EXACT RATE OF DECAY FOR SOLUTIONS TO DAMPED SECOND ORDER ODE’S WITH A DEGENERATE POTENTIAL

. We prove exact rate of decay for solutions to a class of second order ordinary diﬀerential equations with degenerate potentials, in particular, for potential functions that grow as diﬀerent powers in diﬀerent directions in a neigborhood of zero. As a tool we derive some decay estimates for scalar second order equations with non-autonomous damping.

1. Introduction. In this paper we study rate of convergence to equilibrium of solutions to second order ordinary differential equations of the typë u + g(u)u + ∇E(u) = 0, (DP) which describe damped oscillations of a system. We assume that the potential energy E : R n → R + has its only local minimum in the origin and g : R n → R is positive (except in the origin), so the term g(u)u has a damping effect. The scalar case with E(u) = a|u| p , g(s) = b|s| α was studied by Haraux [9] and the vector valued case with E(u) = A 1 2 u p , g(s) ∈ (c 1 |s| α , c 2 |s| α ), A being a symmetric positive linear operator on a Hilbert space H was studied by Abdelli, Anguiano and Haraux [1]. For these cases exact decay rates were derived. Let us mention, that in both cases E satisfies E(u) ∼ u p , ∇E(u), u ∼ u p on a neighborhood of zero (where f ∼ g means cf ≤ g ≤ Cf for some positive constants c, C and ·, · is the scalar product on H).
In [5] similar decay estimates as in [9], [1] were derived with the assumptions formulated in terms of the Lojasiewicz gradient inequality, namely for E satisfying and ∇ 2 E(u) ≤ ∇E(u) 1−2θ 1−θ on a neighborhood of zero. The right inequality in (1) is called the Lojasiewicz gradient inequality. Let us mention that the potential functions E from [9], [1] satisfy (1) and also the condition on ∇ 2 E with θ = 1 p . The goal of this paper is to study degenerate cases, where the above assumptions do not hold, e.g. the behavior of E is not power-like or E does not satisfy the left 532 TOMÁŠ BÁRTA inequality in (1) with the same θ as the right inequality 1 . A prototype of such E is with u = (u 1 , . . . , u k ) ∈ R n (u i ∈ R ni are not necessarily scalars, n i = n) and p 1 ≥ p 2 ≥ · · · ≥ p k ≥ 2 are not all equal. We show that in such cases we obtain the same estimates (from above and from below) as for E(u) = u p1 .
Further, we study the exact decay for the case where u i in (2) are scalars. In the case studied in [9] and [1] the authors have shown that if α > 1− 2 p (i.e. the damping function is smaller than a threshold), then the solutions oscillate and all solutions converge to the origin with the same speed. On the other hand, if α < 1 − 2 p (the damping function is larger than the threshold), then the solutions do not oscillate and there appear solutions with exactly two rates of convergence called fast solutions and slow solutions (see also [2] for existence of slow solutions). We show similar results for the degenerate case, in particular we show that for E given by (2) with u i being scalars, at most n + 1 speeds of convergence occur (depending on p i 's).
While studying the exact decay for solutions to (DP) we look at the equations for single coordinates of ü Since we assume E to be in the special form (2) (a slightly more general case is considered below), these equations are coupled only by the term g(u)u i . So, we consider these coordinate equations as non-autonomous problems where the dependence of g on other coordinatesu i , i = j is hidden in the dependence on t, in particular, g j is defined by g j (s, t) = g((u 1 (t), . . . ,u j−1 (t), s,u j+1 (t), . . . ,u n (t))).
Therefore, we also give results on decay and oscillations for non-autonomous equations of the type (3) that may be of interest on their own. The results for α < 1 − 2 p are again similar to those in [9], [1]. Decay estimates for another type of nonautonomous damping were derived in [3], [7], [10]. The paper is organized as follows. In Section 2 we present basic definitions and assumptions valid throughout the rest of the paper. Section 3 is devoted to the scalar autonomous problems and Section 4 to scalar non-autonomous problems. The results in this section are based on comparison with the autonomous case. The degenerate vector-valued problem (DP) is studied in Section 5.
2. Basic definitions and preliminaries. In this paper we study three types of equations: the scalar autonomous problem the scalar non-autonomous problem and the degenerate vector valued problem (DP). The assumption on g ∈ C(R) for (AP), resp. g ∈ C(R n ) for (DP) is (here |s| denotes the Euclidean norm of s if s ∈ R n ). In the non-autonomous case we assume only g ∈ C(R × R + ), for some α ∈ (0, 1), c g , C g > 0 and all s in any bounded set (with c g , C g depending on the set), and all t ≥ 0 in case of (Gn). The potential function E ∈ C 2 (R) in (AP), (NP) is assumed to satisfy for some p ≥ 2, c E , C E > 0 and all s in a bounded neighbourhood of the origin. In case of (DP) we assume E ∈ C 2 (R n ) is of the form where u = (u 1 , u 2 , . . . , u n ), E i ∈ C 2 (R) satisfy (E) with exponents p i respectively, and p 1 ≥ p 2 ≥ · · · ≥ p n ≥ 2. By a solution to (AP), (NP), (DP) we always mean a classical solution defined on R + . If u (resp. (u 1 , . . . , u n )) is a solution to one of these equations, then v (resp. (v 1 , . . . , v n )) always denotes its velocity, i.e. v =u (resp. v i =u i , i = 1, . . . , n). We denote This function is non-increasing along solutions since whenever u is a solution to any of the studied equations. Sometimes, we write E(t) instead of E(u(t), v(t)). If α ≥ 1 − 2 p (p, α from (E), (Gn), (G)), we speak about the oscillatory case, otherwise we speak about the non-oscillatory case. In the non-oscillatory case, we say that the solution u is a fast solution if it converges to zero and lim t→+∞ v(t) E(u(t)) = +∞ (i.e. the kinetic energy is much bigger than the potential energy of u as t tends to infinity). On the other hand, u si called a slow solution if it converges to zero and lim t→+∞ v(t) E(u(t)) = 0. Let us now present two easy lemmas that show that the fast solutions converge to zero faster than slow solutions and how the speed of convergence depend on the trajectory in the uv plane, i.e. on the ratio of u(t) and v(t) . Let X(a, b) = {u ∈ C 2 ((a, b)) :u > 0 on (a, b)}. By trajectory of u we mean the function Then u 2 needs more time than u 1 to get from x to y, i.e. if u 1 (t 1 ) = x = u 2 (t 2 ) and u 1 (s 1 ) = y = u 2 (s 2 ), then Proof. We have for i = 1, 2 du.

TOMÁŠ BÁRTA
The assertion now follows easily from V u1 ≥ V u2 (resp. V u1 > V u2 on a neighborhood of x).
Dividing by |u(t)| a and integrating from t 0 to t we get in the theorems and lemmas below we mean that there exist C > 0, T > 0 such that the inequality holds for all t ≥ T .
3. Scalar autonomous problem. In this section we study the autonomous problem (AP). We assume that g satisfies (G) and E satisfies (E) with α < 1 − 2 p , which is the non-oscillatory case. We first formulate the main result, Theorem 3.1. In fact, it is a minor generalization of results proved by Haraux in [9]. However, important are the lemmas below leading to the proof of the theorem. They are needed in the next section for investigation of the non-autonomous problem.
Then all solutions converge to zero and do not oscillate (e.g. u, v change sign only finitely many times). Further, any solution to (AP) is either fast or slow. Moreover, every fast solution satisfies and every slow solution satisfies We first show that some sets are positively invariant for solutions of (AP), namely the sets O ε,K , N ε,K , P δ,η defined below.
Then the sets EXACT RATE OF DECAY FOR DAMPED SECOND ORDER PROBLEMS 535 are positively invariant for solutions u of (AP). Moreover, any solution in O ε,K is a slow solution, it enters the set N ε,K , and satisfies (5).
It remains to investigate the upper part of the boundary, i.e. v(t) Lemma 3.3. There exist δ, η > 0 such that the set is positively invariant for (AP) and any solution to (AP) with (u(t 0 ), v(t 0 )) ∈ P δ,η for some t 0 > 0 is a slow solution and satisfies (5).
. Then for any u ∈ [−δ, 0], inequality (6) holds with (ε, K) = (u, K(u)) and therefore (by Lemma 3.2) the set O −u,K(u) is positively invariant. We have for all t ≥ t 0 and by Lemma 3.2 it is a slow solution and satisfies (5).
Lemma 3.4. Let us consider two sequences (u n ), (v n ) satisfying lim u n = 0 and u n < 0, 0 < v n < M |u n | p Proof. Let δ, η be the constants form Lemma 3.3, then obviously u(t n ) ≥ −δ for all n large enough and Proposition 3.5. Let u be a solution to (AP) satisfying u < 0, v =u > 0 on (T 0 , +∞) and (u(t), v(t)) → (0, 0). Then u is either fast solution or slow solution.
In the latter case, u satisfies (5).
Proof. If u is not a fast solution, then there exists M > 0 such that v(tn) for a sequence t n +∞. By Lemma 3.4 we have (u(t n ), v(t n )) ∈ P δ,η for large n. Hence u is a slow solution by Lemma 3.3 and (5) holds.

Proof. By Lemma 3.3, any fast solution satisfies
Proof of Theorem 3.1. Convergence to zero follows from Theorem 4.1 below and absence of oscillations follows from Proposition 4.4 below. Then any solution satisfies u < 0, v > 0 on (T, +∞) or symmetrically u > 0, v < 0. By Proposition 3.5, any solution is slow or fast and slow solutions satisfy (5). By Lemma 3.6, fast solutions satisfy (4).
4. Nonautonomous damping. In this section we study the non-autonomous problem (NP). We keep the assumption (E) and assume that g satisfies (Gn). We show that for g bounded all solutions converge to zero and that they do not oscillate if α < 1 − 2 p . Then we study decay of the non-oscillatory solutions. Theorem 4.1. Let g satisfy (Gn) for some α ≥ 0 and g(s, t) ≤ M for all s from a bounded set and all t ≥ 0. Then any solution to (NP) converges to zero as t → +∞.
Proof. Let u be a solution to (NP).
Proposition 4.4. Let g satisfy (Gn) for some α < 1 − 2 p and let u be a solution to (NP) such that lim t→+∞ u(t) = 0. Then u does not oscillate, i.e. u,u do not change sign on (t 0 , +∞) for some t 0 ≥ 0.
Proof. Let us assume for contradiction that a solution u to (NP) oscillates, i.e. there exists a sequence t n +∞ such that u(t n ) = 0 or v(t n ) = 0. We show that for every ε > 0 there exists T ε such that v(T ε ) = 0 and |u(T ε )| ≤ ε. In fact, if v(t) = 0 on some (T, +∞), then u would be monotone on (T, +∞) and it would be a contradiction with existence of t n . So, there exists a sequence s n +∞ with v(s n ) = 0 and since any solution converges to zero, for large n we have |u(s n )| ≤ ε.
In the following we consider only solutions satisfying u < 0, v =u > 0 on (T, +∞). We now formulate and prove two main theorems of this section. Theorem 4.5 is applied in the next section. In fact, it says that any fast solution converges faster than any slow solution, even for solutions to different problems with the same α (and possibly different p's). Theorem 4.6 says that if the non-autonomous part of the damping is smaller than the natural damping given by the velocity of slow solutions to the corresponding autonomous problem, then the non-autonomous part does not influence the decay.
Proof. Let u be a solution that is not fast. Then there exists M > 0 such that v(tn) ≤ M for a sequence t n +∞. By Lemma 3.4, there exists n ∈ N such that (u(t n ), v(t n )) ∈ P δ,η . Let us consider the solution u 1 of the autonomous problem (AP) with u 1 (t n ) = u(t n ), v 1 (t n ) = v(t n ). By Lemma 3.3, u 1 is a slow solution to (AP) and it satisfies (5) , v(t)) belongs to O ε,K for some ε, K, what we use in the next Theorem).
Let u be a fast solution. Then v(t) > η|u(t)| 1 1−α on some interval (T, +∞) (otherwise, we would proceed as in the first paragraph of this proof and obtain that u is a slow solution). By Lemma 2.2 we have . Therefore, (again by Lemma (2.2)) we obtain v(t) ≤ Ct − 1 α .
Proof. In the proof of Theorem 4.5 we have shown that any slow solution (u(t), v(t)) belongs to O ε,K for some ε, K and all t ≥ t n . Let us set T = t n and take ε > 0 such that v(T ) > εT − p−1 p−2−α . We show that v(t) > εt − p−1 p−2−α for all t > T . In fact, and by Theorem 4.5 a contradiction with the definition of t 0 . Hence, v(t) > εt − p−1 p−2−α holds on (T, +∞), and therefore g(v, t)v < C(ε)v α+1 on (T, +∞). Now, if we compare the solution u with the solution u 2 ofü+C(ε)u α+1 −p|u| p−1 = 0, 5. Degenerate potential. In this section we investigate the problem (DP). We assume that g satisfies (G) and E ∈ C 2 (R n ) is of the form where u = (u 1 , u 2 , . . . , u n ), E i ∈ C 2 (R) satisfy (E) with exponents p i respectively, and p 1 ≥ p 2 ≥ · · · ≥ p n ≥ 2. Then (DP) can be written as the following system of equations for u = (u 1 , . . . , u n ) The equations are coupled only by the term g(u). Below, we often write E(t) instead of E(u(t),u(t)). Let us start with the decay estimates for solutions of (DP).
Theorem 5.1. Let u be a solution to (DP). If α ≥ 1 − 2 p1 , then If α < 1 − 2 p1 , then Remark 5.2. Let us remark that Theorem 5.1 remains valid (with the same proof) if u i are vector valued functions with values in R ni , g : R ni → R, and E i : R ni → R. In this case, |s| in conditions (E), (G) denotes the Euclidean norm of s. We can also assume that E i satisfy (1) and ∇ 2 E(u) ≤ ∇E(u) instead of (E). Then Theorem 5.1 remains valid with a similar proof where we define H j (t) = E j (t) + ε ∇E j (u j (t)) βj ∇E j (u j ), v j with appropriate β j 's, cf. [5].
Proof of Theorem 5.1. Let u = (u 1 , . . . , u n ) be a solution to (DP). Let us define and otherwise. The last term in the definition of H j is estimated by (we write u j instead of u j (t)) where we applied the Young inequality, then E j (u) ∼ u pj and finally 2(β j + 1) ≥ p j and boundedness of E j (u j (t)). It follows that H j (t) ∼ E j (t). Further, we have (we On the other hand, the right-hand side of (15) is bounded from above if 2B ≤ α + 2, Bp j ≤ β j + p j for all j. Here, the best choice (largest possible B) is always B = α+2 2 (for both oscillatory and non-oscillatory coordinates) and we obtain which completes the proof.
From now on, let us assume that u i are scalar valued. For a solution u = (u 1 , . . . , u n ) and any fixed i ∈ {1, 2, . . . , n} let us denote f j (t) = i =ju 2 i (t) ≥ 0. Then u j solves the nonautonomous problem (3) with so (Gn) is satisfied. Moreover, by Theorem 5.1 we know that every solution converges to zero. Now, we can apply the results from the previous section to obtain more gentle properties of solutions. In particular, we show that each solution to (DP) has one of (at most) n + 1 speeds of convergence to the origin that are given by fast and slow solutions of the equations (11). First of all, by Theorem 4.5 we have the following.
Corollary 5.3. Let u = (u 1 , . . . , u n ) be a solution to (DP). If i ∈ {1, . . . , n} is such that α < 1 − 2 pi , then u i does not oscillate. Moreover, for such i, the function u i (as a solution of (3)) is either fast and satisfies So, we speak about a non-oscillatory coordinate if α < 1 − 2 pi and about oscillatory coordinate if α ≥ 1 − 2 pi (we do not know whether the oscillatory coordinates really oscillate) and a non-oscillatory coordinate of a particular solution can be called slow coordinate or fast coordinate. We now show that there appear at most n + 1 different rates of convergence of solutions to (DP), in particular, if there are k non-oscillatory coordinates, then each solution has one of the k + 1 possible decay rates.
pj for all j). Then for any solution to (DP) its energy satisfies for some j ∈ {1, . . . , m}. Moreover, to each of the m + 1 decay rates there exists a solution with this decay.
Proof. The moreover part is easy. If all coordinates except u j are zero, then u j satisfies (AP). Hence, by [9], the function u j decays as t − α 2 if it is an oscillatory coordinate and if it is a non-oscillatory coordinate, then it is a slow solution with α . Existence of slow solutions follows from Lemma 3.2, existence of fast solutions was proved in [9, Theorem 3.4] for g(s) = c|s| α , E(u) = |u| p and the general case can be proved by modifying that proof. It remains to show that no other speeds of convergence appear.