Mildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay

Transcription

1 arxiv:93.273v [math.ap] 6 Mar 29 Mildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay Marina Ghisi Università degli Studi di Pisa Dipartimento di Matematica Leonida Tonelli PISA Italy) ghisi@dm.unipi.it Massimo Gobbino Università degli Studi di Pisa Dipartimento di Matematica Applicata Ulisse Dini PISA Italy) m.gobbino@dma.unipi.it

2 Abstract We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t +. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t +, with the same rate of the solution of the limit problem of parabolic type. Mathematics Subject Classification 2 MSC2): 35B25, 35B4, 35L7, 35L8. Key words: singular perturbation, degenerate Kirchhoff equations, quasilinear hyperbolic equation, weak dissipation, energy estimates.

3 Introduction Let H be a real Hilbert space. For every x and y in H, x denotes the norm of x, and x, y denotes the scalar product of x and y. Let A be a self-adjoint linear operator on H with dense domain DA). We assume that A is nonnegative, namely Ax, x for every x DA), so that for every α the power A α x is defined provided that x lies in a suitable domain DA α ). We consider the Cauchy problem εu ε t) + + t) pu ε t) + A/2 u ε t) 2γ Au ε t) = t,.) u ε ) = u, u ε ) = u,.2) where ε >, p, and γ >. Equation.) is the prototype of all degenerate Kirchhoff equations with weak dissipation εu ε t) + bt)u ε t) + m A/2 u ε t) 2 )Au ε t) = t,.3) where b : [, + ), + ) and m : [, + ) [, + ) are given functions which are always assumed to be of class C or at least locally Lipschitz continuous), unless otherwise stated. It is well known that.) is the abstract setting of a quasilinear nonlocal partial differential equation of hyperbolic type which was proposed as a model for small vibrations of strings and membranes. Equation.3) is called nondegenerate or strictly hyperbolic) when µ := inf mσ) >, σ and mildly degenerate when µ = but m A /2 u 2 ). In the special case of equation.) this assumption reduces to A /2 u..4) Concerning the dissipation term bt)u εt), we have constant dissipation when bt) δ > is a positive constant, and weak dissipation when bt) as t +. Finally, the operator A is called coercive when { } Ax, x ν := inf : x DA), x >,.5) x 2 and noncoercive when ν =. The singular perturbation problem in its generality consists in proving the convergence of solutions of.3),.2) to solutions of the first order problem bt)u ε t) + m A/2 u ε t) 2 )Au ε t) =, u) = u,.6)

4 obtained setting formally ε = in.3), and omitting the second initial condition in.2). The singular perturbation problem gives rise to several subproblems. The first step is of course the existence of global solutions for the limit problem.6). This has been established in [] under very general assumptions. The second step is the existence of a global solution for the hyperbolic problem.3),.2). The third step is the convergence of solutions u ε t) of the hyperbolic problem to the solution ut) of the parabolic problem. The final goal are the so called error-decay estimates which prove in the same time that the difference u ε t) ut) decays to as t + with the same rate of ut)), and tends to as ε +. The second and third step have generated a considerable literature, which we sum up below. Nondegenerate Kirchhoff equations with constant dissipation The first results where obtained in the eighties by E. De Brito [2] and Y. Yamada [2]. They independently proved the global solvability of the hyperbolic problem with initial data u, u ) DA) DA /2 ) under a suitable assumption involving ε, the initial data, and the constant dissipation δ. Once that δ and the initial data are fixed, this condition holds true provided that ε is small enough. The key step in their proofs, as well as in all the subsequent literature, is to show that solutions satisfy an a priori estimate such as m A /2 u ε t) 2 ) ε u m A /2 u ε t) 2 ε ) t) Au εt) bt),.7) which is clearly more likely to be true when ε is small enough. Existence of global solutions without the smallness assumption on ε remains a challenging open problem, as well as the nondissipative case bt). More recently H. Hashimoto and T. Yamazaki [2] obtained optimal error-decay estimates for the singular perturbation problem. Thanks to these estimates, which improve or extend all previous works see [3, ]), this case can be considered quite well understood. Degenerate Kirchhoff equations with constant dissipation The case where mσ) = σ γ with γ ) has been studied in the nineties by K. Nishihara and Y. Yamada [6] see also [2]). The result is the existence of a unique global solution for the mildly degenerate equation provided that ε is small enough. Later on this existence result was extended by the authors [6] to arbitrary locally Lipschitz continuous nonlinearities mσ), and by the first author [4, 5] to non-lipschitz nonlinearities of the form mσ) = σ γ with γ, ). All the quoted papers considered also the asymptotic behavior of solutions, but the estimates proved therein where in general far from being optimal. In the meanwhile 2

5 sharp decay estimates were the subject of a series of papers by T. Mizumachi [3, 4]) and K. Ono [7, 8]), in which however only the special case mσ) = σ was considered. More recently the authors [7] obtained optimal and ε-independent decay estimates for the general case. As expected the result is that solutions of the hyperbolic problem always decay as the corresponding solutions of the limit problem. As for the singular perturbation problem, in [8] the authors proved that u ε t) ut) uniformly in time, but without sharp error-decay estimates, which in this case remain an open problem. From the technical point of view, the difficulty is that in the degenerate case the denominator in.7) may vanish. This cannot happen for t = due to the mild nondegeneracy assumption, but it does happen in the limit as t + due to the decay of solutions. Nondegenerate Kirchhoff equations with weak dissipation Let us come to the nondegenerate case with bt) = +t) p. What complicates things is the competition between the smallness of ε and the smallness of bt). In particular it is no more enough to prove that the left-hand side of.7) is bounded, but it is necessary to prove that it decays with an a-priori fixed rate. This is the reason why this problem has been solved only in recent years in some papers by M. Nakao and J. Bae [5], by T. Yamazaki [22, 23], and by the authors [9]. The result is that for every p [, ], and every u, u ) DA) DA /2 ), the hyperbolic problem has a unique global solution provided that ε is small enough. Moreover the solution decays to as t + as the solution of the limit problem, and optimal errordecay estimates for the singular perturbation can be proved. When p > existence of global solutions for the hyperbolic problem is still an open problem, but in any case solutions cannot decay to as t +. On the other hand, solutions of the limit parabolic problem decay to zero also for p >, faster and faster as p grows. This means that a threshold appears. When p [, ] the smallness of ε is dominant over the smallness of bt), and.3) behaves like a parabolic equation. When p > the smallness of bt) is dominant over the smallness of ε, and.3) behaves like a nondissipative hyperbolic equation. Degenerate Kirchhoff equations with weak dissipation Let us finally come to equation.), which is the object of this paper. Now in.7) the smallness of ε has to compete both with the decay of bt), and with the vanishing of the denominator. So one needs a priori decay estimates for terms whose denominator vanishes in the limit. To our knowledge the only previous result for this equation was obtained at the end of the nineties by K. Ono [9], who proved global existence for the mildly degenerate case when γ =, p [, /3], and of course ε is small enough. We recall that mσ) = σ is the only case where sharp decay estimates were already available in those years. 3

6 In this paper we consider the global solvability of.),.2) with more general values of the parameters, and initial data u, u ) DA) DA /2 ) satisfying.4). Our first result Theorem 2.) concerns the coercive case. Under this assumption we prove that a unique global solution exists provided that γ >, p [, ] and ε is small enough. We also prove sharp decay estimates as t +. Our second result Theorem 2.2) concerns the noncoercive case. In this case we prove that a unique global solution exists provided that γ, p [, γ 2 + )/γ 2 + 2γ )], and ε is small enough. Note that the supremum of this interval is both when γ = and when γ +, but it is strictly smaller than for γ >. We also provide decay estimates for solutions. Finally in both cases we show Theorem 2.3) that for p > solutions of.) provided that they exist, which remains an open problem) do not decay to as t +. From the point of view of global solvability and decay properties these results show that in the coercive case equation.) behaves like the nondegenerate one, exhibiting nondissipative hyperbolic behavior for p >, and parabolic behavior for p [, ]. We point out that this is true also in the non-lipschitz case γ, ). In the noncoercive case with γ ) we have once again hyperbolic behavior for p >, and parabolic behavior for p [, γ 2 + )/γ 2 + 2γ )]. Proofs rely on the techniques introduced in [7] in order to prove sharp decay estimates. When the operator is coercive the decay rate depends only on p and γ. When the operator is noncoercive the decay rate belongs to a range depending on p and γ, but within this range it seems to depend on the initial conditions. The existence of a range of possible decay rates is what in the noncoercive case creates the no-man s land between γ 2 + )/γ 2 + 2γ ) and. What happens when p lies in this interval is not clear yet. In this paper we don t consider the behavior of solutions as ε +, even if all our ε-independent estimates are for sure a first step in this direction. We just mention that a simple adaptation of the arguments of [] and [8] should be enough to prove two types of result: that u ε u uniformly in time without estimates of the convergence rate), and that u ε u in every interval [, T] with an estimate of the error depending on T. On the other hand, obtaining error-decay estimates analogous to the nondegenerate case seems to be a much more difficult task. Apart from the partial results of [8] this problem is still open in the degenerate case, both with constant and with weak dissipation. 2 Statements Our first result concerns the global solvability of the hyperbolic problem and decay properties of solutions in the case of coercive operators. 4

7 Theorem 2. Let H be a Hilbert space, and let A be a nonnegative selfadjoint unbounded) operator with dense domain. Let us assume that A satisfies the coerciveness condition.5). Let γ >, and let p [, ]. Let us assume that u, u ) DA) DA /2 ) satisfy.4). Then there exists ε > such that for every ε, ε ) problem.),.2) has a unique global solution u ε C 2 [, + ); H) C [, + ); DA /2 )) C [, + ); DA)). 2.) Moreover there exist positive constants C and C 2 such that C + t) p+)/γ A/2 u ε t) 2 C + t) p+)/γ Au εt) 2 u ε t) 2 C 2 + t) p+)/γ t ; 2.2) C 2 + t) p+)/γ t ; 2.3) C 2 + t) 2+p+)/γ t. 2.4) Our second result in the counterpart of Theorem 2. in the case of noncoercive operators. Theorem 2.2 Let H be a Hilbert space, and let A be a nonnegative selfadjoint unbounded) operator with dense domain. Let γ, and let p γ2 + γ 2 + 2γ. 2.5) Let us assume that u, u ) DA) DA /2 ) satisfy.4). Then there exists ε > such that for every ε, ε ) problem.),.2) has a unique global solution satisfying 2.). Moreover there exist constants C and C 2 such that C + t) p+)/γ A/2 u ε t) 2 Au ε t) 2 C 2 + t) p+)/γ+) t ; 2.6) C 2 + t) p+)/γ t ; 2.7) u ε t) 2 C 2 + t) [2γ2 + p)γ+p+]/γ 2 +γ) t. 2.8) The last result of this paper concerns the case p >. An analogous result holds true for nondegenerate equations see [9, Theorem 2.3]). 5

8 Theorem 2.3 Let H and A be as in Theorem 2.2. Let m : [, + ) [, + ) be a continuous function. Let b : [, + ), + ) be a continuous function such that + Let u, u ) DA) DA /2 ) be such that u 2 + bs) ds < ) A /2 u 2 mσ) dσ >. 2.) Let us assume that for some ε > problem.3),.2) has a global solution u ε satisfying 2.). Then lim inf u ε t) 2 + A /2 u ε t) 2) >. 2.) t + Remark 2.4 The constants ε, C, C 2 given in Theorem 2. above may be taken as continuous functions of ν, γ, p, u, A /2 u, A /2 u, Au, u, A /2 u. The same is true for the constants ε, C, C 2 given in Theorem 2.2, apart from the fact that in this case there is no dependence on ν. Remark 2.5 Our results can be easily extended to more general Kirchhoff equations. For example with the same technique we can deal with nonlinearities m : [, + ) [, + ) of class C such that c σ γ mσ) c 2 σ γ, c σ γ m σ) c 2 σ γ in a right-hand neighborhood of σ = for suitable positive constants c and c 2. However this generality only complicates proofs without introducing any new idea. Remark 2.6 In Theorem 2. we assume that γ >, while in Theorem 2.2 we assume that γ. Some weaker results can be obtained with similar techniques also when the operator is noncoercive and γ >. For example for every γ > one can prove the global solvability for every p [, γ/γ + 2)] and of course ε small enough). The solution also satisfies 2.6). We sketch the argument in Remark 3.4. Note that when γ the upper bound γ/γ + 2) is always less than the upper bound in 2.5). 3 Proofs Proofs are organized as follows. First of all in 3. we state and prove two simple comparison results for ordinary differential equations, which we need several times in the sequel. Then we prove Theorem 2. and Theorem 2.2. Their proofs have a common part, which we concentrate in 3.2 in the form of an a priori estimate Proposition 3.3). Then in 3.3 we conclude the proof of Theorem 2., and in 3.4 we conclude the proof of Theorem 2.2. Finally, in 3.5 we prove Theorem

9 3. Comparison results for ODEs Numerous variants of the following comparison result have already been used in [4, 5, 6, 7, 8, 9]. Lemma 3. Let T >, let p, and let f : [, T] [, + ) be a function of class C. Let us assume that there exist two constants c >, c 2 such that f t) c + t) pft) + c 2 ft) t [, T]. 3.) Then we have that ft) f) + c2 Proof. From 3.) if follows that c ) 2 + t) 2p t [, T]. 3.2) f t) c 2 + t) pft) + c2 2 2c + t) p, which is equivalent to say that ft) is a subsolution of the differential equation y t) = c 2 + t) pyt) + c2 2 2c + t) p. 3.3) Let gt) denote the right-hand side of 3.2). Then c 2 + t) pgt) + c2 2 2c + t) p g t), which is equivalent to say that gt) is a supersolution of 3.3). Since f) g) the conclusion follows from the standard comparison principle between subsolutions and supersolutions. The next comparison result looks quite technical, but the basic idea is the following. When ft) the differential inequalities 3.5) and 3.7) can be explicitly integrated, thus providing estimates for wt). When ft) is small enough according to 3.4) estimates of the same order can still be proved. The interested reader is referred to [7, Lemma 4.2] for a similar comparison result. Lemma 3.2 Let p, γ >, α >, and T > be real numbers. Let w : [, T] [, + ) be a function of class C with w) >, and let f : [, T] R be a continuous function. Let us assume that { } + s) p fs) ds min 4γ[w)], α + t) p+ t [, T]. 3.4) γ 2p + ) Then we have the following implications. 7

10 ) If w satisfies the differential inequality w t) 2 + t) p [wt)] +γ α + ft)) t [, T], 3.5) then we have the following estimate [ { p + wt) w) max 2, αγ[w)] γ 2) If w satisfies the differential inequality }] /γ + t) p+)/γ t [, T]. 3.6) w t) 2 + t) p [wt)] +γ α + ft)) t [, T], 3.7) then we have the following estimate ] /γ 3αγ [w)]γ wt) w) [ + p + + t) p+)/γ t [, T]. 3.8) Proof. Let yt) be the solution of the Cauchy problem y t) = 2 [yt)] γ+, It is easy to see that yt) = w) + 2γ[w)] γ t ) /γ For every t [, T] let us set y) = w). t > 2γ[w)]. γ Φt) := α p + [ + t) p+ ] + + s) p fs) ds, zt) := y Φt)). First of all we have to prove that zt) is well defined. This is true because if we set { } C := min 4γ[w)], α, γ 2p + ) then from 3.4) we have that ) α [ Φt) p + C + t) p+ ] C C > 2γ[w)]. γ Moreover a simple calculation shows that zt) is a solution of the Cauchy problem z t) = 2 + t) p [zt)] γ+ α + ft)) t [, T], 3.9) z) = w). 3.) 8

11 Proof of statement ) Assumption 3.5) is equivalent to say that wt) is a subsolution of the Cauchy problem 3.9), 3.). The usual comparison principle implies that wt) zt). Now we have to estimate zt). From 3.4) it follows that α + 2γ[w)] γ Φt) + 2γ[w)] γ p + C + 2γ[w)] γ α 2p + ) = 2 + αγ[w)]γ [ + t) p+ ] min p + { 2, αγ[w)]γ p + ) [ + t) p+ ] C) [ + t) p+ ] } + t) p+, where in the last step we exploited the elementary inequality It follows that A + Bx ) min{a, B}x A, B, x. wt) zt) = w) [ + 2γ[w)] γ Φt)] /γ [ { }] /γ p + w) max 2, αγ[w)] γ + t), p+)/γ which is exactly 3.6). ) 4γ[w)] γ Proof of statement 2) Assumption 3.7) is equivalent to say that wt) is a supersolution of the Cauchy problem 3.9), 3.), hence wt) zt) for every t [, T]. Since + 2γ[w)] γ Φt) + t) p+ + 2γ[w)] γ α + 3αγ[w)]γ p + ) + t) p+, ) p + + C + t) p+ the conclusion follows as in the previous case. 3.2 Basic energy estimates In this section we prove some energy estimates and a lower bound for A /2 u ε t). Such estimates don t require the coerciveness of the operator, and they are fundamental both in the proof of Theorem 2. and in the proof of Theorem 2.2. They extend to 9

12 the weakly dissipative equation the estimates stated in [7, section 3.4] in the case of constant dissipation. The estimates involve the following energies F ε t) := ε A/2 u εt) 2 A /2 u ε t) 2γ + Au εt) 2, 3.) P ε t) := ε A/2 u ε t) 2 A /2 u ε t) 2 Au ε t), u ε t) 2 + Au εt) 2 A /2 u ε t) 2γ+4 A /2 u ε t) 2, 3.2) Q ε t) := u ε t) 2 A /2 u ε t) 4γ+2, 3.3) R ε t) := ε A/2 u ε t) 2 A /2 u ε t) + Au εt) 2 2γ+2 A /2 u ε t) ) We point out that the first summand in the definition of P ε t) is nonnegative due to the Cauchy-Schwarz inequality. We state the result in the form of an a priori estimate. We assume that in some interval [, S) there exists a solution of the hyperbolic problem satisfying a given estimate see 3.5) below), and we deduce that this solution satisfies several energy inequalities in the same interval. We point out that all constants do not depend on S. Proposition 3.3 Let H and A be as in Theorem 2.2. Let γ >, p [, ], K > be real numbers, and let u, u ) DA) DA /2 ) satisfy.4). Then there exists positive constants ε, σ, σ with the following property. If ε, ε ), S >, and u ε C 2 [, S); H) C [, S); DA /2 )) C [, S); DA)) is a solution of.),.2) such that A /2 u ε t) and Au εt), u εt) A /2 u ε t) 2 K + t) p t [, S), 3.5) then for every t [, S) we have that F ε t) + + s) p A /2 u εs) 2 A /2 u ε s) 2γ ds F ε); 3.6) P ε t) P ε ); 3.7) Q ε t) Q ε ) + 4P ε ) + t) 2p ; 3.8)

13 + t) 2p R ε t) + + s) p A /2 u ε s) 2 ds A /2 u ε s) 2γ+2 [R ε ) + 2K + )P ε )] + t) p+ ; 3.9) + s) p u εs), Au ε s) ds A /2 u ε s) 2γ+2 σ + t) p+ ; 3.2) A /2 u ε t) 2 σ. + t) p+)/γ 3.2) Proof. Let us set σ := u, Au A /2 u 2γ P )Q ) + 2P )) + 2γ + 3) R ) + 2K + )P )), σ := A /2 u 2 + 3γP ) A /2 u 2γ p + Let us choose ε in such a way that { 4ε, 4ε Kγ + ), σ ε min ) /γ. 3.22) 4γ A /2 u, P ) 2γ 2p + ) }. 3.23) Proof of 3.6) through 3.9) Let us compute the time derivative of the energies 3.) through 3.4). After some computations we find that F ε = 2 + t) + p γε u ε, Au ) ε A /2 u ε 2, 3.24) A /2 u ε 2 A /2 u ε 2γ P ε = 2 + t) + γ + ε, Au ε p 2)ε u A /2 u ε 2 Q ε = 2 ε + t) + 2γ + p )ε u ε, Au ε A /2 u ε 2 R ε = 2 + t) + γ + ε, Au ε p )ε u A /2 u ε 2 ) A /2 u ε 2 A /2 u ε 2 Au ε, u ε 2, 3.25) A /2 u ε 2γ+4 ) Q ε 2 u ε, Au ε ε A /2 u ε 2γ+2, 3.26) ) A /2 u ε 2 A /2 u ε 2γ+2 2 u ε, Au ε Au ε 2 A /2 u ε ) Thanks to assumption 3.5) and the second inequality in 3.23) we have that F εt) A /2 u ε t) 2 ; 3.28) + t) p A /2 u ε t) 2γ P ε t) ; 3.29)

14 Q ε t) ε + t) pq εt) 2 u ε t), Au εt) ε A /2 u ε t) 2γ+2; 3.3) R ε t) 3 A /2 u εt) t) p A /2 u ε t) εt), Au ε t) Au ε t) 2 2γ+2 2 u. 3.3) A /2 u ε t) 4 Integrating 3.28) in [, t] we obtain 3.6). Conclusion 3.7) trivially follows from 3.29). From 3.3) we deduce that Q ε t) ε + t) pq εt) + 2 u εt) Au ε t) ε A /2 u ε t) 2γ+2 ε + t) pq εt) + 2 Pε ) Q ε t). ε Therefore applying Lemma 3. we obtain 3.8). From 3.3) we have that [ + t) 2p R ε t) ] = 2p + t) 2p R ε t) + + t) 2p R ε t) 2p + t) 2p ε A/2 u ε 2 A /2 u ε 2γ+2 + 2p + t)2p Au ε 2 A /2 u ε A /2 u t)p ε 2 A /2 u ε 2 + Au ε 2 u 2γ+2 t)2p ε, Au ε A /2 u ε 2 A /2 u ε 2 =: I t) + I 2 t) + I 3 t) + I 4 t). Since 2p p, and 2pε 2ε /2, we have that I t) + I 3 t) 2pε 3 ) + t) p A /2 u εt) 2 2 A /2 u ε t) + A /2 u εt) 2 2γ+2 t)p A /2 u ε t) 2γ+2. From 3.5), 3.7), and the fact that 2p p we have that I 2 t) + I 4 t) 2K + p) + t) p Au εt) 2 It follows that [ + t) 2p R ε t) ] + t) p A /2 u εt) 2 Integrating in [, t] we obtain 3.9). A /2 u ε t) 2 2K + ) + t)p P ε ). A /2 u ε t) 2γ+2 + 2K + )P ε) + t) p. 2

15 Proof of 3.2) Let us consider the following identity [ + t) p u ε, Au ε A /2 u ε 2γ+2 + t) p u ε, Au ε A /2 u ε 2γ+2 = ] + t) p A /2 u ε 2 A /2 u ε 2γ γ + 2) + t) p u ε, Au ε 2 A /2 u ε p + u 2γ+4 t)p ε, Au ε A /2 u ε 2γ+2 =: J t) + J 2 t) + J 3 t) + J 4 t). 3.32) In order to estimate the integral of the left-hand side, we estimate the integrals of the four terms in the right-hand side. By 3.7) and 3.8) we have that hence u εt), Au ε t) Au εt) A /2 u ε t) 2γ+2 A /2 u ε t) u εt) A /2 u ε t) 2γ+ P ε ) Q ε ) + 4P ε ) + t) 2p Pε )Q ε ) + 2P ε )) + t) p, 3.33) t J s) ds u, Au ) A /2 u + + 2γ+2 t)2p Pε )Q ε ) + 2P ε ). The integral of J 2 t) can be easily estimated using 3.9). As for J 3 t), by Cauchy-Schwarz inequality we have that u εt), Au ε t) 2 A /2 u ε t) 2γ+4 A/2 u εt) 2 A /2 u ε t) 2 A /2 u ε t) 2γ+4 = A/2 u εt) 2 A /2 u ε t) 2γ+2, and therefore we reduce once again to 3.9). Finally, from 3.33) we obtain that J 4 s) ds ) Pε )Q ε ) + 2P ε ) + t) 2p. 2 Plugging all these estimates in 3.32), and recalling once again that + t) 2p + t) p+ for every t, we obtain 3.2). Proof of 3.2) Let us set w ε t) := A /2 u ε t) 2. Then ) w εt) = 2 + t) p [w ε t)] γ+ Auε t) 2 A /2 u ε t) + ε t), Au ε t) 2 ε u, 3.34) A /2 u ε t) 2γ+2 hence by 3.7) ) w ε t) 2 + t)p [w ε t)] γ+ P ) + ε u εt), Au ε t). A /2 u ε t) 2γ+2 3

16 This means that w ε satisfies a differential inequality of the form 3.7) with α := P ), ft) := ε u ε s), Au εs). 3.35) A /2 u ε s) 2γ+2 Thanks to 3.2) and the last inequality in 3.23) we have that { } + s) p fs) ds εσ + t) p+ min 4γ A /2 u, P ) + t) p+, 2γ 2p + ) and therefore the function ft) satisfies assumption 3.4) of Lemma 3.2. From statement 2) of that lemma we obtain 3.2). 3.3 Proof in the coercive case Local maximal solutions Problem.),.2) admits a unique local-in-time solution, and this solution can be continued to a solution defined in a maximal interval [, T), where either T = +, or or lim sup A /2 u εt) 2 + Au ε t) 2) = +, 3.36) t T lim inf t T A/2 u ε t) 2 =. 3.37) We omit the proof of these standard results. The interested reader is referred to [6] see also []). Preliminaries and notations Let ν satisfy.5), and let { }] /γ σ 2 := A /2 u [max 2 p + 2,. νγ A /2 u 2γ Let K be such that K > Au, u ) A /2 u, K > P )Q 2 ) + 2P ) σ2. γ 3.38) Starting with this value of K let us define σ and σ as in the proof of Proposition 3.3, and let us choose ε satisfying 3.23), and the further requirement { } ν σ ε min 2p + ),. 3.39) 4γ A /2 u 2γ 4

17 Let us finally set { S := sup τ [, T) : A /2 u ε t) and Au εt), u ε t) K } t [, τ]. A /2 u ε t) 2 + t) p From the mild nondegeneracy assumption.4) and the first inequality in 3.38) it is easy to see that S >. Moreover in the interval [, S) all the conclusions of Proposition 3.3 hold true. Estimate from above for A /2 u ε t) Let us set w ε t) := A /2 u ε t) 2 as in the proof of Proposition 3.3. Once again w ε t) is a solution of 3.34). Since we are in the coercive case we have that Au ε t) 2 ν A /2 u ε t) 2. Therefore from 3.34) it follows that w ε t) 2 + t)p [w ε t)] γ+ ν + ε u ε t), Au ) εt), A /2 u ε t) 2γ+2 which means that w ε satisfies an inequality of the form 3.5) with α := ν, and ft) defined as in 3.35). Thanks to 3.2) and 3.39) the function ft) satisfies assumption 3.4) of Lemma 3.2. From statement ) of that lemma we obtain that A /2 u ε t) 2 σ 2 + t) p+)/γ t [, S). 3.4) Global existence We prove that S = T = +. Let us assume by contradiction that S < T. By definition of S this means that either A /2 u ε S) 2 = or Au ε S), u ε S) A /2 u ε S) 2 = K + S) p. 3.4) By continuity all the estimates proved so far hold true also for t = S. In particular 3.2) rules out the first possibility in 3.4). On the other hand from 3.33), 3.4), and the second inequality in 3.38), we have that Au ε S), u εs) Au εs) A /2 u ε S) 2 A /2 u ε S) < u εs) A /2 u ε S) 2γ+ A/2 u ε S) 2γ P )Q ) + 2P )) + S) p K + S K + S) p, which rules out the second possibility in 3.4). σ γ 2 + S) p+ 5

18 It remains to prove that T = +. Let us assume by contradiction that T < +. Then the quoted local existence result says that either 3.36) or 3.37) holds true. On the other hand now we know that 3.2) is satisfied for every t [, T), which rules 3.37) out. Moreover from 3.4) we have that A /2 u ε t) 2 is uniformly bounded from above in [, T), hence by 3.6) it follows that also A /2 u ε t) and Au εt) are uniformly bounded from above in [, T). This rules 3.36) out. Decay estimates Let us prove estimates 2.2), 2.3), and 2.4). Now we know that the solution is global, and that all the estimates proved so far hold true for every t. Therefore 2.2) follows from 3.2) and 3.4). Moreover from 3.7) and the coerciveness assumption.5) we have that ν Au εt) 2 A /2 u ε t) 2 P ) t, hence 2.3) follows from 2.2). Finally, 2.4) follows from 3.8) and 3.4). 3.4 Proof in the noncoercive case Local maximal solutions As in the coercive case there exists a unique local-in-time solution which can be continued to a solution defined in a maximal interval [, T), where either T = +, or 3.36) holds true, or 3.37) holds true. Preliminaries and notations Let σ be the constant defined in 3.22), let and let K be such that σ 3 := 6γ + ) u 2 + A /2 u 2γ u 2), σ 4 := 2 A/2 u 2 A /2 u + 2 Au 2 + Au, u + 2γ 2 A /2 36σ γ u 2γ, K > Au, u A /2 u 2, K > [ + γ)σ 3] γ )/γ+) ) u σ4 + 4σ A /2 u 2γ ) Starting with this value of K let us define σ as in the proof of Proposition 3.3, and let us choose ε satisfying 3.23) and the further condition 6ε. As in the coercive case let us finally set { S := sup τ [, T) : A /2 u ε t) and Au εt), u εt) K } t [, τ]. A /2 u ε t) 2 + t) p 6

19 From the mild nondegeneracy assumption.4), and the first inequality in 3.42), it is easy to see that S >. Moreover in the interval [, S) all the conclusions of Proposition 3.3 hold true. In the following we set β = p + γ, and we prove estimates involving the following energies H ε t) := ε u εt) 2 + γ + A/2 u ε t) 2γ+2 ; 3.43) D ε t) := ε + t) p u ε t), u εt) + ) εp u 2 + t) p ε t) 2 ; 3.44) D ε t) := ε + t) 2β u εt), Au ε t) ; 3.45) A /2 u ε t) 2γ G ε t) := + t) β u εt) ) A /2 u ε t) 4γ All the estimates we present are first claimed in the interval [, S). At the end of the proof we show that S = T = +, thus obtaining that all the estimates actually hold true for every t. First order estimate In this section of the proof we show that + t) p+ H ε t) + u ε t) s) u ε s) 2 ds σ 3 t [, S). 3.47) To this end we begin by taking the time derivative of 3.44): D εt) = + t) p A /2 u ε t) 2γ+2 + ε + t) p u εt) 2 + Integrating in [, t] we obtain that + s) p A /2 u ε s) 2γ+2 ds = D ε ) D ε t) + ε εp p) + 2 εp p) 2 u ε t) 2 + t) 2 p. + s) p u εs) 2 ds + u ε s) 2 ds. 3.48) + s) 2 p From our assumptions on ε and p we have that 2ε < /4, 2p p +, εp /2. Therefore D ε t) 2ε 2 + t) 2p u ε t) u εt) 2 εp t) u εt) 2 p 2 u εt) 2 4 ε + t)p+ u ε t) 2 8 u εt) 2. 7

20 Plugging this estimate in 3.48) we obtain that 8 u εt) s) p A /2 u ε s) 2γ+2 ds D ε ) + 4 ε + t)p+ u ε t) 2 + +ε + s) u εs) 2 ds + εp p) 2 u ε s) 2 ds. 3.49) + s) 2 p Let us consider now the energy defined in 3.43). A simple calculation gives that ) [ ] + t) p+ εp + ) H ε = + t) 2 u + t) ε 2 + p + p γ + + t)p A /2 u ε 2γ+2. Let us integrate in [, t]. Using 3.49) and rearranging the terms we obtain that + t) p+ ) p + ε u ε + t)p+ 4 γ + t) 2 + A /2 u ε t) 2γ+2 γ + H ε ) 2 εp + ) ε p + ) + s) u ε γ + s) 2 ds + + p + D ε ) γ + 8 u εt) 2 εp p) t ) u ε s) 2 + ds. 2 + s) 2 p From the smallness assumptions on ε, and the fact that p+)/γ+) 2, it follows that 2 + t)p+ H ε t) + + s) u ε s) 2 ds + p + 8 γ + u εt) 2 H ε ) + 2 D ε ) ) + p + εp p) t γ + 2 In particular we have that u ε s) 2 ds. 3.5) + s) 2 p u ε t) 2 8γ + ) t p + H u ε s) 2 ε) + 2 D ε ) ) + 4ε p) ds, + s) 2 p hence by Gronwall s lemma u ε t) 2 8γ + ) p + H ε) + 2 D ε ) ) exp 8γ + ) p + H ε) + 2 D ε ) ) exp4ε) 6γ + ) H ε ) + 2 D ε ) ). p + 8 4ε p) ) ds + s) 2 p

21 Integrating in [, t] we obtain that u ε s) 2 + p) ds 6γ + s) 2 p p + H ε) + 2 D ε ) ). Coming back to 3.5) we have therefore that 2 + t)p+ H ε t) + + s) u εs) 2 ds + p + 8 γ + u εt) 2 + 8pε) H ε ) + 2 D ε ) ). 3.5) It remains to estimate the right-hand side. This can be easily done because 8pε, and H ε ) + 2 D ε ) ε u 2 + γ + A/2 u 2γ+2 + 2ε u, u + εp) u 2 u 2 + A /2 u 2γ u 2. Plugging this estimate in 3.5), and multiplying by 8γ + ), we obtain 3.47). Second order estimate In this section of the proof we show that + t) β F ε t) t) β + s) 2β p A/2 u ε s) 2 A /2 u ε s) 2γ ds σ 4 t [, S). 3.52) To this end we begin by computing the time derivative of 3.45): D ε t) = + t)2β Au ε t) 2 + ε + t) 2β A/2 u εt) 2 Au ε t) 2γ + 2γε + t) 2β u ε t), Au εt) 2 A /2 u ε t) + 2γ+2 + t) 2β p ε 2β ) u ε t), Au ε t) + t) p A /2 u ε t) 2γ =: I t) + I 2 t) + I 3 t) + I 4 t). 3.53) Let us estimate this derivative from above. To this end in I 2 t) we replace the exponent 2β with the bigger exponent 2β p. The term I 3 t) is nonpositive and can be neglected. In order to estimate I 4 t) we remark that < β 2, hence 2β 3. Due to the smallness of ε we have therefore that ε 2β + t) p 2β ε + + 3ε 2, + t) p 9

22 and thus I 4 t) 2 + t) 2β p A /2 u εt) A /2 u ε t) 2γ 4β + t)2β p A/2 u εt) 2 A /2 u ε t) 2γ + 4β + t)2β p 2 A /2 u ε t) 2γ 2. Since γ we can estimate the last term using 3.2). After some calculations with the exponents we obtain that hence A /2 σ γ u ε t) 2γ 2 + t) p+ β, 3.54) I 4 t) 4β + A/2 u εt) 2 t)2β p A /2 u ε t) 2γ + 4βσ γ Plugging these estimates in 3.53) we have proved that + t) β. D εt) + t) 2β Au ε t) t) 2β p ε + ) A /2 u ε t) 2 4β A /2 u ε t) + 2γ +4βσ γ + t) β. Integrating in [, t] we obtain that + s) 2β Au ε s) 2 ds Using 3.54) once more we have that ε + ) + s) 2β p A/2 u ε s) 2 ds + 4β A /2 u ε s) 2γ + D ε ) D ε t) + 4σ γ + t) β. 3.55) D ε t) ε2 2 + t)2β A/2 u ε t) 2 A /2 u ε t) 2γ t)2β 2 A /2 u ε t) 2γ 2 ε2 2 + t)2β A/2 u ε t) 2 A /2 u ε t) 2γ + 2 σ γ + t) β+p. Since β + p β, plugging this estimate in 3.55) we obtain that + s) 2β Au ε s) 2 ds D ε ) + ε2 2 + A/2 u t)2β ε t) 2 A /2 u ε t) + 9 2γ 2 σ γ + t) β + + ε + ) + s) 2β p A/2 u ε s) 2 ds. 3.56) 4β A /2 u ε s) 2γ 2

23 hence Let us consider now the energy defined in 3.). A simple calculation gives that [ ] + t) 2β 2 F ε = + t) 2β + t) + 2εγ Au ε, u ε p A /2 u ε 2βε ) A /2 u ε t A /2 u ε + 2γ +2β + t) 2β Au ε 2. By definition of S, and the second inequality in 3.23), we have that 2εγ Au εt), u εt) + 2βε A /2 u ε t) 2 + t 2εγK + 2βε) + t) p + t) p, 3.57) [ + t) 2β F ε t) ] + t) 2β p A/2 u ε t) 2 A /2 u ε t) 2γ + 2β + t)2β Au ε t) 2. Let us integrate in [, t]. Using 3.56) and rearranging the terms we obtain that + t) 2β βε)ε A/2 u ε t) 2 A /2 u ε t) + + 2γ t)2β Au ε t) 2 + ) t + 2 2βε + s) 2β p A/2 u ε s) 2 A /2 u ε s) ds F ε) + 2β D 2γ ε ) + Since β 2, and 2βε 4ε /4, it follows that 2 + t)2β F ε t) s) 2β p A/2 u ε s) 2 ds A /2 u ε s) 2γ F ε ) + 2β D ε ) + 9βσ γ + t) β A/2 u 2 A /2 u 2γ + Au Dividing by + t) β /2 we obtain 3.52). +9βσ γ + t) β. Au, u + A /2 8σ γ u 2γ + t) β. Estimate on the derivative Let us consider the energy defined in 3.46). Its time derivative is given by 2 G ε t) = ε + t)β + t) + 4εγ u εt), Au ε t) βε ) u ε t) 2 p A /2 u ε t) 2 + t A /2 u ε t) + 4γ 2 ε + u εt), Au ε t) t)β A /2 u ε t). 2γ 2

24 hence Arguing as in 3.57) we find that 4εγ u ε t), Au εt) + βε A /2 u ε t) 2 + t t) p, G εt) 2ε + t) pg εt) + 2 ε + t)β/2 Au ε t) G ε t). Thanks to 3.52) we have therefore that hence by Lemma 3. G εt) 2ε + t) pg εt) + 2 σ4 G ε t), ε G ε t) G ε ) + 6σ 4 + t) 2p t [, S). 3.58) Global existence We prove that S = T = +. Let us assume by contradiction that S < T. Then by continuity all the estimates proved so far hold true also for t = S. Moreover by definition of S we have the alternative 3.4). The first possibility can be ruled out using 3.2) exactly as in the coercive case. In order to rule out the second possibility we consider the inequality Au ε S), u ε S) u ε Au A /2 u ε S) 2 ε S) S) A /2 u ε S) 2γ A/2 u ε S) 2γ ) Let us estimate the three factors. From 3.52) we have that σ4 Au ε S) + S) β/2. From 3.58) we have that u ε S) A /2 u ε S) 2γ ) Gε ) + 6σ 4 + S) 2p u + S) β/2 A /2 u + 4 σ 2γ 4 + S) β/2 p. Since γ the last factor in 3.59) can be estimated using 3.47). We obtain that A /2 u ε S) 2γ 2 [γ + )σ 3 ] γ )/γ+) + S) p+)γ )/γ+). Plugging all these estimates in 3.59), and recalling 3.42), we obtain that Au ε S), u ε S) A /2 u ε S) 2 < K + S) p + S)2p β p+)γ )/γ+). 22

25 If p satisfies 2.5), then the last exponent is less than or equal to zero, and this is enough to rule out the second possibility in 3.4). It remains to prove that T = +. So let us assume by contradiction that T < +. Then the quoted local existence result says that either 3.36) or 3.37) holds true. On the other hand as in the coercive case we have that 3.2) is satisfied for every t [, T), which rules 3.37) out. Moreover from 3.47) we have that A /2 u ε t) 2 is uniformly bounded from above in [, T), hence by 3.6) it follows that also A /2 u ε t) and Au ε t) are uniformly bounded from above in [, T). This rules 3.36) out. Decay estimates Let us prove estimates 2.6), 2.7), and 2.8). Now we know that the solution is global, and that all the estimates proved so far hold true for every t. Therefore the lower bound in 2.6) follows from 3.2), while the upper bound follows from 3.47). Moreover 2.7) follows from 3.52). Finally, 2.8) follows from 3.58) and 2.6). Remark 3.4 A careful inspection of the proofs reveals that 3.47) was proved without using the assumption γ. At this point one can modify 3.59) as follows: Au ε S), u ε S) Au εs) A /2 u ε S) 2 A /2 u ε S) u ε S) A /2 u ε S) 2γ+ A/2 u ε S) 2γ. Now we can estimate the first and second factor using 3.33) as we did in the coercive case, and then estimate the last factor using 3.47). All these inequalities require neither the coerciveness of the operator, nor γ. We end up with an estimate such as Au ε S), u ε S) K A /2 u ε S) 2 + S) p + S)2p γp+)/γ+) for a suitable constant K. The last exponent is less than or equal to zero provided that p γ/γ + 2). This is the key point of the proof of global solvability for γ > and p [, γ/γ + 2)] without coerciveness assumptions. We leave the details to the interested reader. 3.5 Proof of Theorem 2.3 In analogy with 3.43) let us set H ε t) := ε u εt) 2 + A /2 u εt) 2 mσ) dσ. Assumption 2.) is equivalent to say that H ε ) >. Moreover we have that H ε t) = 2bt) u ε t) 2 2 ε bt)h εt) t, 23

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