A HYPERBOLIC PROBLEM WITH NONLINEAR SECOND-ORDER BOUNDARY DAMPING. G. G. Doronin, N. A. Lar kin, & A. J. Souza

Transcription

1 Electronic Journal of Differential Equations, Vol. 1998(1998), No. 28, pp ISSN: URL: or ftp or (login: ftp) A HYPERBOLIC PROBLEM WITH NONLINEAR SECOND-ORDER BOUNDARY DAMPING G. G. Doronin, N. A. Lar kin, & A. J. Souza Abstract. The initial boundary value problem for the wave equation with nonlinear second-order dissipative boundary conditions is considered. Existence and uniqueness of global generalized solutions are proved. 1. Introduction In [1], J.L. Lions considers nonlinear problems on manifolds in which the unknown ω satisfies the Laplace equation in a cylinder Q and a nonlinear evolution equation of the form ω ν + ω tt + ω t ρ ω t = (1.1) on the lateral boundary Σ of Q. Hereν is an outward normal vector on Σ. This problem models water waves with free boundaries ([2], [3]). The boundary condition ω ν + ω t ρ ω t = (1.2) arises when one studies flows of a gas in channels with porous walls [4, 5]. The presence of the second derivative with respect to t in the boundary condition is due to internal forces acting on particles of the medium at the outward boundary. Motivated by this, we study in the present paper the wave equation with the nonlinear boundary condition and with the initial data u tt u = f in Q (1.3) u ν + K(u)u tt + u t ρ u t = on Σ (1.4) u(x, ) = u t (x, ) =. (1.5) The term K(u)u tt models internal forces when the density of the medium depends on the displacement Mathematics Subject Classifications: 35A5, 35L2, 35A35, 49M15. Key words and phrases: Initial Boundary-Value Problem, Faedo-Galerkin Method. c 1998 Southwest Texas State University and University of North Texas. Submitted May 22, Published October 3, Partially supported by the Conselho Nacional de Desenvolvimento Tecnológico e Científico, Brazil, under Grants /95-, 4542/97-9 and /

2 2 G. G. Doronin, N. A. Lar kin, & A. J. Souza EJDE 1998/28 In [1] it is shown that (1.1) can be replaced by the evolution equation u tt + A(u)+ u t ρ u t = on Σ, where A is a linear positive self-adjoint operator. In that sense, the expression (1.2) looks like a semilinear hyperbolic equation on the manifold Σ. Equation (1.4) also behaves as a hyperbolic equation with nonlinear principal operator. Generally speaking, quasilinear hyperbolic equations do not have global regular solutions. There are examples of blow-up at a finite time. (See, for instance, [6].) Nevertheless, the presence of linear damping allows proof of the existence of global solutions for small initial data ([7]). Moreover, a nonlinear damping makes it possible to prove global existence theorems for some quasilinear wave equations without restrictions on a size of the initial conditions ([8], [9]). Here we use the ideas from [8] to prove the existence of global generalized solutions to the problem (1.3)-(1.5). We exploit the Faedo-Galerkin method, a priori estimates and compactness arguments. Uniqueness is proved in the one-dimensional case. We consider the classical wave equation only to simplify calculations. Similar results hold for a second-order evolution equation of the form u tt + A(t)u + F (u, u t )=f, where A(t) is a linear, strictly elliptic operator, and F (u, u t ) is a suitable function of u and u t. Moreover, hyperbolic-parabolic or elliptic equations also may be considered. 2. The Main Result For T>, let Ω be a bounded open set of R n with sufficiently smooth boundary Γ and Q= Ω (,T). We consider the hyperbolic problem u tt u = f(x, t), (x, t) Q; (2.1) ( u ν + K(u)u tt + u t ρ u t) =; Σ 1 u Σ = ; (2.2) u(x, ) = u t (x, ) =. (2.3) Here K(u) is a continuously differentiable positive function; ν is the outward unit normal vector on Γ; Γ = Γ ;Γ = ;Σ i =Γ i (,T) (i =,1); ρ (1, ). We denote by H 1 (Ω) the Sobolev space H 1 (Ω) with the condition u Γ =; (u, v)(t) = Ω u(x, t)v(x, t) dx; u is the norm in L2 (Ω): u 2 (t) = (u, u)(t); u = n i=1 2 u/ x 2 i. Definition. A function u(x, t) such that u L (,T;H 1 (Ω)), u t L (,T;H 1 (Ω)) L ρ+2 (Σ 1 ), u tt L (,T;L 2 (Ω) L 2 ( )), u(x, ) = u t (x, ) =

3 EJDE 1998/28 A hyperbolic problem 3 is a generalized solution to (2.1)-(2.3) if for any functions v H 1 (Ω) L ρ+2 (Γ) and ϕ C 1 (,T)withϕ(T) = the following identity holds: T (u [ tt,v)(t)+( u, v)(t)+ ut ρ u t K (u)u 2 ] t vdγ ϕ(t)dt T ϕ (t) K(u)u t vdγdt = T (f,v)ϕ(t)dt. We consider functions K(u) satisfying the assumptions (2.4) <K K(u) C(1 + u ρ ), (2.5) ρ K (u) ρ 1 C(1 + K(u)). (2.6) These conditions mean that the density of the medium can not increase too rapidly as a function of displacement. The condition (2.6) appears quite naturally because functions with polynomial growth, such as K(u) =1+ u s with 1 s ρ, satisfy it. The inequality K(u) K means that the vacuum is forbidden. The main result of this paper is the following. Theorem. Let the function K(u) satisfy assumptions (2.5) and (2.6) and suppose f(x, t) H 1 (,T; L 2 (Ω)). Then for all T>there exists at least one generalized solution to the problem (2.1)-(2.3). If n =1, this solution is unique. Proof. We prove the existence part of the Theorem by the Faedo-Galerkin method. First, we construct approximations of the generalized solution. Then we obtain a priori estimates necessary to guarantee convergence of approximations. Finally, we prove the uniqueness in the one-dimensional case. 3. Approximate solutions Let {w j (x)} be a basis in H 1 (Ω) L ρ+2 ( ). We define the approximations u N (x, t) = g i (t)w i (x), (3.1) where g i (t) are solutions to the Cauchy problem (f,w j )(t) =(u N tt,w j )(t)+( u N {, w j )(t)+ K(u N )u N tt + u N t ρ u N t i=1 } wj dγ; (3.2) g j () = g j() = ; j =1,..., N. (3.3) It can be seen that (3.2) is not a normal system of ODE; therefore, we can not apply the Caratheodory theorem directly. To overcome this difficulty, we have to prove that the matrix A defined by (Ag ) j = g j (t)+ { K(u N ) i=1 g i (t)w i(x) } w j (x) dγ ; j=1,..., N (3.4)

4 4 G. G. Doronin, N. A. Lar kin, & A. J. Souza EJDE 1998/28 has an inverse. Multiplying (3.2) by g j (t) and summing over j, we obtain the quadratic form N q (g 1,..., g N )= (g j )2 + j=1 i=1 K(u N )w i w j dγ g The condition K(u) K > implies that for any g (t) q = (g j )2 + j=1 K(u N ) j=1 g j w j 2 dγ j=1 (g i g j. j )2 + K u N tt 2 L 2 ( ) >. Hence, the quadratic form q is positive definite and all eigenvalues of the symmetric matrix A in (3.4) are positive. Thus, (3.2) can be reduced to normal form and, by the Caratheodory theorem, the problem (3.2),(3.4) has solutions g j (t) H 3 (,t N ) and all the approximations (3.1) are defined in (,t N ). 4. A priori estimates Next, we need a priori estimates to show that t N = T and to pass to the limit as N. To simplify the exposition, we omit the index N whenever it is unambiguous to do so. Multiplying (3.2) by 2g j and summing from j =1toj=N,weobtain 2(f,u t )(t) = d dt ( ut 2 + u 2) (t)+ 2 u t ρ+2 dγ + { d dt } ( ) K(u)u 2 t K (u)(u t ) 3 dγ. Integrating with respect to τ from to t, weget 2 t Notice that (f,u τ )dτ = ( u t 2 + u 2) (t) t +2 { u τ ρ+2 12 } K (u)(u τ ) 3 dγ dτ + t 2 u τ { u 2 τ ρ 12 } K (u)u τ dγ dτ t { } 2 u τ 2 u τ ρ ε u τ ρ C(ε) K ρ (u) ρ 1 dγ dτ, K(u)u 2 t dγ.

5 EJDE 1998/28 A hyperbolic problem 5 where ε is an arbitrary positive number. From now on, we denote by C all constants independent of N. Fixing ε =1/2, taking into account (2.6), and applying the Cauchy-Schwarz inequality, we get ( ut 2 + u 2) t (t)+ u τ ρ+2 dγ dτ + K(u)u 2 t dγ t ( f 2 + u τ 2) t (τ) dτ + C u τ 2 (1 + K(u)) dγ dτ. (4.1) Note that K(u) C (1 + K(u)) where 2C =min{1,k }. Therefore, for the function E 1 (t) = ( u t 2 + u 2) (t)+c (1 + K(u)) u t 2 dγ we have from (4.1) the inequality E 1 (t) C 1+ t E 1 (τ)dτ. By Gronwall s lemma, we conclude that, for all t (,T)andforallN 1, E 1 (t) C. This and (4.1) give that for all t (,T), t u τ ρ+2 dγ dτ C, K(u N )(u N t ) 2 dγ C, where C does not depend on N. In order to obtain the second a priori estimate, we observe that u tt () f (); (4.2) u 2 tt(x, )dγ f 2 /K(). (4.3) Indeed, multiplying (3.2) by g j (), summing over j, and setting t =,weobtain (u tt,u tt )() + K()u 2 tt (x, ) dγ =(f,u tt)()

6 6 G. G. Doronin, N. A. Lar kin, & A. J. Souza EJDE 1998/28 which implies (4.2). Consequently, K()u 2 tt (x, ) dγ f () u tt () f 2 (), which gives (4.3). Differentiating (3.2) with respect to t, multiplying by g j, and summing over j, we obtain the identity (f t,u tt )(t) = 1 d ( utt 2 + u t 2) (t) 2dt { + K(u)utt u ttt + K (u)u t u 2 tt +(ρ+1) u t ρ u 2 } tt dγ. Notice that and K(u)u tt u ttt = 1 2 K (u)u t u 2 tt dγ ε ε d ( ) K (u) u 2 1 dt tt 2 K (u)u t u 2 tt u t ρ u tt 2 dγ+c(ε) u t ρ u tt 2 dγ+c(ε) K (u) ρ ρ 1 utt 2 dγ (1 + K(u)) u tt 2 dγ. Setting ε = ρ, wehave 1d u tt 2 + u t 2 + 2dt ( f t 2 + u tt 2) (t)+c K(u)u 2 tt dγ (t)+ (1 + K(u)) u tt 2 dγ. u t ρ u tt 2 dγ (4.4) Defining E 2 (t) as E 2 (t)= u tt 2 + u t 2 + C (1 + K(u)) u tt 2 dγ (t) and taking into account (4.2), (4.3), we reduce (4.4) to the form E 2 (t) C 1+ t E 2 (τ)dτ. By Gronwall s lemma, for all t (,T),N 1weobtain E 2 (t) C.

7 EJDE 1998/28 A hyperbolic problem 7 Taking into consideration that u Σ =, we obtain the following statements u N L (,T;H 1 (Ω)); u N t L (,T;H 1 (Ω)) L ρ+2 (Σ) L (,T;L 2 (Γ)); u N tt L (,T;L 2 (Ω) L 2 (Γ)); (4.5) t un t 1+ρ/2 L 2 (Σ); K 1/2 (u N )u N tt L (,T;L 2 (Γ)). 5. Passage to the limit Multiply (3.2) by ϕ C 1 (,T)withϕ(T) = and integrate with respect to t from to T. After integration by parts, we obtain T (un tt,w j )+( u N, w j )+ T T ϕ (t) ϕ(t) u N t ρ u N t w j dγ ϕ(t)dt K(u N )u N t w j (x) dγ dt + ϕ(t)k(u N )u N t T K (u N )(u N t ) 2 w j dγ dt = T (f,w j )ϕ(t)dt. Because of (4.5) we can extract a subsequence u µ from u N such that: Therefore, u µ u weakly star in L (,T;H 1 (Ω)); u µ t u t weakly star in L (,T;H 1 (Ω)) L ρ+2 (Σ); u µ tt u tt weakly star in L (,T;L 2 (Ω) L 2 (Γ)); u µ,u µ t u, u t a.e. on Σ. u µ t ρ u µ t L q (Σ), q =(ρ+2)/(ρ+1)>1, and converges a.e. on Σ; K(u µ )u µ t L q (Σ), and converges a.e. on Σ; K (u µ )(u µ t ) 2 L q (Σ), and converges a.e. on Σ. (5.1) Thus, we are able to pass to the limit in (5.1) to obtain T (u ( tt,w j )+( u, w j )+ ut ρ u t K (u)u 2 ) t wj dγ ϕ(t)dt T ϕ (t) K(u)u t w j dγ dt = T (f,w j )ϕ(t)dt. (5.2)

8 8 G. G. Doronin, N. A. Lar kin, & A. J. Souza EJDE 1998/28 It can be seen that all the integrals in (5.2) are defined for any function ϕ(t) C 1 (,T),ϕ(T) =. Taking into account that {w j (x)} is dense in H 1 (Ω) L ρ+2 (Γ), we conclude that (2.4) holds. If n =1,2, one can get more regular solutions. In this case u L (,T;L q (Γ)) for any q [1, ). Hence, K(u)u tt L (,T;L p (Γ)), where p is an arbitrary number from the interval [1, 2). This allows us to rewrite (5.2) in the form T T (f,w j )dt = (u tt,w j )+( u, w j )+ (K(u)u tt + u t ρ u t ) w j dγ ϕ(t) dt. Taking into account that almost every point t (,T) is a Lebesgue point and that w j (x) are dense in H 1 (Ω) and therefore in L q (Γ), we obtain (u tt,v)(t)+( u, v)(t)+ {K(u)u tt + u t ρ u t }vdγ=(f,v)(t), where v is an arbitrary function from H 1 (Ω). 6. Uniqueness Let n =1. Letuand v be two solutions to (2.1)-(2.3), and set z(x, t) =u(x, t) v(x, t). Then for fixed t, for every function φ H 1 (Ω), we have + (z tt,φ)(t)+( z, φ)(t) {K(u)z tt + v tt (K(u) K(v)) + u t ρ u t v t ρ v t }φdγ=. Since z t (x, t) L (,T;H 1 (Ω)), we may take φ = z t, and this equation can be reduced to the inequality 1 d E(t)+ K(u)(z t ) 2 dγ 2 dt + {v tt z t (K(u) K(v)) 12 } K (u)u t (z t ) 2 dγ. Here we set E(t) = z t 2 (t)+ z 2 (t) and use the monotonicity of u t ρ u t,the differentiability of K, and the regularity of K(u)u tt (see the end of previous section). Condition (2.6) then implies that 1 d E(t)+ K(u)(z t ) 2 dγ 2 dt C max(1 + K(u)) ρ 1 ρ ut (z t ) 2 1 { dγ+ zt 2 + v tt 2 K(u) K(v) 2} dγ 2 C z t 2 dγ+max K(u) K(v) 2 v tt 2 dγ C 1 z t 2 L 2 ( ) + C 2 v tt 2 L 2 ( ) z 2 C( ).

9 EJDE 1998/28 A hyperbolic problem 9 Integrating from to t, using (2.5) and the Sobolev embedding theorem ([1]), we obtain z t 2 (t)+ z 2 (t)+ z t 2 L 2 ( ) (t) C t [ ] z t 2 L 2 ( ) (τ)+ z 2 (τ) dτ. This implies that z =andu=va.e. in Q. The proof of the Theorem is completed. Remark. We use homogeneous initial conditions (2.3) for technical reasons. Nonhomogeneous initial data also can be considered without any restrictions on their size ([1]). In fact, suppose that initial conditions are imposed as follows u(x, ) = u (x), u t (x, ) = u 1 (x), x Ω. Using the transformation v(x, t) = u(x, t) u (x) u 1 (x) t, we obtain the problem v tt v = F (x, t) in Q; (6.1) v ν + φ ν + K(v + φ)v tt + v t + u 1 ρ (v t + u 1 )= onσ 1 ; (6.2) v + φ = onσ ; (6.3) v(x, ) = v t (x, ) = in Ω. (6.4) Here φ(x, t) =u (x)+u 1 (x) tand F (x, t) =(f+ φ)(x, t) are given functions. It is clear that for regular solutions the compatibility conditions u ν + K(u )(f + u )+ u 1 ρ u 1 Γ1 =; u Γ = need to be satisfied. This implies that conditions (6.2)-(6.4) are also compatible. If (u,u 1 )(x) H 2 (Ω), than F (x, t) H 1 (,T;L 2 (Ω)). Moreover, if u 1 L ρ+2 ( ), then we are able to obtain necessary a priori estimates and to pass to the limit by the method of Sections 4 and 5. Of course, the use of conditions (6.2), (6.3) in place of (2.2) complicates calculations, but does not affect the final result. Acknowledgments. We thank the referee for his/her valuable suggestions. REFERENCES [1] Lions, J.-L., Quelques méthodes de résolution des problèmas aux limites non linéaires, Paris, Dunod, [2] Garipov, R.M., On the linear theory of gravity waves: the theorem of existence and uniqueness, Archive Rat. Mech. Anal., 24 (1967), [3] Friedman A., Shinbrot M., The initial value problem for the linearized equations of water waves, J. Math. Mech., 17 (1967), [4] Couzin A.T., Lar kin N.A., On the nonlinear initial boundary value problem for the equation of viscoelasticity, Nonlinear Analysis. Theory, Methods and Applications, 31 (1998),

10 1 G. G. Doronin, N. A. Lar kin, & A. J. Souza EJDE 1998/28 [5] Greenberg J.M., Li Ta Tsien, The Effect of Boundary Damping for the Quasilinear Wave Equation, J. of Differential Equations, 52 (1984), [6] Roždestvenskii, B.L., Janenko, N.N., Systems of Quasi-linear Equations and their Applications to Gas Dynamics, Providence, AMS, [7] Matsumura A., Global existence and asymptotics of the solutions of second-order quasilinear hyperbolic equations with first-order dissipation, Publ. Res. Inst. Sci. Kyoto Univ., 13 (1977), [8] Larkin, N.A., Global solvability of boundary value problems for a class of quasilinear hyperbolic equations, Siber. Math. Journ., 22, (1981), [9] Frota, C., Larkin, N.A., On local and global solvability of nonlinear mixed problems modeling vibrations of a string, Proc. of 7-th Intern. Col. on Diff. Equats., August 1966, Plovdiv, Bulgaria, VSP Edition, (1997), [1] Sobolev, S.L., Some Applications of Functional Analysis in Mathematical Physics, Providence, AMS, G. G. Doronin Current address: DME-CCT-UFPB, CEP , Campina Grande, PB, Brazil. Permanent address: 639, Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia address: gleb@dme.ufpb.br N. A. Lar kin Departamento de Matemática, Fundação Universidade Estadual de Maringá, CEP 972-9, Maringá, PR, Brazil address: nalarkin@gauss.dma.uem.br A. J. Souza Departamento de Matemática e Estatística, Universidade Federal da Paraíba, CEP , Campina Grande, PB, Brazil address: cido@dme.ufpb.br