Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions

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1 Tahamtani and Peyravi Boundary Value Problems, :9 RESEARCH Open Access Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions Faramarz Tahamtani * and Amir Peyravi * Correspondence: tahamtani@shirazu.ac.ir Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 7454, Iran Abstract This paper is concerned with investigating the spatial behavior of solutions for a class of hyperbolic equations in semi-infinite cylindrical domains, where nonlinear dissipative boundary conditions imposed on the lateral surface of the cylinder. The main tool used is the weighted energy method. Mathematics Subject Classification () 35B4, 35L5, 35L35 Keywords: Hyperbolic equation, Nonlinear boundary conditions, Phragmén-Lindelöf type theorem, Asymptotic behavior Introduction The aim of this paper is to study the spatial asymptotic behavior of solutions of the problem determined by the equation u tt = u t au t u, (x, t) (, ), (:) where a is a positive constant and where = {x R n : x n R, x =(x,..., x n ) Ɣ xn R n }, Ɣ = {(x, x n ) : x n = }. When we consider equation (.), we impose the initial and boundary conditions u(x,) = u t (x,) =, x, (:) u(x,,t) = h (x, t), u v (x,,t) =h (x, t), (x, t) Ɣ (, ), (:3) u =, u = f ( ) u, (x, t) (, ), (:4) ν where ν is the outward normal to the boundary and = {x R n : x Ɣ xn, x n < }, Tahamtani and Peyravi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Tahamtani and Peyravi Boundary Value Problems, :9 Page of 9 where Ɣ is a map from R into family of bounded domains in R n- with sufficiently smooth boundary Γ such that < m inf In the sequel, we are using Ɣ sup Ɣ m <. = {x R n :< x n <}, R = {x R n : <x n < }, and assume f satisfies v F(v) = f (ξ)dξ αvf (v) >, α>, v R, (:5) vf (v) γ v p, p >, γ>, v R. (:6) In recent years, much attention has been directed to the study of spatial behavior of solutions of partial differential equations and systems. The history and development of this question is explained in the work of Horgan and Knowles []. The interested reader is referred to the papers [-9] and the reviews by Horgan and Knowles [,,]. The energy method is widely used to study such results. Spatial growth or decay estimates for nontrivial solutions of initial -boundary value problems in semi-infinite domains with nonlinearities on the boundary have been studied by many authors. Since 98, when Edvard Phragmén and Ernst Lindelöf published their idea [], many authors have obtained spatial growth or decay results by Phragmén-Lindelöf theorems. In [3], Horgan and Payne proved some these types of theorems and showed the asymptotic behavior of harmonic functions defined on a three-dimensional semi-infinite cylinder when homogeneous nonlinear boundary conditions are imposed on the lateral surface of the cylinder. Payne and Schaefer [4] proved such results for some classes of heat conduction problems. In [5], Quintanilla investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions, (see also [6,7]). Under nonlinear dissipative feedbacks on the boundary, Nouria [8] proved a polynomial stability for regular initial data and exponential stability for some analytic initial data of a square Euler-Bernoulli plate. For the used methodology, one can see [9,] where the stabilities are investigated in the cases bounded and unbounded feedbacks for some evolution equations. Recently, Celebi and Kalantarov [] established a Phragmén-Lindelöf type theorems for a linear wave equation under nonlinear boundary conditions. In our study, we establish Phragmén-Lindelöf type theorems for equation (.) with nonlinear dissipative feedback terms on the boundary. Our study is inspired by the results of []. For the proof of our results, we will use the following Lemma. Lemma []Let ψ be a monotone increasing function with ψ() = and lim z ψ(z) =. Then (z) > satisfying (z) < ψ( (z)), z>, tends to when z.

3 Tahamtani and Peyravi Boundary Value Problems, :9 Page 3 of 9 (i) If ψ(z) cz m for some c and m > for z z, then m lim inf m ϕ(z) >. z z (ii) If ψ(z) cz for some c and z z, then lim inf ( ϕ(z) exp z ) >. z c Spatial estimates With the solutions of (.-.4) with h i (x, t) =,i =, is naturally associated an energy function E() = u t u t u uf ( u)dsdη, (:) Ɣ η where. Ω denotes the usual norm in L (Ω). A multiplication of equation (.) by u t, integrating over Ω and using (.3-.5): d u t u F( u)dsdη a u t Since Ɣ η u t = (u t, u xn x n x n ) Ɣ (u txn, u xn x n ) Ɣ (u t, u txn ) Ɣ. (u t, u xn x n x n ) Ɣ = (u txn, u xn x n ) Ɣ, we obtain d u t u Ɣ η u t =(u txn, u xn x n ) Ɣ (u t, u txn ) Ɣ. F( u)dsdη a u t (:) Let δ >. Multiplying (.) by δu, integrating over Ω, and adding to (.), we obtain d { u t u δ(u, u t ) aδ u δ u Ɣ η F( u)dsdη (a δ) u t u t δ u δ Ɣ η uf ( u)dsdη =(u txn, u xn x n ) Ɣ (u t, u txn ) Ɣ δ ) Ɣ δ(u, u txn ) Ɣ. (:3)

4 Tahamtani and Peyravi Boundary Value Problems, :9 Page 4 of 9 Integrating (.3) with respect to t over (, T) and using (.5), one can find u t u δ u aδ u δ(u, u t ) α uf ( u)dsdη (a δ) δ Ɣη Ɣη u t δ uf ( u)dsdη u [δ )Ɣ δ(u, u txn )Ɣ ]. u t [(u txn, u xn x n )Ɣ (u t, u txn )Ɣ ] (:4) On exploiting (.) and the inequality ( 4 ) ut δ u δ(u, u t ),theestimate (.4) takes the form T σ E() [(u txn, u xn x n )Ɣ (u t, u txn )Ɣ ] T [ )Ɣ (u, u txn )Ɣ ], (:5) by choosing δ = a, δ = min{, a }, σ =max{ a δ, δ }. Now we find upper bounds for the right hand side of (.5). Using the Young s and Schwartz inequalities, we have (u txn, u xn x n ) Ɣ u t Ɣ u Ɣ, (:6) (u t, u txn ) Ɣ u t Ɣ u t Ɣ, (:7) ) Ɣ u xn Ɣ u xn x n Ɣ. (:8) By the Poincaré inequality, it is not difficult to see v D λ v D D vda. (:9) D

5 Tahamtani and Peyravi Boundary Value Problems, :9 Page 5 of 9 Inserting (.9) into (.8), we get ) Ɣ λ u Ɣ Ɣ Ɣ uda u x n x n Ɣ, (:) where Δ and are Laplacian and gradient operators in R n-, respectively, Γ isthe area of Γ and l is the Poincaré constant. Now, we recall the inequality D vda r D v ds I D v da, (:) from [3] where r =sup D x and I = D x da. Using (.) and the Hölder s inequality to estimate the boundary integral Ɣ uda in (.), we obtain ) Ɣ M u Ɣ γ p M u p da p u xn x n Ɣ, (:) where M = λ I/ m /, M = rl(p )/p m / γ /p,suchthatr =sup r, l =inf l, I =sup I, L =sup L and m =inf Γ inwhichl istheareaof Γ. From (.6) the inequality (.) yields ) Ɣ M u Ɣ M p uf ( u)da u xn x n Ɣ. (:3)

6 Tahamtani and Peyravi Boundary Value Problems, :9 Page 6 of 9 Consequently p uf ( u)ds u x n x n da = Ɣ p uf ( u)ds Ɣ p p u x n x n da p p μp uf ( p u)ds p μ( p) Ɣ u x n x n da p p, where the Young s inequality ε α ε β ε =(αγ ) ε βγ ε ( ε) ε εαγ ( ε)βγ ε, for < ε <, μ = p p and g = μ p have been used. Therefore, p uf ( u)ds u x n x n da where Ɣ N(p) uf ( u)ds N(p) = p p p (p). By using (.3) and (.4), we get Ɣ u x n x n da p p, (:4) ) Ɣ M M Ñ(p) u Ɣ uf ( u)ds u x n x n da Ɣ p p, (:5)

7 Tahamtani and Peyravi Boundary Value Problems, :9 Page 7 of 9 where Ñ(p) = [ N(p) ] p p. From (.5), it is easy to see ) Ɣ M M CÑ(p) u Ɣ uf ( u)ds where C is a positive constant. Next, we exploit Poincaré inequality to estimate (u, u txn ) Ɣ ρ u Ɣ u Ɣ p p, u t Ɣ, (:7) where r is the Poincaré constant. Now, from the inequalities (.5-.7), (.6), and (.7), one can find E() [ σ u t Ɣ 3 ( ) ] σ u t Ɣ σ M ρ u Ɣ uf ( u)ds σ M CÑ(p) uf ( u)ds [ ut Ɣ u Ɣ u t Ɣ ] Upon inserting (.) into the right hand side of (.8), we may write an inequality in the form ( ) 5 E() σ M ρ E ()σm CÑ(p) [E p ()] p. (:9) p At this point, by the inequality (.9), the function ψ(z) = α z α z p satisfies in the hypothesis of the Lemma. Therefore, we have proved the following theorem. Theorem Let u(x, t) be a nontrivial solution of (.) - (.4) with h i (x, t) =,i =, under the conditions (.5) and (.6). Then and where lim inf E() p p >, p (,), lim inf E() exp( ) >, p [, ), c c =max { ( ) } 5 σ M ρ, σ M CÑ(p). p p.

8 Tahamtani and Peyravi Boundary Value Problems, :9 Page 8 of 9 Theorem Consider the equation (.) subject to the conditions u(x,,t) =h (x, t) and u v (x,,t) =h (x, t)for x Î Γ.IfE( ) is finite, then lim u t R u t R u R =. (:) proof By the same manner followed in theorem, it is easy to find the inequality (a δ) ( 3 δ u t R ) T u t R δ u t Ɣ [δ( λ u R λ )] u t Ɣ u Ɣ, where l is the Poincaré constant. Choosing δ Î (, a), h = min{a -δ, δ, } and we obtain where γ = η max{ 3 δ,δ( λ λ )}, Ẽ() γ Ẽ (), (:) Ẽ() = u t R u t R u R. Thus, (.) follows from (.). Authors contributions The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 3 April Accepted: 3 August Published: 3 August References. Horgan, CO, Knowles, JK: Recent developments concerning Saint-Venant s principle. In: Wu TY, Hutchinson JW (eds.) Advances in Applied Mechanics, vol. 3, pp Academic Press, New York (983). Celebi, AO, Kalantarov, VK, Tahamtani, F: Phragmén-Lindelöf type theorems for some semilinear elliptic and parabolic equations. Demonstratio Mathematica. 3, (998) 3. Flavin, JN: On Knowles version of Saint-Venant s principle in two-dimensional elastostatics. Arch Ration Mech Anal. 53, (974). doi:.7/bf Flavin, JN, Knops, RJ, Payne, LE: Decay Estimates for the Constrained Elastic Cylinder of Variable Cross Section. Quart Appl Math. XLVII, (989) 5. Flavin, JN, Knops, RJ: Asymptotic behaviour of solutions to semi-linear elliptic equations on the half cylinder. Z Angew Math phys. 43, 45 4 (99). doi:.7/bf Flavin, JN, Rionero, S: Qualitative Estimates for Partial Differential Equations, An Introduction. CRC Press, Roca Raton (996) 7. Horgan, CO: Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flow. Quart Appl Math. 47, (989) 8. Knowles, JK: On Saint-Venant s principle in the two-dimensional linear theory of elasticity. Arch Ration Mech Anal., 3 44 (966) 9. Knowles, JK: An energy estimate for the biharmonic equation and its application to Saint-Venant s principle in plane elastostatics. Indian J Pure Appl Math. 4, (983)

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