Convergence to equilibrium for solutions of an abstract wave equation with general damping function

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Convergence to equilibrium for solutions of an abstract wave equation with general damping function Tomáš Bárta a,∗ , Eva Fašangová b a Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague,

Sokolovská 83, 18675 Praha 8, Czech Republic b Technische Universität Dresden, Institut für Analysis, Fachrichtung Mathematik, Zellescher Weg 12-14,

01062 Dresden, Germany Received 11 November 2014; revised 9 September 2015

Abstract We prove convergence to a stationary solution as time goes to infinity of solutions to abstract nonlinear wave equation with general damping term and gradient nonlinearity, provided the trajectory is precompact. The energy is supposed to satisfy a Kurdyka–Łojasiewicz gradient inequality. Our aim is to formulate conditions on the function g as general as possible when the damping is a scalar multiple of the velocity, and this scalar depends on the norm of the velocity, g(|ut |)ut . These turn out to be estimates and a coupling condition with the energy but not global monotonicity. When the damping is more general, we need an angle condition. © 2015 Elsevier Inc. All rights reserved. Keywords: Abstract wave equation with damping; Convergence to equilibrium; Łojasiewicz inequality

1. Introduction This work has been inspired by a result of Chergui presented in [4], where the following semilinear damped wave equation utt (t, x) + |ut (t, x)|α ut (t, x) = u + f (x, u(t, x)), * Corresponding author.

E-mail address: [email protected] (T. Bárta). http://dx.doi.org/10.1016/j.jde.2015.10.003 0022-0396/© 2015 Elsevier Inc. All rights reserved.

t ≥ 0, x ∈  ⊂ RN ,

(1)

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with zero boundary conditions on ∂ is studied. It is proved that every bounded solution has relatively compact trajectory and that for certain values of α every solution with relatively compact trajectory converges to a stationary solution. The set of admissible α’s depends on the Łojasiewicz exponent of the operator  + f (x, ·). The main goal of this paper is to study the above equation with a more general damping term g(|ut |)ut resp. G(u, ut ) instead of |ut |α ut and to obtain convergence to equilibrium for solutions with relatively compact trajectory. We prove our result in a more general setting assuming an abstract gradient operator E  (u) instead of u + f (x, u). Thus, we study the equation utt (t) + g(|ut (t)|)ut (t) = E  (u(t)),

t > 0,

(2)

where the damping is g(|ut |)ut , ut is the velocity and g is a scalar function. Our analysis shows also the way how to generalise the result to a more general model, an anisotropic, inhomogeneous medium where the damping need not point into the direction of the velocity, that is the equation utt (t) + G(u(t), ut (t)) = E  (u(t)),

t > 0.

(3)

In this formulation we obtain a generalisation of [1, Theorem 4] for the ordinary differential equation u(t) ¨ + G(u(t), u(t)) ˙ = E  (u(t)),

t > 0,

(4)

for u : [0, +∞) → RN , for more general damping than in [1]; see also [3]. We denote V := H01 (), H := L2 (), V ∗ := H −1 (), where  ⊂ RN is open and bounded. Let E ∈ C 2 (V ) and let g : [0, +∞) → [0, +∞) be given. Consider the problem (2) with given initial values u(0) = u0 ∈ V , ut (0) = u1 ∈ H . Let us assume that there exists a solution u ∈ C 1 ([0, +∞), H ) ∩C([0, +∞), V ) such that |ut |2 g(|ut |) ∈ L1 ((0, +∞), L1 ()) and assume that the trajectory {(u(t), ut (t)) : t ≥ 0} is relatively compact in V × H . Then there exists a sequence tn → +∞ such that (u(tn ), ut (tn )) converges to some (ϕ, ψ) ∈ V × H and one can show that ψ = 0 (see [4]). The question we are interested in is whether lim u(t) = ϕ ?

t→+∞

In [4, Theorem 1.4] Chergui gave a positive answer to this question for the equation (1) under suitable assumptions on f provided α satisfies the following two conditions: θ 1. 0 < α < 1−θ , where θ is a Łojasiewicz exponent depending on E, 0 < θ ≤ 12 , 4 2. α < N−2 .

The first condition says that the damping term g(|ut |)ut is not too small near zero (which seems to be a reasonable condition). It also estimates the growth at infinity but it can be seen from the proof that one does not need this estimate. The second condition says that the growth of g at +∞ is not too fast and it stems from a Sobolev embedding needed in the proof. It also means that the growth of g at zero is not too small, but we show that this estimate at zero is not necessary. From the physical interpretation we would say that the bigger is the damping term, the better is the convergence or the stabilisation effect.

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In general, convergence can fail. For example, for linear damping in equation (1) (i.e. α = 0, resp. g(|ut |)ut = ut ) see [8], for the damping g(|ut |)ut = |ut |ut in equation (4), N = 1, monotone right-hand side E  see [7]. In this paper we give a positive answer to the question for equations (2), (3) and (4) under suitable assumptions. First we formulate the assumptions on g, E and a coupling condition between them, and we formulate our main result for equation (2). In Section 3 we formulate an equivalent set of assumptions. In Section 4 we prove the result for equation (2). In Section 5 we formulate the assumptions on G, E and prove the main result for equation (3). Finally we mention a corollary for the equation (4). 2. Main result for the equation with a scalar damping function First of all we fix the notation. Let  ⊂ RN be a bounded domain. We work with the Hilbert spaces H := L2 () resp. L2 (, Rn ) with norm and scalar product denoted by  ·  and ·, · , and V := H01 () resp. H01 (, Rn ) with norm  · V . We have the continuous embedding V → H , we identify H with its dual H ∗ , and we denote V ∗ the dual space to V . In this way we have V → H → V ∗ and after identification v, u V ∗ ,V = v, u for u ∈ V ⊂ H, v ∈ H ⊂ V ∗ . The norm and the scalar product on V ∗ are denoted by  · ∗ and ·, · ∗ . We denote by K : V ∗ → V the duality mapping given by v, u ∗ = v, Ku V ∗ ,V ,

u, v ∈ V ∗ .

For N > 2 the space V is embedded into Lq () resp. Lq (, Rn ) provided q ≤ 2N define p := N+2 for N > 2 and p := 1 for N = 1, and have 1 ≤ p < 2,

2N N−2 .

We



V → Lp → H → Lp → V ∗ with p  =

2N N−2

for N > 2, p  = ∞ for N = 1, and v, u V ∗ ,V = v, u Lp ,Lp = v, u ,

u ∈ V , v ∈ H.

For N = 2 the above embeddings hold for all p ∈ (1, 2) (and corresponding p  ) but not for p = 1. Since there is no minimal value of p in this case, we fix the value of p later, see the text below condition (g3). The norm on Lp is denoted by .p . We usually denote real numbers by s, r, vectors in Rn by z, w, and x ∈ RN . By |.| we denote the Euclidean norm on Rn (resp. absolute value on R). Letters u, v are used for members of V ∗ (and its subspaces V , H ) or for functions of two variables, e.g. u : [0, +∞) → H . If u is a ˙ for the time function of t ∈ R and x ∈ RN , we often write u(t) instead of u(t, ·), and ut = du dt = u derivative. By BV (ϕ, ε) we denote the open ball in V with radius ε and centered at ϕ. Now we introduce the assumptions. We say that a function  : [0, +∞) → [0, +∞) has property (KL) if it is nondecreasing, sublinear (i.e. √ (s + r) ≤ (s) + (r) for all r, s ≥ 0) and satisfies (s) > 0 for all s > 0, and (s) ≤ C s for all s ∈ [0, τ ] for some constants τ > 0 and C > 0. For example, the function (s) = s 1−θ with θ ∈ [0, 12 ] has property (KL). Remarks. Since the assumptions on  below ((e1) and (h2)) involve only arguments near zero, we could define the property (KL) on a neighbourhood of zero only (any such function can be extended by a constant to [0, +∞) such that it has the above properties on the whole [0, +∞)).

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The sublinearity assumption could be weakened to (s + r) ≤ C((s) + (r)) for some C > 0 and all r, s ≥ 0, and our results would remain valid. Our assumptions on the operator E are the following. (E) Assume that E ∈ C 2 (V ) satisfies: (e1) there exists a function  with property (KL) such that 1 is integrable in a neighbourhood of zero and such that E satisfies the Kurdyka–Łojasiewicz gradient inequality with the function  in a neighbourhood of the critical points of E, i.e., for each ϕ ∈ N := {ψ ∈ V : E  (ψ) = 0} there exist η, C > 0 such that E  (u)∗ ≥ C(|E(u) − E(ϕ)|),

u ∈ BV (ϕ, η);

(5)

(e2) for all u ∈ V , the operator KE  (u) ∈ L(V ) extends to a bounded linear operator on H and sup KE  (u)L(H ) is finite whenever u ranges over a compact subset of V .  In [4] Chergui works with E  (u) = u + f (x, u) which corresponds to E(u) =  12 |∇u(x)|2 + u F (x, u)dx, where F (x, u) := 0 f (x, s) ds (n = 1). By [4, Corollary 1.2], this function E satisfies the Łojasiewicz gradient inequality E  (u)∗ ≥ C|E(u) − E(ϕ)|1−θ

(6)

with some θ ∈ [0, 12 ) in a neighbourhood of N , provided f satisfies certain assumptions. The Łojasiewicz inequality (6) is a special case of the Kurdyka–Łojasiewicz inequality (5) with the function (s) = s 1−θ , θ being the Łojasiewicz exponent. It is easy to see that Chergui’s operator satisfies (e2) as well. The conditions (e1) and (e2) (with (6) instead of (5)) appear also in [5], where linear damping is considered. Now we formulate the assumptions on the damping function. (G) The function g : [0, +∞) → [0, +∞) is continuous on (0, +∞) and there exists τ > 0 such that (g1) there exists C2 > 0 such that g(s) ≤ C2 on [0, τ ), (g2) there exists C3 > 0 such that C3 ≤ g(s) on [τ, +∞), (g3) if N = 2 then there exist C4 > 0 and α > 0 such that g(s) ≤ C4 s α on [τ, +∞), and if 4 N > 2 then there exists C4 > 0 such that g(s) ≤ C4 s α on [τ, +∞) with α = N−2 . If N > 2, then p =

α+2 α+1

holds. We set p :=

α+2 α+1

for N = 2, too.

(H) For τ from condition (G) there exists a function h : [0, +∞) → [0, +∞), which is concave and nondecreasing on [0, τ ] and satisfies (h1) g ≥ h on [0, τ ], 1 (h2) the function s → (s)h((s)) belongs to L1 ((0, τ )), √ (h3) the function ψ : s → sh( s) is convex on [0, τ 2 ]. Note that no monotonicity of the damping function g is assumed, only some estimates from 4 above and from below. Clearly, Chergui’s damping function g(s) = s α , α < N−2 satisfies (G)

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and it also satisfies (H) with h(s) = s α for (s) = s 1−θ , θ ∈ (0, 12 ), where our condition (h2) θ corresponds to Chergui’s condition 0 < α < 1−θ . This is the condition coupling the damping function g with the operator E. We want to achieve that the damping term g(|v|)v lies in the largest Lp -space satisfying   V → Lp , i.e., Lp → V ∗ , for v ∈ Lp . This is guaranteed by the growth condition (g3). If N = 2, then there is no largest Lp -space with this property. However, there is an optimal (smallest) 2 Orlicz space L , (t) = et satisfying V → L (see [6]). It would be a subject of further research to work with this embedding and to extend the results of this paper for exponentially growing functions g in case N = 2. Our main result is formulated for solutions in the following sense. We say that u ∈ 1,1 2,1 ([0, +∞), V ) ∩ Wloc ([0, +∞), H ) is a strong solution to (2) if (2) holds in V ∗ for almost Wloc every t > 0. The omega-limit set of u is ωV (u) = {ϕ ∈ V : there exists a sequence tn  +∞ such that lim u(tn ) − ϕV = 0}. n→∞

Condition (g1) and the choice of p imply that g(|ut (t)|)ut (t) ∈ Lp → V ∗ for almost every t > 0 for a strong solution. We analogously define a strong solution of equation (3) under the assumptions on G in Section 5. Theorem 2.1. Let E and g satisfy (E), (G) and (H). Let u be a strong solution to (2) such that {(u(t), ut (t)) : t ≥ 0} is relatively compact in V × H and ϕ ∈ ωV (u). Then limt→+∞ (u(t) − ϕV + ut (t)) = 0. Remarks. Condition (H), which estimates g from below on a neighbourhood of zero, is more complicated than the others, but this condition is trivial if lim infs→0+ g(s) > 0, since then a small constant function h works ((h2) holds since 1/ is integrable due to (e1)). If lim infs→0+ g(s) = 0, then necessarily h(0) = 0. Condition (h2) says √ that the growth of h at zero must be steep enough. In fact, together with the condition (s) ≤ c s from property (KL) we  (0) = +∞. Finally, let us mention have that h+ (0) = +∞ and if lims→0+ g(s) = 0, then also g+ that every function       1 αn 1 α1 1 α2 h(s) := s α ln . . . ln . . . ln ln ln s s s with α ∈ (0, 1), n ∈ N, αi ∈ R satisfies condition (h3). √ This can be shown by computing the first derivative of h and the second derivative of s → sh( s). 3. An equivalent set of assumptions ˜ (H), ˜ ()) and show that these In this section we introduce another set of assumptions ((G), assumptions are equivalent to assumptions (G), (H). These new assumptions are motivated by the proof of Theorem 1.4 in [4]. Reading that proof carefully and analysing the assumptions needed lead us to this set and we prove the assertion of Theorem 2.1 under these new assumptions in the next section. Here we show that the old assumptions imply the new ones. And we also show the opposite implication which says in some sense that these assumptions are the best possible if we want to use the method from [4]. Here are the new assumptions:

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˜ The function g : [0, +∞) → [0, +∞) is continuous on (0, +∞) and there exists τ > 0 such (G) that (g3) holds. ˜ There exists a function h˜ : [0, +∞) → [0, +∞), which is positive and concave on (0, +∞) (H) and satisfies ˜ g ≥ h˜ on [0, +∞), (h1) 1 ˜ the function s → (h2) belongs to L1 ((0, 1)), ˜ (s)h((s)) √ ˜ the function ψ : s → s h( ˜ s) is convex on [0, +∞). (h3) () There exists a Young function γ : [0, +∞) → [0, +∞) (convex with γ (0) = 0, lims→+∞ γ (s) = +∞) such that (γ 1) there exists D1 > 0 such that γ (g(s)s) ≤ D1 g(s)s 2 for s ≥ 0, (γ 2) there exists D2 > 0 such that γ (s) ≥ D2 s 2 on a neighbourhood of zero, 1

(γ 3) the function γ˜ : s → γ (s p ) is convex on [0, +∞). (γ 4) for every K > 0 there exists C(K) such that γ (Ks) ≤ C(K)γ (s) holds for all s ≥ 0. We say that a function f : [0, +∞) → [0, +∞) has property (K) if for every K > 0 there exists C(K) such that f (Ks) ≤ C(K)f (s) for all s ≥ 0. So, (γ 4) says that γ has property (K). Typically, nondecreasing functions with polynomial growth do have this property, while functions with exponential growth do not. ˜ implies that Lemma 3.1. Condition (H) ˜ (h5) ˜ (h6) ˜ (h7) ˜ (h8)

h˜ is nondecreasing on [0, +∞), ˜ s h˜ ± (s) ≤ h(s) for s ∈ [0, +∞), ˜h has property (K), ψ has property (K).

˜ ˜ follow immediately from concavity and positivity of h. ˜ holds with ˜ (h7) Proof. (h5), (h6) ˜ C(K) = 1 for K ≤ 1 since h is nondecreasing and C(K) = K for K > 1 since h˜√is con˜ ˜ ˜ ˜ cave and √ h(s)√≥ 0. We show √ that (h8) follows from (h7). In fact, ψ(Ks) = Ks h( Ks) ≤ ˜ KsC( K)h( s) = KC( K)ψ(s). 2 Lemma 3.2. Denote by δ the convex conjugate function to γ . Then (γ 2) is equivalent to δ(s) ≤ d3 s 2 on a neighbourhood of zero for some d3 > 0. Proof. By definition δ(r) = sups≥0 (rs − γ (s)). From the shape of γ it follows that the maximizer s0 of rs − γ (s) is small if r is small. Hence, for small r, maxs≥0 (rs − γ (s)) ≤ 2 maxs≥0 (rs − D2 s 2 ) = (2Dr )2 . For the converse implication we compute γ (s) = maxr≥0 (sr − 2

δ(r)) ≥ maxr≥0 (sr − d3 r 2 ) =

s2 . (2d3 )2

2

Proposition 3.3. Given a function g : [0, +∞) → [0, +∞) and a function  : [0, +∞) → ˜ (H), ˜ (). [0, +∞), the set of assumptions (G), (H) is equivalent to the set of assumptions (G), ˜ (H), ˜ () imply (G) and (H). For the moment, τ > 0 is arbitrary, Proof. First we show that (G), later it will be chosen small enough. The condition (g3), upper bound on [τ, +∞), follows from ˜ on a neighbourhood of infinity [K, +∞) and from continuity of g on the compact interval (G)

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[τ, K]. The condition (g2), lower bound on [τ, +∞), follows from positivity and concavity of h˜ ˜ Concerning condition (g1), upper bound on [0, τ ), we distinguish two cases. and inequality (h1). The first case lims→0+ sg(s) = 0 leads to a contradiction. In fact, taking sk → 0, sk > 0 with sk g(sk ) ≥ c > 0 and dividing the inequality in (γ 1) by g(s)s we obtain γ (sk g(sk )) ≤ D1 sk . sk g(sk ) Here the right-hand side tends to zero as k → ∞ and the left-hand side does not since γ (r) ≥ ar for r ∈ [c, +∞) for some a > 0 (γ is increasing and convex on a neighbourhood of +∞). In the second case lims→0+ sg(s) = 0 we have γ (g(s)s) ≥ D2 s 2 g(s)2 and γ (g(s)s) ≤ D1 g(s)s 2 , hence 1 g(s) ≤ D D2 for s ∈ [0, τ ), provided τ > 0 is small enough. Condition (H) follows immediately by ˜ ) on (τ, +∞). taking h := h˜ on [0, τ ] and constant h(τ ˜ (H) ˜ and (). Now we prove that (G), (H) imply (G), ˜ (G) follows immediately from (g3). To show () let us define  γ (s) :=

for s ∈ [0, τ ) for s ∈ [τ, +∞),

c1 s 2 s p − τ p + c1 τ 2

p−2

where the constant c1 > 0 will be chosen such that c1 < τ p−2 , c1 ≤ p2 τ p , and even smaller if necessary, see later. This function is nonnegative, continuous, and convex since p ≥ 1 and γ− (τ ) = 2c1 τ < pτ p−1 = γ+ (τ ). Therefore γ is a Young function, and the property (γ 2) holds trivially. The property in (γ 4) holds with C(K) = 1 provided K ≤ 1, since γ is increasing. For K > 1 we distinguish three cases. – If s < Kτ , then γ (Ks) = c1 K 2 s 2 = K 2 γ (s). – If s > τ , then γ (Ks) = K p s p − τ p + c1 τ 2 = ≤

K p s p − τ p + c1 τ 2 γ (s) s p − τ p + c1 τ 2

K p r p − τ p + c1 τ 2 γ (s). p p 2 r∈(τ,+∞) r − τ + c1 τ

– If s ∈ [ Kτ , τ ], then γ (Ks) =

sup

p p p +c τ 2 K p s p −τ p +c1 τ 2 1 γ (s) ≤ maxr∈[ Kτ ,τ ] K r −τ c1 s 2 c1 r 2

γ (s).

Now, (γ 4) is proven with C(K) being the maximum of the three factors on the right-hand sides above. Concerning (γ 3), the function  γ˜ (s) :=

2

c1 s p s − τ p + c1 τ 2

is convex, since p < 2 and γ˜− (τ ) =

2c1 p2 −1 p τ

for s ∈ [0, τ p ) for s ∈ [τ p , +∞)

≤ 1 = γ˜+ (τ ).

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Now we look for a suitable constant D1 > 0 in order to satisfy (γ 1). We distinguish the cases. – If g(s)s < τ then for s < τ by (g1) we have γ (g(s)s) = c1 g(s)2 s 2 ≤ c1 C2 g(s)s 2 , while for 2 2 s ≥ τ we have γ (g(s)s) = c1 g(s)2 s 2 = c1 g(s)s s g(s)s ≤ c1 g(s)s . – If g(s)s ≥ τ then we have γ (g(s)s) = g(s)p s p − τ p + c1 τ 2 ≤ g(s)p s p = g(s) (g(s)s)p−2 g(s)s 2 ≤ C2 τ p−2 g(s)s 2 for s < τ , using (g2) and p − 2 < 0. For s ≥ τ and N ≥ 2 we use (g3) and the fact that p > 1, α(p − 1) + p − 2 = 0 and obtain p−1 p−2  p−1 γ (g(s)s) ≤ g(s)p−1 s p−2 g(s)s 2 ≤ C4 s α s g(s)s 2 = C4 g(s)s 2 . For s ≥ τ and N = 1 we have p = 1 and γ (g(s)s) ≤ g(s)p s p = 1s g(s)s 2 ≤ τ1 g(s)s 2 . p−1

Thus we can take D1 = max{c1 C2 , c1 , C2 τ p−2 , C4 , τ1 }. ˜ If h(0) > 0, then g is bounded from below on [0, +∞) by a positive Finally we turn to (H). ˜ and conditions (h2), ˜ constant and we define h˜ to be this constant. This function satisfies (h1) ˜ (h3) are obvious. If h(0) = 0, take δ ∈ (0, τ ) such that h(δ) < C3 and h (δ) > 0. Then h (δ) < h(δ) δ since h is concave and h(0) = 0. Let us define  ˜ := h(s)

h(s) 2 h(δ) 2

+ ( 1δ

−

1 h (δ)δ 2 s) 2

for s ∈ [0, δ) for s ∈ [δ, +∞).

˜ In any case h˜ is positive and concave on (0, +∞). Further, h(s) ≤ h(δ) for all s and we can ˜ as follows. For s ∈ (0, δ) we have h(s) ˜ ≤ h(s) ≤ g(s). For s ∈ [δ, τ ] we have h(s) ˜ ≤ verify (h1) ˜ ≤ h(δ) ≤ C3 ≤ g(s). h(δ) ≤ h(s) ≤ g(s). For s ∈ (τ, +∞) we have h(s) √ ˜ the convexity of the function ψ : s → s h( ˜ s). On [0, δ 2 ) convexity follows We turn to (h3), from (h3). For s > δ 2 we have     2  1 h (δ)δ h(δ) 1 h (δ)δ 2 1 h (δ)δ 2 − 3 ψ  (s) = s = + −s2 s 2 > 0. 2 δ 2 2 4 2 For s = δ 2 we have  ψ− (δ 2 ) =

  h(δ) 1 h(δ) 1 1 h+ (δ)δ 2  (δ 2 ). + δ 2 h− (δ) = + − = ψ+ 2 4δ 2 δ 2δ 2

˜ follows immediately from (h2). Consequently, ψ is convex on [0, +∞). Condition (h2)

2

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4. Proof of convergence for the equation with scalar damping ˜ (H), ˜ () hold. We denote by c∗ , cp Let the assumptions (E), (G), (H) or equivalently (E), (G), ∗ ∗ p and cp the constants of the embeddings H → V , H → L and Lp → V ∗ , respectively. We start with the following lemma (compare to [4, Proposition 1.5]). Lemma 4.1. The following assertions hold for a strong solution u to (2) which satisfies the assumptions of Theorem 2.1: (i) the function t → g(|ut (t)|)|ut (t)|2 belongs to L1 ((0, +∞), L1 ()), (ii) limt→+∞ ut (t) = 0, (iii) ϕ ∈ ωV (u) implies E  (ϕ) = 0. Proof. Multiplying the equation (2) by ut (t) with respect to the duality ., . V ∗ ,V we have utt , ut V ∗ ,V + g(|ut |)ut , ut = E  (u), ut V ∗ ,V

(7)

and integrating over [s, T ] we obtain 

   T  1 1 2 2 g(|ut (t)|)|ut (t)|2 dt ≤ 0. ut (T ) − E(u(T )) − ut (s) − E(u(s)) = − 2 2 s 

This implies that 21 ut (·)2 − E(u(·)) is nonincreasing. Relative compactness of the trajectory of u then yields (i). Part (ii) follows from [2, Theorem 2.8]. To prove (iii) let us fix ϕ ∈ ωV (u) and tn → +∞ such that u(tn ) → ϕ in V . Then tn +s

u(tn + s) = u(tn ) +

ut (r)dr. tn

Since the integral tends to zero in H by (ii), relative compactness of the trajectory implies that u(tn + s) → ϕ in V for every s ∈ [0, 1]. The following equalities hold in V ∗ (the second equality follows from Lebesgue dominated convergence theorem and the last one from (ii)): 

1

E (ϕ) =

1



E (ϕ)ds = lim

n→∞

0

E  (u(tn + s))ds

0

1 = lim

(utt (tn + s) + g(|ut (tn + s)|)ut (tn + s)) ds

n→∞ 0

⎛ = lim ⎝ut (tn + 1) − ut (tn ) +

tn +1

g(|ut (s)|)ut (s)⎠ ds

n→∞

tn tn +1

= lim

g(|ut (s)|)ut (s)ds .

n→∞ tn

⎞

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10

In the following estimates we use the embedding Lp → V ∗ , the notation s,τ := {x ∈  : |ut (x, s)| < τ }, the assumptions (g1) and (g3), the relation α(p − 1) + p = 2 (if N = 1, the p−1 estimates hold with C4 replaced by 1/τ ), Hölder’s inequality, the embedding H → Lp and 1

Jensen’s inequality (the function s → s p is concave): tn +1

tn +1

∗

g(|ut (s)|)ut (s)ds g(|ut (s)|)ut (s)p ds =

≤ cp

∗

tn

⎛

= cp∗

tn +1

⎜ ⎝

tn



tn +1

⎜ ⎝

tn

p



p C2 |ut (s)|p

g(|ut (s)|)

p⎟

g(|ut (s)|)|ut (s)| ⎠ ds ⎞1



p

⎟ p−1 C4 g(|ut (s)|)|ut (s)|2 ⎠

+

ds

\s,τ

⎛

tn +1

p−1

\s,τ

s,τ

⎜ ⎝ ⎝ C2

tn

≤ cp∗ C2 cp

p

g(|ut (s)|) |ut (s)| + p

⎛

≤ cp∗

⎞1



s,τ

⎛

≤ cp∗

tn



⎞1

⎛

p

p−1 p

|ut (s)|p ⎠ + C4

⎝

 tn +1



⎞1⎞ p 2⎠ ⎟ g(|ut (s)|)|ut (s)| ⎠ ds



p−1 p

ut (s)ds + cp∗ C4

tn

⎛ t +1⎛ ⎞ ⎞ p1 n  ⎝ ⎝ g(|ut (s)|)|ut (s)|2 ⎠ ds ⎠ . tn



The last terms tend to zero by (ii) and (i), and consequently E (ϕ) = 0.

2

Lemma 4.2. There exists a constant Ch˜ > 0 such that 2 ˜ h(v ∗ )v ≤ Ch˜

 g(|v(x)|)|v(x)|2 dx,

v ∈ H.



˜ The following computation holds, since by Lemma 3.1 the function Proof. Let ψ be from (h3). h˜ is nondecreasing (first inequality) and has property (K) (second inequality): 2 2 ˜ ∗ v)v2 ≤ C(c∗ )h(v)v ˜ ˜ = C(c∗ )ψ(v2 ). h(v ∗ )v ≤ h(c

Since ψ has property (K) we have     1 1 ψ(v2 ) = ψ || v2 ≤ C(||)ψ v2 . || ||

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11

˜ we have By Jensen’s inequality (ψ is convex) and assumption (h1)  ψ



⎛

1 1 v2 = ψ ⎝ || ||



⎞ |v|2 ⎠ ≤



1 ||

 ψ(|v|2 ) = 

Altogether, the assertion follows with Ch˜ =

1 ||

 

1 ∗ || C(||)C(c ).

1 2 ˜ h(|v|)|v| ≤ ||

 g(|v|)|v|2 . 

2

˜  be given as in the assumptions and ε > 0 Proof of Theorem 2.1. Let the functions h, (small enough), which will be specified later. For a strong solution u of (2) let us denote v(t, x) := ut (t, x) and for convenience we abbreviate M = E  . Let us assume that h˜ is everywhere differentiable (the other case is discussed at the end of the proof). We define s E(u(t), v(t)) := (H (u(t), v(t))),

where (s) := 0

1 dξ, ˜ (ξ )h((ξ ))

s ≥ 0,

and 1 ˜ H (u, v) = v2 + E(ϕ) − E(u) − ε h(v ∗ ) M(u), v ∗ , 2

u ∈ V , v ∈ H.

(It follows from the computations below that H (u(t), v(t)) ≥ 0.) We show that E is nonincreasing along solutions and that −

d E(u(t), v(t)) ≥ cv(t)∗ dt

holds for almost all t ∈ [0, +∞) such that u(t) ∈ BV (ϕ, η), where η is taken from condition (e1), because then the convergence u(t) → ϕ as t → +∞ follows from [2, Corollary 2.9]. Let us fix t > 0 and write (u, v) instead of (u(t), v(t)). We take the scalar product in V ∗ of the equation (2) with v and with M(u), vt , v ∗ + g(|v|)v, v ∗ = M(u), v ∗ , vt , M(u) ∗ + g(|v|)v, M(u) ∗ = M(u), M(u) ∗ . Inserting this and (7) into the derivative below we compute (here we use that u is a strong solu˜ which guarantees that g(|v|)v ∈ Lp () → V ∗ , since v ∈ V → Lp ()): tion, and (G) d d d H (u(t), v(t)) = H (u(t), v(t)), ut V ∗ ,V + H (u(t), v(t)), vt dt du dv  ˜ = − M(u), ut V ∗ ,V − ε h(v ) M (u)ut , v ∗ ∗ v, vt ∗ ˜ + vt , v − ε h˜  (v∗ ) M(u), v ∗ − ε h(v ∗ ) M(u), vt ∗ v∗ 1 = − g(|v|)v, v V ∗ ,V − ε h˜  (v∗ ) M(u), v 2∗ v∗

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T. Bárta, E. Fašangová / J. Differential Equations ••• (••••) •••–•••

1  ˜ + ε h˜  (v∗ ) g(|v|)v, v ∗ M(u), v ∗ − ε h(v ∗ ) M (u)v, v ∗ v∗ ˜ ˜ − ε h(v ∗ ) M(u), M(u) ∗ + ε h(v ∗ ) g(|v|)v, M(u) ∗

(8)

1 (here and in what follows, if v(t) = 0 then any term containing v has to be replaced by 0). In ∗ the last equality we keep the first and fifth terms and estimate the other terms from above. The first term is  − g(|v|)v, v V ∗ ,V = − g(|v|)v, v Lp ,Lp = − g(|v(t, x)|)|v(t, x)|2 dx. 

˜ The third term can be estimated The second term is less or equal to zero, due to Lemma 3.1 (h5). ˜ the Cauchy–Schwarz inequality and the embedding Lp () → with the help of Lemma 3.1 (h6), V ∗ as follows |εh˜  (v∗ )

1 ∗ ˜ g(|v|)v, v ∗ M(u), v ∗ | ≤ ε h(v ∗ )M(u)∗ cp g(|v|)vp . v∗

The last, sixth term is estimated by (here we use again the Cauchy–Schwarz inequality and Lp () → V ∗ ) ∗ ˜ ˜ |εh(v ∗ ) g(|v|)v, M(u) ∗ | ≤ ε h(v ∗ )M(u)∗ cp g(|v|)vp .

The fourth term is rewritten and is estimated with the help of the Cauchy–Schwarz inequality, (e2) and relative compactness of the trajectory u, then by Lemma 4.2, and finally by choosing ε small enough, as follows    ˜ ˜ ˜ | − ε h(v ∗ ) M (u)v, v ∗ | = ε h(v ∗ )| KE (u)v, v V ,V ∗ | = ε h(v ∗ )| KE (u)v, v |   1 2 ˜ g(|v|)|v|2 ≤ g(|v|)|v|2 . ≤ ε h(v ∗ )Cv ≤ εCCh˜ 4 



Altogether, we have d H (u(t), v(t)) ≤ − dt



2 ˜ g(|v(t, x)|)|v(t, x)|2 dx − ε h(v ∗ )M(u)∗



˜ + 2εcp∗ h(v ∗ )M(u)∗ g(|v|)vp +

1 4

 g(|v(t, x)|)|v(t, x)|2 dx.

(9)



We show that the third term on the right-hand side of (9) can be dominated by the sum of the first and second terms. By Young’s inequality we have (δ is the convex conjugate to γ )  M(u)∗ g(|v|)vp ≤ δ

   1 M(u)∗ + γ Kg(|v|)vp . K

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Since M(u)∗ is bounded, and (γ 2) we have

M(u)∗ K

13

is uniformly small if K is large enough, and by Lemma 3.2



 1 d3 δ M(u)∗ ≤ 2 M(u)2∗ . K K Moreover, we have (here the first and the third steps are the definition of γ˜ , the second step is Jensen’s inequality (γ˜ is convex), the fourth step is property (K) for γ and the last step uses (γ 1)): ⎛ ⎞  γ (Kg(|v|)vp ) = γ˜ ⎝ K p |g(|v|)v|p ⎠ 

1 ≤ ||





  1 γ˜ ||K p |g(|v|)v|p = ||

1 1  C K|| p ≤ ||

 γ (|g(|v|)v|) ≤ 



  1 γ || p K|g(|v|)v|



1 1  C K|| p D1 ||

Hence, 1 d3 1  M(u)∗ g(|v|)vp ≤ 2 M(u)2∗ + C K|| p D1 || K

Taking K so large that 2cp∗ Kd32 ≤ 2εcp∗

1 2

 g(|v|)|v|2 . 

 g(|v|)|v|2 . 

and ε so small that

1 D1  1 ˜ C K|| p sup h(v(s) ∗) ≤ || 2 s≥0

we obtain ε˜ 1 2 ˜ 2εcp∗ h(v ∗ )M(u)∗ g(|v|)vp ≤ h(v ∗ )M(u)∗ + 2 2

 g(|v|)|v|2 . 

Inserting this inequality into (9) we obtain  d ε˜ 1 2 g(|v(t)|)|v(t)|2 − h(v(t) H (u(t), v(t)) ≤ − ∗ )M(u(t))∗ ≤ 0. dt 4 2 

In particular, t → H (u(t), v(t)) is nonincreasing. Since H (u(tn ), v(tn )) → H (ϕ, 0) = 0 by Lemma 4.1, we have H (u(t), v(t)) ≥ 0 and the definition of E(u(t), v(t)) is correct. We can further estimate using Lemma 4.2   d c˜ 2 2 2 ˜ H (u(t), v(t)) ≤ −ch(v ∗ ) v + M(u)∗ ≤ − h(v ∗ ) (v + M(u)∗ ) dt 2 with the constant c = min{ 4C1 , 2ε }. h˜

(10)

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14

Now we compute the derivative d d 1 · H (u(t), v(t)) E(u(t), v(t)) = ˜ dt (H (u(t), v(t)))h((H (u(t), v(t)))) dt

(11)

and see that E is nonincreasing along solutions. We estimate the right-hand side. Let us assume that t > 0 is such that u = u(t) ∈ BV (ϕ, η). We can write 1 ˜ (H (u, v)) ≤ ( v2 ) + (|E(ϕ) − E(u)|) + (εh(v ∗ )M(u)∗ v∗ ) 2 1 2 2 ˜ ˜ ≤ (v2 ) + M(u)∗ + (ε h(v ∗ )M(u)∗ ) + (ε h(v ∗ )v∗ ) C ≤ c1 (v + M(u)∗ ) , since  is nondecreasing and sublinear (first inequality), by the Kurdyka–Łojasiewicz gradient inequality, the Cauchy–Schwarz inequality, monotonicity and sublinearity of  (second inequal√ ˜ ity), since (s) ≤ C s and by boundedness of h(v ∗ )) (last inequality). Using Lemma 3.1 (h˜ is nondecreasing and has property (K)) we have ˜ ˜ (H (u, v))h((H (u, v))) ≤ c1 C(c1 )(v + M(u)∗ )h(v + M(u)∗ ).

(12)

Since v + M(u)∗ ≥ c∗ v∗ and since the function s → ˜ s is nondecreasing (this follows h(s) ˜ if h˜ is differentiable), we have, using also property (K) of h, ˜ from (h6) v + M(u)∗

˜ h(v + M(u)∗ )

≥

c∗ v∗

˜ ∗ v∗ ) h(c

≥

c∗ v∗ . ˜ C(c∗ )h(v ∗)

(13)

Altogether, inserting the estimates (10), (12) and (13) into the equality (11) we obtain the estimate for the derivative of E along solutions: −

c˜ 2 d 2 h(v∗ )(v + M(u)∗ ) ≥ c2 v(t)∗ E(u(t), v(t)) ≥ ˜ dt c1 C(c1 )(v + M(u)∗ )h(v + M(u)∗ )

for t satisfying u(t) ∈ BV (ϕ, η). Thus, the proof is done for h˜ everywhere differentiable. If h˜ is not everywhere differentiable, we need to correct two places in the above proof. First, it is not clear that dtd E(u(t), v(t)) exists for almost all t . But in fact, if dtd v(t)∗ = 0, ˜ has bounded difference quotients on a neighbourhood of t . ˜ then dtd h(v(t) ∗ ) = 0, since h d If dt v(t)∗ = 0, we can compute the left and right derivatives. If dtd v(t)∗ < 0, then d ˜ d d ˜ d ˜ ˜ dt + h(v(t)∗ ) = h− (v(t)∗ ) dt (v(t)∗ ) and dt − h(v(t)∗ ) = h+ (v(t)∗ ) dt (v(t)∗ ) and these two derivatives are equal in all points with countably many exceptions since h˜ is con˜ cave. We can proceed similarly provided dtd v(t)∗ > 0. Therefore h(v(t) ∗ ) (hence also E(u(t), v(t))) has a time derivative everywhere except on a countable set. Second, the equalities (8) hold only in the points t where h˜  (v(t)∗ ) exists. However, in the points where dtd v(t)∗ = 0, the equalities hold if we replace the terms containing h˜  by zeros

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15

˜ (since dtd h(v(t) ∗ ) = 0) and these zeros can be estimated as the original terms, so the rest of the proof remains unchanged also for these points. As we have shown in the previous paragraph, this already covers all t’s except countably many and the proof is completed. 2 5. Equation with general damping In this section we replace the assumptions (G), (H) by the following ones. The function E is the same as above. (GG) The function G : Rn × Rn → Rn is continuous and there exists τ > 0 such that (gg1) there exists C2 > 0 such that |G(w, z)| ≤ C2 |z| for all z ∈ BRn (0, τ ), w ∈ Rn , (gg2) there exists C3 > 0 such that C3 |z| ≤ |G(w, z)| for all z ∈ Rn \ BRn (0, τ ), w ∈ Rn , (gg3) if N = 2 then there exist C4 > 0, α > 0 such that |G(w, z)| ≤ C4 |z|α |z| for all 4 z ∈ Rn \ BRn (0, τ ), w ∈ Rn ; if N > 2 then the inequality holds with α = N−2 , (gg4) there exists C5 > 0 such that G(w, z), z ≥ C5 |G(w, z)||z| for all w, z ∈ Rn . (HH) For τ from condition (GG) there exists a function h : [0, +∞) → [0, +∞), which is concave and nondecreasing on [0, τ ] and satisfies (hh1) |G(w, z)| ≥ h(|z|)|z| for all z ∈ BRn (0, τ ), w ∈ Rn , 1 (hh2) the function s → (s)h((s)) belongs to L1 ((0, τ )), √ (hh3) the function ψ : s → sh( s) is convex on [0, τ 2 ]. Note that (h2), (h3) remained unchanged, (g1), (g2), (g3), (h1) were naturally reformulated for a function G(w, z) corresponding to g(|z|)z. The angle condition (gg4) was added. It means that the angle between the direction of the velocity and the direction of the damping stays away from π 2 and generalises the condition that g > 0. As in Section 3 we can define a global lower bound ˜ function h. Theorem 5.1. Let E and G satisfy (E), (GG) and (HH). Let u be a strong solution to (3) such that {(u(t), ut (t)) : t ≥ 0} is relatively compact in V × H and let ϕ ∈ ωV (u). Then limt→+∞ (u(t) − ϕV + ut (t)) = 0. Proof. The computations in Sections 3 and 4 remain valid with g(|z|)z replaced by G(w, z), ˜ g(|z|)|z|2 replaced by G(w, z), z and with three further changes. First, the inequality in (h1) n ˜ has to be replaced by |G(w, z)| ≥ h(|z|)|z|, z, w ∈ R . Second, in Proposition 3.3, (γ 1) can be proved as follows (with the help of the angle condition (g4)). – If |G(w, z)| < τ then we have γ (|G(w, z)|) = c1 |G(w, z)|2 ≤

c1 |G(w, z)| c1 G(w, z), z max{C2 , 1} G(w, z), z . ≤ C5 |z| C5

– If |G(w, z)| ≥ τ then we have γ (|G(w, z)|) = |G(w, z)|p − τ p + c1 τ 2 ≤ |z| |G(w, z)| ≤

|G(w, z)|p−1 |z|

1 p−1 1 max{C2 τ p−2 , C4 , } G(w, z), z . C5 τ

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T. Bárta, E. Fašangová / J. Differential Equations ••• (••••) •••–•••

  2≤ 2 ˜ Third, in the proof of Lemma 4.2 we replace the estimate  h(|v|)|v|  g(|v|)|v| by (for u ∈ V , v ∈ H)   ˜h(|v|)|v|2 ≤ 1 G(u, v), v . 2 C5 



6. Ordinary differential equation We may change the setting of Sections 2, 3 and 4 in the following way. Let V = H = V ∗ = and all the norms and scalar products are the norm and the scalar product in RN . We take p = 1 (the only purpose of p was to make the embedding V ∗ → Lp () continuous, now the Lp -norm is replaced by the norm in RN ). The growth condition (g3) is not needed here, since it was needed only to show condition (γ 1) in case p > 1. Condition (e2) holds trivially in this finite-dimensional setting. Of course, all integrals over  and the variable x have to be erased in the above sections. In this way we can obtain the following result. RN

Theorem 6.1. Let the functions E : RN → R, G : RN × RN → RN satisfy (e1), (gg1), (gg2), 2,1 (gg4) of (GG), and (HH). Let u ∈ W 1,∞ ((0, +∞), RN ) ∩ Wloc ([0, +∞), RN ) be a solution to ˙ = 0. (4) and let ϕ ∈ ω(u). Then limt→+∞ (u(t) − ϕ + u(t)) This result generalises our result [1, Theorem 4]. There it is assumed that G is estimated by multiples of a radially symmetric concave function g˜ from below and from above, i.e. that 2 ≤ G(w, z), z ≤ C g(|z|)|z| 2 , and we had a condition on ∇G. Moreover, we assumed cg(|z|)|z| ˜ ˜  to be concave but in fact it is sublinearity what was needed in [1, Theorem 4]. In [3] the ˙ was considered. damping |u(t)| ˙ α u(t) References [1] T. Bárta, R. Chill, E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system, Monatsh. Math. 166 (2012) 57–72. [2] T. Bárta, Convergence to equilibrium of relatively compact solutions to evolution equations, Electron. J. Differential Equations 2014 (81) (2014) 1–9. [3] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differential Equations 20 (3) (2008) 643–652. [4] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity, J. Evol. Equ. 9 (2009) 405–418. [5] R. Chill, A. Haraux, M.A. Jendoubi, Applications of the Łojasiewicz–Simon gradient inequality to gradient-like evolution equations, Anal. Appl. 7 (2009) 351–372. [6] A. Cianchi, Optimal Orlicz–Sobolev embeddings, Rev. Mat. Iberoam. 20 (2004) 427–474. [7] A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, 1991. [8] M.A. Jendoubi, P. Poláˇcik, Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. Roy. Soc. Edinburgh Sect. A 133 (5) (2003) 1137–1153.