An N-order Iterative Scheme for a Nonlinear Wave Equation Containing a Nonlocal Term

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1 Filomat 3:6 7) DOI.98/FIL76755N Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: An N-order Iterative Scheme for a Nonlinear Wave Equation Containing a Nonlocal Term Le Thi Phuong Ngoc a Bui Minh Tri b Nguyen Thanh Long b a University of Khanh Hoa Nguyen Chanh Str. Nha Trang City Vietnam. b Department of Mathematics and Computer Science University of Natural Science Vietnam National University Ho Chi Minh City 7 Nguyen Van Cu Str. Dist. 5 Ho Chi Minh City Vietnam. Abstract. In this paper we consider an initial - boundary value problem for a nonlinear wave equation containing a nonlocal term. Using a high order iterative scheme the existence of a unique weak solution is proved. Furthermore the sequence established here converges to a unique weak solution at a rate of order N N ).. Introduction In this paper we consider the following initial - boundary value problem for a nonlinear wave equation u tt u xx = f x t u ut) ) x Ω = ) < t < T.) u t) = u t) = ux ) = ũ x) u t x ) = ũ x).3) µ f ũ ũ are given functions and the nonlinear term f x t u ut) ) contains a nonlocal term ut) = u x t)dx. Eq..) constitutes a case relatively simpler of a more general equation namely ) u tt x µx t u u x )u x = f x t u ux u t u u x ) x Ω = ) < t < T.4) it has its origin in the nonlinear vibration of an elastic string Kirchhoff 5]) for which the associated equation is ρhu tt = P + Eh L u ) L y t) y dy u xx.) Mathematics Subject Classification. 35L; 35L7; 35Q7 Keywor. Nonlinear wave equation containing a nonlocal term Faedo - Galerkin method the convergence of order N Received: 3 April 5; Accepted: October 5 Communicated by Naseer Shahzad addresses: ngoc966@gmail.com Le Thi Phuong Ngoc) tri.hcmuns@gmail.com Bui Minh Tri) longnt@gmail.com Nguyen Thanh Long)

2 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) here u is the lateral deflection ρ is the mass density h is the cross section L is the length E is Young s modulus and P is the initial axial tension. In ] Carrier also established a model of the type L ) u tt = P + P u y t)dy u xx P and P are constants. In ] Medeiros has studied Eq..4) with f = f u) = bu b is a given positive constant and Ω is a bounded open set of R 3. In 4] Hosoya and Yamada also have considered Eq..4) with f = f u) = δ u α u δ > α are given constants. In 3] Ficken and Fleishman established the unique global existence and stability of solutions for the equation u xx u tt αu t βu = εu 3 + γ ε >. Rabinowitz 4] proved the existence of periodic solutions for u xx u tt αu t = f x t u u x u t ) ε is a small parameter and f is periodic in time. In 8] Long and Diem have studied the linear recursive scheme associated with the nonlinear wave equation u tt u xx = f x t u u x u t ) < x < < t < T associated with.3) and the following mixed conditions u x t) h u t) = u x t) + h u t) = h > h are given constants. Afterwar this result has been extended in 9] ] to the nonlinear wave equation with the Kirchhoff - Carrier operator. In ] the following equation u tt µt u u x )u xx = f x t u u x u t u u x ) < x < < t < T associated with the mixed homogeneous conditions was studied. By the linear recursive scheme and by a standard argument existence of a local solution was proved. On the other hand an asymptotic expansion was established. In ] 5] a high order iterative scheme was established in order to get a convergent sequence at a rate of order N N ) to a local unique weak solution of a nonlinear Kirchhoff Carrier wave equation as follows u tt µt ut) u x t) ) x Ax)u x) = f x t u) < x < < t < T associated with the mixed homogeneous conditions. Based on the above problems we consider Prob..).3). With the assumption f C N ] R + R R + ) and some other conditions we shall establish a high order iterative scheme in order to get a convergent sequence at a rate of order N to a local unique weak solution of Prob..).3). By the fact that we associate with Eq..) a recurrent sequence {u m } defined by u m t u m = x Di 3 Dj 4 f x t u m u m ) u m u m ) i u m u m ) j i+j N < x < < t < T u m satisfying.).3) for all m and the first term u =. This result is a relative generalization of 8] - ] ] 3] 5].

3 . The High Order Iterative Method Le Thi Phuong Ngoc et al. / Filomat 3:6 7) First we denote the usual function spaces used in this paper by the notations L p = L p ) H m = H m ). Let be either the scalar product in L or the dual pairing of a continuous linear functional and an element of a function space. The notation stan for the norm in L and we denote by X the norm in the Banach space X. We call X the dual space of X. We denote by L p T; X) p for the Banach space of real functions u : T) X measurable such that and u L p T;X) = T ut) p X dt ) /p < + for p < u L T;X) = ess sup ut) X for p =. <t<t Let ut) u t) = u t t) = ut) u t) = u tt t) = üt) u x t) = ut) u xx t) = ut) denote ux t) u t x t) ux t) u t x x t) ux t) respectively. With f C k ] R x + R R + ) f = f x t u z) we put D f = f x D f = f t D 3 f = f u D 4 f = f z and Dα f = D α...dα 4 4 f α = α... α 4 ) Z 4 + α = α α 4 = k D ) f = f. We then have the following lemma the proof of which can be found in ]. Lemma.. The imbedding H C ]) is compact and i) v C ]) v H for all v H ii) v C ]) v x for all v H. Now we make the following assumptions: H ) ũ H H and ũ H H ) f C N ] R + R R + ) with f t z) = f t z) = t z. Fix T >. For each M > given we define two constants K M f ) K M f ) as follows K f M) = sup{ f x t u z) : x t T u M z M } K M f ) = α N K D α f M). For every T T ] and M > we put WM T) = {v L T; H H ) : v t L T; H ) and v tt L Q T ) with v L T;H H ) v t L T;H ) v tt L Q T ) M} W M T) = {v WM T) : v tt L T; L )}.) with Q T = ) T). We shall choose as first term u suppose that u m W M T) and associate with problem.).3) the following variational problem: Find u m W M T) m ) so that u mt) v + u mx t) v x = F m t) v v H u m ) = ũ u m) = ũ.).3)

4 F m x t) = i+j N Le Thi Phuong Ngoc et al. / Filomat 3:6 7) Di 3 Dj 4 f u m ]u m u m ) i u m t) u m t) ) j.4) here we use the following notations f u] = f x t u ut) ) D i f u] = D i f x t u ut) ) i = 3 4. Then we have the following theorem. Theorem.. Let H ) H ) hold. Then there exist a constant M > depending on ũ ũ and a constant T > depending on ũ ũ f such that for u there exists a recurrent sequence {u m } W M T) defined by.3).4). Proof. The proof consists of several steps. Step : The Faedo - Galerkin approximation introduced by Lions 7]). Let us consider a special basis of H formed by the eigenfunctions w j of the operator = u x : Put w j = λ j w j w j H H w j x) = sinjπx) λ j = jπ j = ) u m t) = k j= c mj t)w j the coefficients c satisfy the system of nonlinear differential equations mj ü m t) w j + u mxt) w jx = F m t) w j j k u m ) = ũ k u m ) = ũ k.6).7) in which ũ k = k j= α j w j ũ strongly in H H ũ k = k j= β j w j ũ strongly in H.8) F m x t) = i+j N with the notations D ij f = Di 3 Dj 4 f = D ij f u m ]u m u m ) i u m t) um t) ) j.9) i+j f u i z j i + j N D f = f. Let us suppose that u m satisfies.). Then we have the following lemma. Lemma.3. Let H ) H ) hold. For fixed M > and T > then the system.7) -.9) has a unique solution u m t) on an interval T m ] T]. Proof of Lemma.3. The system of Eqs..7) -.9) is rewritten in the form c mj t) + λ j c mi t) = F m t) w j j k c ) = α mj ċ ) = β j mj and it is equivalent to the system of integral equations c mj t) = α j j..) cosλ j t) + λ j β sinλ j j t) + t λ sinλ j jt s)) F m s) w j.) for j k. Omitting the indexs m k it is written as follows c = Lc].)

5 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) Lc] = L c]... L k c]) c = c... c k ) L j c]t) = q j t) + N j c]t) q j t) = α j cosλ j t) + λ j β j sinλ j t) N j c]t) = t λ sinλ j jt s)) Fc]s) w j j k Fc]t) = D ij f u m ]ut) u m ) i ut) u m t) ) j i+j N ut) = k j= c jt)w j. For every T m ρ} c X = sup ct) ct) = k j= t T m T] and ρ > that will be chosen later we put X = C T m ]; R k) S = {c X : c X c j t) for each c = c... c k ) Y. Clearly S is a closed nonempty subset in X and we have the operator L : X X. In what follows we shall choose ρ > and T m that L : S S is contractive. i) First we note that for all c = c... c k ) S > such ut) ct) c X ρ ut) C Ω) ct) ρ.3) so Nc]t) k t Fs). λ On the other hand by Fc]x t) i+j N i+j N i+j N i+j N ut) u m i ut) u m t) j ut) C Ω) + M ) i ut) + u m t) ) j ) i ρ ) j ρ + M + M ) i+j ρ + M we have Nc] X k λ T m K M f ) Hence we obtain i+j N Lc] X α + λ β + T D ) ) ρ ρ M = k λ K M f ) ii) We now prove that m D ) ρ i+j N Lc]t) Ld]t) X k λ T m D ) ρ ) i+j ρ + M. ρ M )..4) ) i+j ρ + M..5) ρ M ) c d X c d S.6)

6 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) D ) ) ρ ρ M = KM f ) i+j N Proof of.6) is as follows. Let c d S put ut) = k j= c jt)w j ut) = k j= c jt)w j. For all t T m ] we have ) i+j ) ρ + M im + i + j)ρ..7) Lc]t) Ld]t) = Nc]t) Nd]t) k t λ Fc]s) Fd]s)..8) On the other hand Fc]x t) Fd]x t) = = + i+j N i+j N i+j N i+j N D ij f u m ]ut) u m ) i ut) u m t) ) j D ij f u m ]vt) u m ) i vt) u m t) ) j D ij f u m ] ut) u m ) i vt) u m ) i] ut) u m t) ) j D ij f u m ]vt) u m ) i ut) u m t) ) j vt) u m t) ) j ]..9) We also note that a i b i = a b) i ν= a ν b i ν for all a b R i =... we deduce from.3) that ut) u m ) i vt) u m ) i i = ut) vt) u m ) ν=ut) ν vt) u m ) i ν ut) vt) i ν= i c d X ν= ut) u m ν vt) u m i ν ρ + M ) ν ρ + M ) i ν.) = i ρ + M ) i c d X. Similarly ut) u m t) ) j vt) u m t) ) j = ut) vt) j ut) u m t) ) ν vt) u m t) ) j ν ν= ut) vt) j j ρ c d X ν= ν= = jρ ρ + M ) j c d X. ut) u m t) ν vt) u m t) j ν ρ + M ) ν ρ + M ) j ν).)

7 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) It implies that Fc]x t) Fd]x t) +K M f ) +K M f ) +K M f ) i+j N i+j N i+j N i+j N i+j N i+j N c d X ut) u m ) i vt) u m ) i ut) u m t) j vt) u m i ut) u m t) ) j vt) u m t) ) j ) i i ρ + M c d X ρ + M) j ρ + M ) i jρ ρ + M ) j c d X i ρ + M ) i +j c d X ρ + M ) i+j jρ c d X i+j N = D ) ) ρ ρ M c d X ) i+j ) ρ + M im + i + j)ρ.) D ) ) ρ ρ M defined as in.7). It follows from.8).) that.6) hol. Choosing ρ > α + β λ and T m T] such that < T m D ) ρ ρ M ) ρ α λ β and k λ T m D ) ρ ρ M ) <..3) Therefore it follows from.4).6) and.3) that L : S S is contractive. We deduce that L has a unique fixed point in S i.e. the system.7).9) has a unique solution u m t) on an interval T m ]. The proof of Lemma.3 is complete. The following estimates allow one to take T m = T independent of m and k. Step : A priori estimates. Put m t) = p m t) + q m t) + t m s) ü p m t) = u m t) + u mxt) q m t) = u mxt) + u m t) Then it follows from.7) and.4) that.4) m t) = m ) + t F m s) u m s) + t F mx s) u mxs) + t ü m s) = S m ) + 3 j= J j..5) We shall estimate step by step all the terms J J J 3 and m ). The term J. Using the inequalities a + b) p p a p + b p ) for all a b p and s q + s p s q p].6)

8 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) we get from.9) that F m x t) KM f ) Hence i+j N i+j N i+j N i+j N i+j N i+j N + M N 3) F m t) KM f ) + M N 3) u m u m i u m t) um t) j u m + u m ) i u m t) + um t) ) j ) i ) j m t) + M m t) + M ) i+j m t) + M i+j i+j m t)) + M i+j i+j + m t) ) ] N M N 3 i+j N i+j N ξ M) + m t) ) ] N 3 i+j + m t) ) ] N 3. i+j + m t) ) ] N 3.7).8) ξ M) = K M f ) + M N 3) Using the inequality i+j N i+j..9) s q + s N s q N ] N = max{n N 3} N.3) we get from.8).3) that J = t F m s) u ξ M) t = ξ M) t 4ξ M) t ξ M) t m s) t F m s) u m s) + m s) ) ] N 3 m s) m s) + m s) ) N + m s) ) N ] + m s) ) N ] ].3) ξ M) = 4ξ M).

9 The term J. By.9) we have Le Thi Phuong Ngoc et al. / Filomat 3:6 7) F mxt) = D f u m ] + D 3 f u m ] u m ] + D D ij f u m ] + D 3 D ij f u m ] u m u m u m ) i + i+j N i+j N u m t) ) j um t) D ij f u m ]iu m u m ) i u mx u m ).3) u m t) ) j um t). Hence F mxt) KM f ) + M) + K M f ) + M) +K M f ) i+j N i+j N ) i M i + m t) M + + M) + K M f ) + M) +K M f ) i+j N M i + m t) + M) + K M f ) + M) +N )K M f ) + M) i+j N i+j N +N )K M f ) + M) = N + M)K M f ) N + M)K M f ) i+j N i+j N i+j N ) i+j i+j N i+j M i+j + i+j M i+j + i+j N N + M)K M f ) + M N ) ξ M) + m t) ) ] N M + ) i+j m t) ) ) j m t) M + m t) M + ) i+j m t) i+j i+j M i+j + m t)) m t)) i+j m t)) i+j i+j i+j M i+j + m t)) i+j M i+j + m t)) i+j i+j + M N + + m t) ) ] N i+j N i+j + m t) ) ] N.33) ξ M) = N + M)K M f ) + M N ) i+j N i+j..34)

10 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) Using the inequality.3) we get from.33) that J = t F mx s) u mxs) t F mx s) u mxs) ξ M) t + m s) ) ] N m s) = ξ M) t m s) + m s) ) ] N 4ξ M) t ξ M) t + m s) ) N ] + m s) ) N ].35) ξ M) = 4ξ M). The term J 3. Equation.7) can be rewritten as follows ü m t) w j u m t) w j = F m t) w j j k..36) Hence it follows after replacing w j with ü m t) and integrating that J 3 = t ü m s) t u m s) + t + t S m s) + ξ M) t t S m s) + 4ξ M) t t S m s) + 4ξ M) t + ξ M)) t + F m s) ) N 3 ] m s) ) ] N 3 + m s) ) ] N 3 + m s) ) N ] ξ 3 M) t + m s) ) ] N m s).37) with ξ 3 M) = + ξ M)). Now we need an estimate on the term m ). We have m ) = ũ k + ũ kx + ũ kx + ũ k..38) By means of the convergences in.8) we can deduce the existence of a constant M > independent of k and m such that m ) M /. Finally it follows from.5).3).35).37).39) that.39) m t) M + T ξm) + ξm) t S m s) ) N for t T m T.4) ξm) = ξ M) + ξ M) + ξ 3 M). Then by solving a nonlinear Volterra integral inequality.4) based on the metho in 6]) the following lemma is proved.

11 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) Lemma.4. There exists a constant T > independent of k and m such that m t) M t T] for all k and m..4) By Lemma.4 we can take constant T m = T for all k and m. Therefore we have u m WM T) for all k and m..4) Step 3: Convergence. From.4) we can extract from {u m } a subsequence {u k i) m } such that u k i) m u m in L T; H H ) weak* u k i) m u m in L T; H ) weak* ü k i) m u m in L Q T ) weak u m WM T)..43).44) We can easily check from.7).8).43).44) that u m satisfies.3).4) in L T) weak. On the other hand it follows from.3) and u m WM T) that u m = u m + F m L T; L ) hence u m W M T) and the proof of Theorem. is complete. Next we put W T) = {v L T; H ) : v L T; L )} then W T) is a Banach space with respect to the norm see 7]): v W T) = v L T;H ) + v L T;L ). Then we have the following theorem. Theorem.5. Let H ) H ) hold. Then there exist constants M > and T > such that i) Prob..).3) has a unique weak solution u W M T). ii) The recurrent sequence {u m } defined by.3).4) converges at a rate of order N to the solution u strongly in the space W T) in the sense u m u W T) C u m u N W T).45) for all m C is a suitable constant. Furthermore we have the estimation u m u W T) C T β Nm for all m C T and < β < are positive constants depending only on T. Proof. Put v m = u m+ u m it is clear that v m satisfies the variational problem v mt) v + v mx t) v x = F m+ t) F m t) v v H v m ) = v m) =.46).47) F m x t) = i+j N D ij f u m ]u m u m ) i u m t) u m t) ) j..48) Taking v = v m in.47) after integrating in t we get σ m t) = t Fm+ s) F m s) v ms).49)

12 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) with σ m t) = v mt) + v mx t)..5) On the other hand by using Taylor s expansion for the function f x t u m u m t) ) around the point x t um u m t) ) up to order N we obtain f u m ] f u m ] = f x t u m u m t) ) f x t u m u m t) ) = i+j N D ij f u m ]v i m + D ij f η m ]v i m um t) u m t) ) j um t) u m t) ) j.5) η m ] = x t u m + θv m θ u m t) + θ) u m t) ) < θ <. Hence it follows from.4).5) that F m+ t) F m t) = D ij f u m ]v i m um+ t) u m t) ) j i+j N + D ij f η m ]v i m Then we deduce from.5) that F m+ t) F m t) + K M f ) + K M f ) i+j N i+j N i+j N i+j N um t) u m t) ) j..5) v mxt) i u m+ t) + u m t) ) j u m+ t) u m t) j v m i W T) u mt) + u m t) ) j u m t) u m t) j v mxt) i+j M) j v m i W T) M)j v m t) j γ T v mx t) + γ T v m N W T) γ T = K M f ) i+j N v mxt) i+j M) j v mx t) + K M f ) j M i+j v mx t) + K M f ) j M i+j γ T = K M f ) M)j v m i+j W T) M)j v m N W T) Then we deduce from.49).5) and.53) that σ m t) = t Fm+ s) F m s) v ms) t F m+s) F m s) v m s) t ) γt v mx s) + γ T v m N v W T) m s) t γ T v mxs) v m s) t + γt v m N W T) v ms) T γ T v m N W T) + γ T + γ T ) t σ ms)..53) M)j..54).55)

13 Le Thi Phuong Ngoc et al. / Filomat 3:6 7) By using Gronwall s lemma we obtain from.55) that v m W T) T γ T e Tγ T+ γ T) vm N W T) µ T v m N W T).56) µ T is the constant given by µ T = T γ T e Tγ T+ γ T)..57) Hence we obtain from.56) that u m u m+p W T) β) µ T ) N β Nm.58) for all m and p. We take T > small enough such that β = µ T ) N M <. It follows that {u m } is a Cauchy sequence in W T). Then there exists u W T) such that u m u strongly in W T). It is similar to argument used in the proof of Theorem. we obtain that u W M T) is a unique weak solution of Prob..).3). Passing to the limit as p + for m fixed we get the estimate.46) from.58). This completes the proof of Theorem.5. Remark. In order to construct a N order iterative scheme we need the condition f C N ] R + R R + ). Then we get a convergent sequence at a rate of order N to a local unique weak solution of problem and the existence follows. This condition of f can be relaxed if we only consider the existence of solution it is not necessary that f C ] R + R R + ) see ]. Acknowledgements. The authors wish to express their sincere thanks to the referees for the suggestions remarks and valuable comments. References ] R. A. Adams Sobolev Spaces Academic Press NewYork 975. ] G.F. Carrier On the nonlinear vibrations problem of elastic string Quart. J. Appl. Math ) ] F. Ficken B. Fleishman Initial value problems and time periodic solutions for a nonlinear wave equation Comm. Pure Appl. Math. 957) ] M. Hosoya Y. Yamada On some nonlinear wave equation I: Local existence and regularity of solutions J. Fac. Sci. Univ. Tokyo. Sect. IA Math ) ] G. R. Kirchhoff Vorlesungen über Mathematiche Physik: Mechanik Teuber Leipzig 876 Section ] Lakshmikantham V Leela S Differential and Integral Inequalities Vol.. Academic Press NewYork ] J. L. Lions Quelques méthodes de résolution des problèmes aux limites non-linéaires Dunod-Gauthier -Villars Paris ] N. T. Long T. N. Diem On the nonlinear wave equation u tt u xx = f x t u u x u t ) associated with the mixed homogeneous conditions Nonlinear Anal. TMA. 9 ) 997) ] N.T. Long A. P. N. Dinh T. N. Diem Linear recursive schemes and asymptotic expansion associated with the Kirchhoff-Carrier operator J. Math. Anal. Appl. 67 ) ) ] N. T. Long On the nonlinear wave equation u tt Bt u u x )u xx = f x t u u x u t u u x ) associated with the mixed homogeneous conditions J. Math. Anal. Appl. 36 ) 5) ] L. A. Medeiros On some nonlinear perturbation of Kirchhoff-Carrier operator Comp. Appl. Math ) ] L. T. P. Ngoc L. X. Truong N. T. Long An N order iterative scheme for a nonlinear Kirchhoff-Carrier wave equation associated with mixed homogeneous conditions Acta Mathematica Vietnamica 35 ) ) ] E. L. Ortiz A. P. N. Dinh Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the Tau method SIAM J. Math. Anal ) ] P. H. Rabinowitz Periodic solutions of nonlinear hyperbolic differential equations Comm. Pure. Appl. Math. 967) ] L. X. Truong L. T. P. Ngoc N. T. Long High-order iterative schemes for a nonlinear Kirchhoff - Carrier wave equation associated with the mixed homogeneous conditions Nonlinear Anal. TMA. 7 - ) 9)