Georgian Mathematical Journal Volume 3 (26), Number 3, 397 4 GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION TO A PETROVKY EQUATION WITH GENERAL NONLINEAR DIIPATION AND OURCE TERM NOUR-EDDINE AMROUN AND ABBE BENAIA Abstract. We consider the nonlinearly damped semilinear Petrovsky equation u 2 xu + g(u ) = b u u p 2 on [, + [ and prove the global existence of its solutions by means of the stable set method in H 2 () combined with the Faedo Galerkin procedure. Furthermore, we study the asymptotic behavior of solutions when the nonlinear dissipative term g does not necessarily have a polynomial growth near the origin. 2 Mathematics ubject Classification: 35L45, 93C2, 35B4, 35L7. Key words and phrases: General nonlinear dissipation, nonlinear source, global existence, decay rate, multiplier method.. Introduction We consider the initial boundary value problem u 2 xu + g(u ) = b u u p 2 in [, + [, (P ) u = ν u = on Γ [, + [, u(x, ) = u (x), u (x, ) = u (x) on, where is a bounded domain in R n with a smooth boundary = Γ. For the problem (P ) when g(s) = δ s m 2 s (m ),. A. Messaoudi [7] obtained relations between m and p for which the global existence or alternatively finite time blow up takes place. More precisely, he showed that solutions with any initial data continue to exist globally in time if m p and blow up in finite time if m < p and the initial energy is negative. To prove the global existence he used a new method introduced by Georgiev and Todorova [2] based on the fixed point theorem. In [3], for a wave equation ( x u instead of 2 xu in (P )) Ikehata by using the stable set method due to attinger [] proved that a global solution exists with no relation between p and m, and Todorova [] proved that an energy decay rate is E(t) ( + t) 2/(m 2) for t, for which she used the general method on energy decay introduced by Nakao [9]. Unfortunately, the methods used by Messaoudi and Todorova do not seem to be applicable to the case of more general functions g. Our purpose in this paper is to give the global solvability in the class H 2 and the energy decay estimates of solutions to the problem (P ) when g(s) does not IN 72-947X / $8. / c Heldermann Verlag www.heldermann.de
398 N.-E. AMROUN AND A. BENAIA necessarily have a polynomial growth near zero and a source term of the form b y p 2 y with a small parameter b. As proved in [4] and [], a decay rate of the global solution depends on the polynomial growth near zero of g(s). We use some ideas from [6] (see also []) introduced in the study of decay rates of solutions to the wave equation u tt x u + g(u t ) = in R +. o, to obtain global decaying solutions to the problem (P ), we use the argument combining the Galerkin approximation scheme (see [5]) with the concept of a stable set in H 2 and the method in [6] to derive a decay rate of the solution. We conclude this section by stating our plan and giving some notations. In ection 2 we formulate some lemmas needed for our arguments. ections 3 and 4 are devoted to the proof of the global existence and decay estimates for the problem (P ). Throughout this paper all the functions considered are real-valued. We omit the space variable x of u(t, x), u t (t, x) and simply denote u(t, x), u t (t, x) by u(t), u (t), respectively, when no confusion arises. Let l be a number with 2 l. We denote by l the L l norm over. In particular, the L 2 norm is denoted 2. ( ) denotes the usual L 2 inner product. We use the familiar function spaces H 2, H 4. 2. Preliminaries Let us state the precise hypotheses on p and g. (H) Assume that 2 < p (n =, 2, 3, 4) or 2 < p 2n 2 n 4 (n 5). () (H2) g is an odd increasing C function and c s g(s) c 2 s r if s with r (n =, 2, 3, 4) or r n + 4 (n 5), n 4 where c and c 2 are positive constants. We first state three well known lemmas that will be needed later. Lemma 2. (obolev Poincaré inequality). Let q be a number with 2 q < + (n =, 2, 3, 4) or 2 q 2n/(n 4) (n 5), then there is a constant C = C(, q) such that u q C u 2 for u H 2 (). (2) We denote by c various positive constants which may be different at different occurrences. Lemma 2.2 ([6]). Let E : R + R + be a non-increasing function and φ : R + R + an increasing C function such that φ() = and φ(t) + as t +.
GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION 399 Assume that there exist σ and ω > such that + E +σ (t)φ (t) dt ω Eσ ()E(), < +. Then ( ) + σ σ E(t) E() + ωσφ(t) t, if σ >, E(t) ce()e ωφ(t) t, if σ =. Remark 2.. A weight function φ(t) was sufficiently used by Martinez [6], and Mochizuki and Motai [8] to establish a decay rate of solutions to a hyperbolic PDE. Lemma 2.3 ([6]). There exists an increasing function φ : R + R such that φ is concave and φ(t) + as t +, φ (t) as t +, and + φ (t) ( g (φ (t)) ) 2 dt < +. In order to state and prove our main results, we first introduce the following notation: Then we can define the stable set as where we use w instead of w(, t). I(t) = I(u(t)) = x u(t) 2 2 b u(t) p p, J(t) = J(u(t)) = 2 xu(t) 2 2 b p u(t) p p, E(t) = E(u(t), u (t)) = J(t) + 2 u (t) 2 2. H = { w H 2 () I(w) > } {}, 3. Global Existence Throughout this section we assume u H 4 () H and u H 2 () L 2r (). We employ the Galerkin method to construct a global solution. Let T > be fixed and denote by V m the space generated by {w, w 2,..., w m }, where the set {w m ; m N} is a basis of L 2, H 2 and H 4 H 2. We construct approximate solutions u m (m =, 2, 3,...) in the form u m (t) = m g jm w j, j=
4 N.-E. AMROUN AND A. BENAIA where g jm (j =, 2,..., m) are determined by the following ordinary differential equations: (u m(t), w j ) + ( x u m (t), x w j ) + (g(u m(t)), w j ) = (b u m (t) p 2 u m (t), w j ), j m, (3) m u m () = u m = (u, w j )w j u in H 4 H 2 as m +, (4) u m() = u m = j= m (u, w j )w j u in H 2 L 2r as m +. (5) j= By virtue of the theory of ordinary differential equations, system (3) (5) has a unique local solution which is extended to a maximal interval [, T m [ (with < T m + ) by the Zorn lemma, since the nonlinear terms in (3) are locally Lipschitz continuous. Note that u m (t) is a C 2 -function. In the next step, we obtain a priori estimates for the solution so that it can be extended outside [, T m [ to obtain one solution defined for all t >. We can utilize a standard compactness argument for the limiting procedure and it suffices to derive some a priori estimates for u m. But this procedure allows us to employ the energy method for a smooth solution u(t) to the problem (P ) (the results should be in fact applied to approximated solutions). Remark 3.. By multiplying the first equation of (P ) by u (t), integrating over, and using integration by parts and the boundary conditions we get E (t) = g(u (t))u (t) dx t [, T ). Lemma 3.. Assume that (H) holds. Let u(t) be a solution with the initial data {u, u } satisfying u H and u L 2 (). If {u, u } satisfies ( ) (p 2)/2 2p η = b C p p 2 E(u, u ) >, (6) then u(t) H for all t [, + ) and there exists a constant M = M( x u 2, u 2 ) > such that x u(t) 2 2 + u (t) 2 2 M for t, and t g(u (s))u (s) ds M for t. (7) Proof. ince I(u ) >, it follows from the continuity of u(t) that I(u(t)) (8)
GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION 4 for some interval near t =. Let tmax be a maximal time (possibly tmax = T m ), when (8) holds on [, tmax). On the other hand, hence J(t) = 2 xu(t) 2 2 b p u(t) p p x u(t) 2 2 = p 2 2p xu(t) 2 2 + p I(u(t)) p 2 2p xu(t) 2 2 Using (2), (6), and (9), we deduce that Therefore we get t [, tmax); 2p p 2 J(t) 2p p 2 E(t) 2p p 2 E(u, u ) t [, tmax). (9) b u(t) p p b C p x u(t) p 2 = b C p x u(t) p 2 2 x u(t) 2 2 ( ) (p 2)/2 2p b C p p 2 E(u, u ) x u(t) 2 2 < x u(t) 2 2 t [, tmax); () x u(t) 2 2 b u(t) p p > on [, tmax). This implies that we can take tmax = T m. Furthermore, by the fact that the energy is non-increasing we have E(u, u ) E(t) = 2 xu(t) 2 2 b p u(t) p p + 2 u (t) 2 2 since I(u(t)), and hence = p 2 2p xu(t) 2 2 + p I(u(t)) + 2 u (t) 2 2 p 2 2p xu(t) 2 2 + 2 u (t) 2 2 on [, tmax), x u(t) 2 2 + u (t) 2 2 C E(u, u ) on [, tmax). () These estimates imply that the (approximated) solution u(t) exists globally in [, + ). This ends the proof of Lemma 3.. Estimate () yields x u m is bounded in L loc(, ; L 2 ). (2) Lemma 3.2. There exists K > such that g(u m) L r+ r ( [,T ]) K for all m N.
42 N.-E. AMROUN AND A. BENAIA Proof. If we define and where Q = [, T ], then from (H2): g(u m(x, t)) r+ r A m = {(x, t) Q\ u m(x, t) } B m = {(x, t) Q\ u m(x, t) > }, dx dt = g(u m(x, t)) r+ r dx dt + g(u m(x, t)) r+ r dx dt A m Hence, by (7), we have B m sup g(s) r+ r dx dt + c 2 s B m g(u m(x, t)) r+ r g(u m(x, t)) u m(x, t) dx dt. dx dt Q sup g(s) r+ r + c 2 M for m N s which completes the proof. Here Q denotes the Lebesgue measure in R n+. Lemma 3.3. There exists a constant M such that for all m N. Proof. From (3) we obtain u m(t) 2 + x u m(t) 2 M u m() 2 2 xu m 2 + g(u m ) 2 + f(u m ) 2 2 xu m 2 + g(u m ) 2 + k x u m p 2, where we set f(u) = bu u p 2. Using the Gagliardo Nirenberg inequality, we have f(u m ) 2 c 2 xu m p 2. ince g(u m ) is bounded in L 2 () by (H2), from (4) and (5) we obtain u m() 2 C. Differentiating (3) with respect to t, we get (u m(t) + 2 xu m(t) + u m(t)g (u m) u mf (u m ), w j ) =.
GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION 43 Multiplying it by 2g jm(t) and summing over j from to m give d ( ) u dt m(t) 2 2 + x u m(t) 2 2 + 2 u 2 m(t)g (u m(t)) dx 2b(p ) u m(t) u m(t) u m (t) p 2 dx. (3) Next, we are are going to analyze the term on the right-hand side of (3). p 2 Making use of the generalized Hölder inequality, observing that + + 2(p ) 2(p ) =, using Lemmas 2. and 3. we conclude that 2 u m(t)u m(t)f (u m (t)) dx b(p ) u m(t) p 2 2(p ) u m(t) 2(p ) u m(t) 2 C x u m (t) p 2 2 x u m(t) 2 u m(t) 2 C 2 ( x u m(t) 2 2 + u m(t) 2 2), (4) where C and C 2 are positive constants independent of m and t [, T ]. Combining (3) and (4) we deduce d ( ) u dt m(t) 2 2 + x u m(t) 2 2 + 2 u 2 mg (u m) dx ( C 2 u m(t) 2 2 + x u m(t) 2) 2, Integrating the last inequality over (, t) and applying Gronwall s lemma, we obtain u m(t) 2 2 + x u m(t) 2 2 e ( ) C 2T u m() 2 2 + x u m() 2 2 for all t R +. Therefore we conclude that u m is bounded in L loc(, ; L 2 ), (5) x u m is bounded in L loc(, ; L 2 ). (6) Furthermore, we claim that Indeed, it follows from (5) and (6) that u m is precompact in L 2 (, ; L 2 ). (7) u m is bounded in L loc(, ; H 2 ) and u m(t) is bounded L loc(, ; L 2 ()). (8) Applying a compactness argument, (7) follows. Applying the Dunford Pettis theorem we conclude from (2), Lemma 3.2, (5) and (6) replacing, if needed, the sequence u m with a subsequence that u m u weak-star in L loc(, ; H 2 ), (9) u m u weak-star in L loc(, ; H 2 ),
44 N.-E. AMROUN AND A. BENAIA u m u weak-star in L loc(, ; L 2 ), (2) g(u m) χ weak-star in L q+ q ( (, T )) for suitable functions u L (, T ; H 2 ) and χ L q+ q ( (, T )) for all T. We have to show that u is a solution of (P ). Lemma 3.4. For each T >, g(u ) L (Q) and g(u ) L (Q) K, where K is obtained in Lemma 3.2. Proof. By (H2) and (7) we have g(u m(x, t)) g(u (x, t)) a.e. in Q, g(u m(x, t))u m(x, t) g(u (x, t))u (x, t) a.e. in Q. Hence, by (7) and Fatou s lemma we have u (x, t)g(u (x, t)) dx dt K for T >. (2) Now, using (2), the proof follows similarly to Lemma 3.2. Lemma 3.5. g(u m) g(u ) in L ( (, T )). Proof. Let E [, T ] and set { E = (x, t) E; g(u m(x, t)) }, E 2 = E \ E, E where E is the measure of E. If M(r) := inf{ s ; s R and g(s) r}, then g(u m) dxdt ( ( )) E + M u mg(u m) dxdt. E E Applying (7) we deduce that sup m E E 2 g(u m) dxdt as E. From Vitali s convergence theorem we deduce that g(u m) g(u ) in L ( (, T )), hence and this implies that g(u m)v dx dt g(u m) g(u ) weak star in L r+ r (Q), g(u )v dx dt for all v L r+ (, T ; H 2 ) (22) as m +. Using the compactness of H 2 in L 2, we see that b u m p 2 u m v dx dt b u p 2 uv dx dt for all v L r+ (, T ; H 2 ) (23)
GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION 45 as m +. It follows at once from (8), (9), (2), (22) and (23) that for each fixed v L r+ (, T ; H 2 ) (u m + 2 xu m + g(u m) b u m p 2 u m )v dx dt (u + 2 xu + g(u ) b u p 2 u)v dx dt as m +. Hence (u + 2 xu + g(u ) b u p 2 u)v dx dt =, v L r+ (, T ; H). 2 Thus the problem (P ) admits a global weak solution u such that u W, (, T ; H()) 2 W 2, (, T ; L 2 ()). The uniqueness of this solution is a consequence of the monotonicity of g and that f is a locally Lipschitz function. 4. Asymptotic Behavior Before stating and proving the decay result, we start with Lemma 4.. uppose that (2) holds and u H and u L 2 () satisfy (6). Then b u(t) p p ( η) x u(t) 2 2 Proof. It suffices to rewrite () as { [ ( ) (p 2)/2 ] } 2p b u(t) p p b C p p 2 E(u, u ) x u(t) 2 2. Theorem 4.. uppose that () holds and u H and u L 2 () satisfy (6). Then the solution satisfies the decay estimates ( ( )) 2 E(t) c G, t where G(s) = sg(s). If, in addition, s g(s)/s is non-decreasing on [, µ] for some µ >, then we have ( ( )) 2 E(t) c g. t Examples. ) If g(s) = e /sp for < s <, p >, then we have c E(t) (ln t). 2/p
46 N.-E. AMROUN AND A. BENAIA 2) If g(s) = e e/s for < s <, then we have E(t) c (ln(ln t)) 2. Proof of Theorem 4.. We multiply the first equation of (P ) by Eφ u, where φ is a function satisfying all the hypotheses of Lemma 2.3. We obtain = Eφ u(u 2 xu + g(u ) b u p 2 u) dx dt ince = [Eφ + + Eφ uu dx Eφ ] T (E φ + Eφ ) ( u 2 + x u 2 2b ) p u p ( ) 2 b p u p dx dt. ( b 2 ) p uu dx dt 2 dx dt + u p dx ( η) p 2 p Eφ x u 2 dx ( η) p 2 2p p p 2 E(t) = 2( η)e(t), Eφ u 2 dx dt ug(u ) dx dt we deduce that 2η E 2 φ dt [Eφ + 2 Eφ ] T uu dx + u 2 dx dt Eφ (E φ + Eφ ) ug(u ) dx dt + uu dx dt Eφ ug(u ) dx dt [Eφ + 2 Eφ uu dx ] T + u 2 dx dt + c(ε) (E φ + Eφ ) Eφ uu dx dt g(u ) 2 dx dt u
GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION 47 + ε Eφ u 2 dx dt + Eφ ug(u ) dx dt u u > for every ε >. Moreover, using the Hölder inequality, Lemma 2. and the Young inequality, we obtain Eφ ug(u ) dx dt u > Eφ ( ) ( u r+ (r+) dx g(u ) (r+) r ) r (r+) dx dt c ( E 3 2 φ u > ) r T u g(u (r+) ) dx dt φ E 3 2 ( E ) r (r+) dt c u > ( ) φ (E 3 2 r r+ ) ( E ) r r (r+) E r+ dt c(ε ) φ ( E E) dt + ε φ E (r+) ( 3 2 (r+)) r dt c(ε )E() 2 + ε E() (r ) 2 φ E 2 dt. Choosing ε and ε small enough, we deduce that E 2 φ dt [Eφ + c Eφ ] T uu dx + u 2 dx dt (E φ + Eφ ) uu dx dt ce() + c Eφ u 2 dx dt. Majorizing the last term of the above inequality, we have Eφ u 2 dx dt = Eφ u 2 dx dt + Eφ u 2 dx dt 2
48 N.-E. AMROUN AND A. BENAIA + Eφ u 2 dx dt, where, for t, 3 := {x, u h(t)}, 2 := {x, h(t) < u h()}, 3 := {x, u > h()}, and h(t) := g (φ (t)), which is a positive non-increasing function satisfying h(t) as t +. ince ( ) Eφ u 2 dx dt c E(t)φ (t) h(t) 2 ds dt ce() φ (t)(g (φ (t))) 2 dt ce(), for x 2 we have φ (t) = g(h(t)) g(u ) (as g is non-decreasing) and hence Eφ u 2 dx dt E g(u ) u 2 dx dt 2 2 h() E u g(u ) dx dt h() 2 E()2, for s h() (as g(s) cs) we have Eφ u 2 dx dt c 2 Eφ u g(u ) dx dt then we deduce that 3 c ce() 2, E( E ) dx dt E 2 φ dt ce(),
GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION 49 and, thanks to Lemma 2.2, we obtain E(t) c φ(t) t. Let s be such that g( s ) ; since g is non-decreasing, we have hence and Thus ψ(s) + (s ) g ( ) s s φ(t) s ( s φ ) = G ( s g ( s G ( s ) ) with t := G ( ). s φ(t) G ( ). t ) s s, Now define H(s) := g(s), where H is non-decreasing, H() =, then we use s the function h(t) := H (φ (t)). On 2 there holds φ (t)u 2 H(u ) u 2 = u g(u ), and the same calculations as above with yield φ (t) = + t H ( ) ds s ( ( )) 2 E(t) c g. t Remark 4.. We can extend all the results obtained above to the case p = 2. But we need some modification of Lemma 3., in that case the smallness of plays an essential role in our argument. Indeed, J(u(t)) 2 xu 2 2 2 bc2 x u 2 2 2 ( bc2 ) x u 2 2. If the condition b C 2 < is fulfilled, then we find a similar result to Lemma 3.. This condition implies that is small in some sense.
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