously,

∂u_k

∂ν (t, x) ≤C˜

∂u_k

∂n(t, x)

|x−x0| forx∈∂Ω, x→x0,

where ˜C is some constant depending only on the curvature of ∂Ω at x₀. Conse- quently,

(5.3)

△u_k−∂²u_k

∂n²

(t, x₀) ≤C˜

∂u_k

∂n (t, x₀) . Since|∇u_k(t,·)|attains its maximum atx₀, we have ∂²u_k

∂n² (t, x₀)≤0. This inequal- ity together with (5.2) and (5.3) imply|∇u_k(t, x₀)|^q−1≤C, which gives the desired˜ estimate.

(ii) Ifq≥pthen it follows from [F] that the functionψ(x) :=α^2/(p−1)e^α(

P

ixi+C)

is a supersolution foruifαand C are large enough. Hence,u(t, x)≤max_Ωψ for anyt < T andx∈Ω. Now the assertion (ii) follows from (i).

Note that choosing ϕ(x) := min{ψ(x), Kdist (x, ∂Ω)} with K sufficiently large we obtain a supersolution ϕ for u(t,·), t ≥ t₀, which gives us an apriori bound

|∇u|=

∂u

∂n

≤K on the boundary∂Ω.

(iii) Letq≥pandut≥0. Then

△u=ut+|∇u|^q−λu^p ≥ |∇u|^q−C1 for some C1 >0, and, consequently,

X

i,j

u²_xixj ≥C₂(△u)²≥C₃|∇u|^2q−C₄ for someC₂, C₃, C₄>0.

By the note in the proof of (ii), the functionw:= ¹₂|∇u|²is bounded on∂Ω so that the last inequality together with (5.1), the boundedness of u and the maximum principle imply the boundedness ofwin [0,∞)×Ω.

6. Stationary solutions for (DGT).

Throughout this section we suppose that Ω is a smoothly bounded domain inR^N, N ≥1. By a (stationary) solution we mean always a classical positive stationary solution.

Lemma 6.1. Letq≥p, λ₁ >0. Then there exists K=K(λ₁)>0such that any positive stationary solutionuof(DGT) withλ≤λ₁ fulfils kuk_C¹_(Ω)≤K.

Proof: We shall use similar arguments as in the proof of Theorem 5.1 (ii), (iii).

One can easily find a function

ϕ(x) =ϕ_α(x) = min{ψ_α(x), Kdist (x, ∂Ω)}, whereψα(x) =α^2/(p−1)e^α(

P

ixi+C) and K=K(α) is a continuous nondecreasing function ofα, lim

α→∞K(α) = +∞, such that forα≥α₀,ϕis a strict supersolution for (DGT) with anyλ≤λ₁. Now suppose that uis a positive stationary solution of (DGT) with λ ≤ λ₁ which does not lie below ϕα0. Choosing α₁ := inf{α; ϕα ≥ u} we have ϕα1 ≥ u and either ∂ϕα1

∂n (x₁) = u(x₁) for some x₁ ∈ ∂Ω or ϕα1(x₂) =u(x₂) for somex₂∈Ω. Since both possibilities lead to the contradiction with the maximum principle, we haveu≤ϕα0, i.e. we have an apriori bound (say C₁) foruinL^∞(Ω) and an apriori bound for

∂u

∂n

=|∇u|on∂Ω.

Puttingw:= ¹₂|∇u|²and assuming thatwattains its maximum at somex₀ ∈Ω, we get by (5.1) (withwt= 0,△w(x₀)≤0,wxj(x₀) = 0) and (DGT)

2λ₁pC₁^p−1w(x₀) ≥2λpu^p−1(x₀)w(x₀)≥X

i,j

u²_xixj(x₀)

≥C₂ △u(x₀)2

=C₂ |∇u(x₀)|^q−λu^p(x₀)2

≥C₃w^q(x₀)−C₄,

which gives an apriori bound forw(x₀).

Remark 6.1. The apriori bound inC¹(Ω) and standard regularity results for the stationary problem related to (DGT) imply also an apriori bound in W^2,r(Ω) for any r >1 so that the set of positive stationary solutions for λ≤ λ₁ is relatively compact inC¹(Ω).

Theorem 6.1. Let q ≥ p. Then there exists λ₀ > 0 such that the stationary problem corresponding to(DGT)

(i) does not have positive solutions for λ < λ₀,

(ii) has at least one positive solution forλ=λ0 and at least two positive solu- tions forλ > λ0.

Proof: To prove (i) suppose the contrary, i.e. there exist solutionsun with λ= λn↓0. By Lemma 6.1, these solutions are uniformly bounded inC(Ω) by some con- stantC₁. Denoting byν_nthe norm ofu_ninW^1,2(Ω), multiplying the (stationary)

equation in (DGT) byu_n and integrating by parts we get

(6.1)

cν_n² ≤ Z

Ω|∇u_n|²dx=− Z

Ω|∇u_n|^qu_ndx+λ_n Z

Ω

u^p+1_n dx

≤λnC₁^p−1 Z

Ω

u²_ndx≤λnC₁^p−1ν_n²

for suitablec >0, which gives us a contradiction.

(ii) Suppose that (DGT) has a positive stationary solutionu0 for some λ0 >0 and letλ > λ₀. Then u₀ ∈ W^2,r(Ω) by Remark 6.1 and u₀ is a (strict) subsolu- tion for (DGT). By the maximum principle, ut ≥ 0 for the solution u of (DGT) starting atu₀. Consequently, the function u(t,·) is bounded inW^1,∞(Ω) by The- orem 5.1 (iii). Standard parabolic regularity results imply now the boundedness of u(t,·) in W^2−ε,r(Ω) for any r > 1 and ε > 0 so that the orbit {u(t,·)}t≥0 is relatively compact inC¹(Ω). Sinceu_t≥0, we haveu(t,·)→u˜ ast→+∞, where

˜

uis a positive stationary solution of (DGT).

To see that (DGT) has a positive stationary solution at least for some λ, let u0 be a nonnegative C²(Ω)-function such that u0 = 0 on ∂Ω, △u0 ≥ |∇u|^q in a neighbourhoodU of∂Ω andu₀≥ε >0 in Ω\U (It is sufficient to chooseu₀(x) :=

w(dist (x, ∂Ω)) for x close to ∂Ω, where w is the solution of O.D.E. w(0) = 0, w^′(0) =C ≫1,w^′′(y) = 2w^′q(y) fory ∈(0, δ],u₀(x) :=w(δ) for dist (x, ∂Ω)> δ, and then regularizeu₀ in the δ/2-neighbourhood of {x; dist (x, ∂Ω) = δ}.). Then u₀ is a subsolution for (DGT) if λis sufficiently large, hence (similarly as above) we get the existence of a positive stationary solution.

Until now, we have shown the existence of a λ₀ > 0 such that the stationary problem corresponding to (DGT) has

(j) no solutions forλ < λ0,

(jj) at least one solution forλ > λ0.

To prove the existence of a solution forλ=λ₀, let u_n be solutions corresponding toλ_n↓λ₀. Due to the apriori bounds (Lemma 6.1 and Remark 6.1) we know that u_n converge to some nonnegative stationary solution of (DGT) with λ =λ₀. To showu6≡0, suppose the contrary. Then similarly as in (6.1) we get

cν_n² ≤ Z

Ω|∇un|²dx≤λn

Z

Ω

u^p+1_n dx≤λn

Z

Ω

u²_ndx max

Ω

u^p−1_n

≤λnν_n²max

Ω u^p−1_n

and sinceun→0 (even inC¹(Ω)), we get a contradiction.

Now let λ > λ₀ and let u_λ be the positive stationary solution which we have got as the limit of the solution ˆu of (DGT) starting atu₀ (= positive stationary solution corresponding toλ0). ChooseK >0,K >sup_t≥0ku(t,ˆ ·)k_C1(Ω)and letf_K

be a smooth cut-off function for the functiony 7→y^q; more precisely, f_K(y) =y^q fory ∈[0, K],f_K(y) =K^q+ 1 fory ≥K+ 1,f_K^′ >0 on [K, K+ 1),f_K(y)≤y^q for anyy≥0. Consider the problem

(DGT)_K







ut=△u−f_K(|∇u|) +λu^p in (0,∞)×Ω,

u= 0 on (0,∞)×∂Ω,

u(0, x) =u₁(x) x∈Ω,

where 0≤ u1 ≤u_λ, u1 ∈ W^2−ε,r(Ω)∩W₀^1,r(Ω) for somer > N/(1−2ε). Since f_K(y) =y^q fory ≤ ku(t,ˆ ·)k_C¹_(Ω), the functionu_λ is a positive stationary solution of (DGT)_K and ˆu(t,·) is a nondecreasing solution of (DGT)_K connecting u₀ to u_λ. Moreover, 0 is a stable stationary solution of (DGT)_K and one can easily find a positive function ˜u0such that∂˜u₀

∂n >0 on∂Ω and the solution of (DGT)_Kstarting at ˜u0 tends (in a monotone way) to 0 ast→ ∞. DenotingS^τu1:=u(τ,·) whereu is the solution of (DGT)_Kstarting atu₁, we get thatS^τis (for anyτ >0) an order- preserving discrete semigroup which maps the order interval [0, u_λ]⊂ W^2−ε,r(Ω) into a relatively compact subset of [0, u_λ]. Moreover, 0 or u_λ is an equilibrium of S^τ which is stable from above or from below, respectively. Due to [AH, Lemma 5], there exists another equilibriumu^τ of S^τ which lies between 0 andu_λ. Since u^τ lies neither aboveu0 nor below ˜u0, we have

(6.2) min{ku^τ−u_λk_C¹_(Ω),ku^τk_C¹_(Ω)} ≥c₀>0

for some c₀ which is independent of τ. The variation of constants formula and a straightforward estimate imply that the set{u^τ}τ∈(0,τ0)is bounded inW^2−ε,r(Ω) and hence we may find a sequenceτ_k↓0 such thatu^τk →u_K inW^2−2ε,r(Ω). Due to (6.2),u_Kis a positive stationary solution of (DGT)_Kwhich lies in [0, u_λ]\{0, u_λ}. Putw_K := ¹₂|∇u_K|². We show thatw_K≤ ¹2K² forKsufficiently large so thatu_K is also a stationary solution of (DGT).

Since w_K ≤ ¹₂|∇u_λ|² ≤ ¹₂K² on the boundary ∂Ω, we may assume that w_K attains its maximum at somex₀ ∈Ω. Supposew_K(x₀)> ¹₂K². Using an analogue to (5.1) we get, similarly as in the proof of Lemma 6.1,

(6.3) C1w_K(x0) ≥2λpu^p−1_K (x0)w_K(x0)≥X

(u_K)²_xixj(x0)≥C2 △u_K(x0)2

=C2 f_K(|∇u_K(x0)|)−λu^p_K(x0)2

≥C3K^2q−C4

On the other hand, due to theL^r-estimates for the stationary problem correspond- ing to (DGT)_K we have

ku_KkW^2,r(Ω) ≤C₅+C₆kf(|∇u_K|)kL^r(Ω)

≤C5+C6(K^q+ 1)^(r−1)/rZ

Ωf(|∇u_K|)dx1/r

≤C₇+C₈K^q(r−1)/r,