Mem. Differential Equations Math. Phys. 36 (2005), T. Kiguradze

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1 Mem. Differential Equations Math. Phys. 36 (2005), T. Kiguradze and EXISTENCE AND UNIQUENESS THEOREMS ON PERIODIC SOLUTIONS TO MULTIDIMENSIONAL LINEAR HYPERBOLIC EQUATIONS (Reported on June 20, 2005) In R n consider the linear hyperbolic equations u (m) = p α(x α)u (α) + p α(x α)u (α) + q(x), () α E m α O m u (m) = p 0 (x)u + q(x), (2) where n 2, x = (x,..., x n) R n, m = (m,..., m n) Z n + and α = (α,..., α n) Z n + are multi indeces, and u (α) = α + +α n u x α.... xαn n We make use of following notations and definitions. Z + is the set of all nonnegative integers; Z n + is the set of all multiindices α = (α,..., α n); α = α + + α n; 0 = (0,..., 0) Z n +. The inequalities between the multiindices α = (α,..., α n) and β = (β,..., β n) are understood componentwise. It will be assumed that m > 0. If for some multiindex α = (α,..., α n) we have α i = = α ik = 0 (i < < i k ), and α j,..., α jn k > 0 (j < < j n k ), {j,..., j n k } = {,..., n} \ {i,..., i k }, then by x α (by x α ) denote the vector (x i,..., x ik ) R k (the vector (x j,..., x jn k ) R n k ). If α > 0, then in equation () by p α(x α) we understand a constant function. A multiindex α Z n + will be called even, if all its components are even. A multiindex α Z n + will be called odd, if α is odd. By E m and O m, respectively, denote the sets of all even and odd multiindices not exceeding m and different from m, i.e., E m = { α Z n + \ {m} : α m, α,..., α n are even }, O m = { α Z n + \ {m} : α m, α + + α n is odd }. By S m denote the set of nonzero multiindices α = (α,..., α n) whose components either equal to the corresponding components of m, or equal to 0, i.e., S m = { α 0 : α i {0, m i } (i = 0,..., n) }. Let ω = (ω,..., ω n) R n b a vector with positive components. Then by denote the rectangular box [0, ω ] [0, ω n] in R n. Moreover, for an arbitrary multiindex α, similarly as we did above, introduce the vectors ω α = (ω i,..., ω ik ) R k and ω α = (ω j,..., ω jn k ) R n k, and the rectangular boxes α = [0, ω i ] [0, ω ik ] in R k and α = [0, ω j ] [0, ω jn k ] in R n k Mathematics Subject Classification. 35L35, 35B0. Key words and phrases. Periodic solution, multidimensional, linear hyperbolic equation.

2 43 We say that a function z : R n R is ω periodic, if z(x,..., x j + ω j,..., x n) z(x,..., x n) (j =,..., n). It will be assumed that the functions p α (α E m O m ) and q, respectively, are ω α periodic and ω periodic continuous functions. Let l = (l,..., l n) Z n +. By Cl denote the space of continuous functions u : R n R, having continuous partial derivatives u (α) (α l). By a solution of equation () (equation (2)) we will understand a classical solution, i.e., a function u C m satisfying equation () (equation (2)) everywhere in R n. In the case, where n = 2, m = m 2 = (n = 2, m = m 2 = 2) sufficient conditions for existence and uniqueness of (ω, ω 2 ) periodic solutions of equation () are given in [ 3, 6 8] (in [9, 0]). In the general case the problem on ω periodic solutions to equations () and (2) are little investigated. In the present paper optimal sufficient conditions of existence and uniqueness of ω periodic solutions to equation () (equation (2)) are given. Similar results for higher order nonlinear ordinary differential equations were obtained by I. Kiguradze and T. Kusano [5]. We consider equations () and (2) in two cases, where m is either even, or odd. Also note that equations () and (2) do contain partial derivatives with even or odd (according to the above definitions) multiindices only (e.g., neither of m and α can equal to (,,, )). Theorem. Let m be even, and let ( ) m + α 2 p α(x α) 0 for x R n α E m, (3) R n \ I p0 = R n, (4) where I p0 = {x R n : p 0 (x) = 0}. Then equation () has at most one ω-periodic Theorem 2. Let m be odd, and let there exist j {, 2} such that along with (4) the inequality α j+ ( ) 2 p α(x α) 0 for x R n α E m (5) holds. Then equation () has at most one ω-periodic Theorems and 2 almost immediately follow from the following lemma. Lemma. Let u C m be an ω periodic function. Then u (α) (x) u(x) dx α = ( ) α ( ) 2 u α2 (x) 2 dx α α α for α E m, u (α) (x) u(x) dx α = 0 α for α O m. One can easily prove the lemma using integration by parts and taking into consideration ω periodicity of u. Proof of Theorem. All we need to prove is that if q(x) 0, then equation () has only a trivial ω periodic Indeed, let q(x) 0, and let u be an arbitrary ω periodic solution of equation (). After multiplying equation () by u and integrating over the rectangular box, by Lemma and condition (3), we get ( u ( m2 )(x) 2 + ( p α(x α) α2 u )(x) 2) dx = 0. (6) α E m (4) and (6) immediately imply that u(x) 0. We omit the proof of Theorem 2, since it is similar to the proof of Theorem.

3 44 Theorem 3. Let m be even, and let 0 ( ) m 2 p 0 (x) < (2π) m ω m, R n \ I ωn p0 = R n. (7) Then equation (2) has at most one ω-periodic To prove the theorem along with Lemma we need the following Lemma 2. Let m be even, and let u C m be an ω periodic function. Then ( ) m2 u (x) 2 dx ωm ω n (2π) m u (m) (x) 2 dx. (8) This lemma immediately follows from Wirtinger s inequality ([4], Theorem 258). Proof of Theorem 3. Assume the contrary: let q(x) 0 and equation (2) have a nontrivial ω periodic solution u. Then we have and u (m) (x) = p 0 (x)u(x) (9) u (m) (x) 2 = p 0 (x)u(x) 2. (0) Multiplying (9) by u, integrating over, by Lemma, we get p 0 (x) u(x) 2 ( m2 ) dx = u (x) 2 dx. () Integrating (0) over and assuming that u(x) 0, by condition (8), we get u (m) (x) 2 dx = p 0 (x)u(x) 2 dx< (2π) m ω m p ωn 0 (x) u(x) 2 dx. (2) On the other hand, from (8) and () we get the inequality p 0 (x) u(x) 2 ω m dx ω n (2π) m u (m) (x) 2 dx, which contradicts to (2). The obtained contradiction completes the proof of the theorem. Remark. In Theorem 3 condition (7) is optimal and it cannot we weakened: strict inequality cannot be replaced by an unstrict one. Indeed, consider the equation where l is a constant. If u (m) = l u, (3) 0 < l < ( ) m (2π) m ω m, ωn then by Theorem 3 equation (3) has only a trivial However, if l = ( ) m (2π) m ω m ωn ( l = 0 ), then it is obvious that the function ( 2π ) ( 2π ) (u(x) ) u(x) = sin x sin x n = ω ω n is a nontrivial ω solution of equation (3). Below we formulate existence theorems.

4 45 Theorem 4. Let m be even, and let along with (3) the inequalities ( ) m + α 2 p α(x α) dx α < 0 for α S m, (4) α p 0 (x) dx 0 hold. Then equation () has one and only one ω-periodic Theorem 5. Let m be the only odd component of the the multiindex m, and let there exist j {, 2} such that along with (5) the inequalities α j+ ( ) 2 p α(x α) dx α < 0 for α S m, (5) α ω2 ωn ( ) j... p 0 (x, x 2,..., x n) dx 2... dx n < 0 for x R 0 0 hold. Then equation () has one and only one ω-periodic Remark 2. In Theorems 4 (Theorem 5) condition (4) (condition (5)) is essential and it cannot be weakened. If for at least one α S m p α(x α) 0, then equation () may not have an ω periodic To verify this, consider the equation u (2,2,2) = u (2,2,0) + u (2,0,2) + u (0,2,2) u (0,2,0) u (0,0,2) + sin 2 (x ) u. (6) In the case, where n = 3, m = m 2 = m 3 = 2 and ω = ω 2 = ω 3 = π, this equation satisfies all of the conditions of Theorem 4, except condition (4). For α = (2, 0, 0) we have p α(x 2, x 3 ) 0. As a result equation (6) has no (π, π, π) periodic Assume the contrary: let equation (6) have a (π, π, π) periodic solution u. By Theorem, it is unique, and therefore is independent of x 2 and x 3. Hence u satisfies the equation sin 2 (x ) u = 0. But the latter equation has only a discontinuous The obtained contradiction proves that equation (6) has no (π, π, π) periodic Theorem 6. Let m be even, and let Theorem 7. Let m be even, and let 0 < ( ) m 2 p 0 (x) < (2π) m ω m. (7) ωn ( ) m 2 p 0 (x) < 0 for x R n. (8) Theorem 8. Let m be odd, and let there exist a number j {, 2} such that ( ) j p 0 (x) < 0 for x R n. (9) Remark 3. In Theorems 6, 7 and 8 the requirement of additional regularity of functions p 0 and q is sharp. If this condition is violated, then equation (2) may not have a ω periodic classical Indeed, consider the equation u (m) = p 0 (x 2,..., x n) u p 2 0 (x 2,..., x n),

5 46 where m is even, and p 0 (x 2,..., x n) is an arbitrary continuous (ω 2,..., ω n) periodic function satisfying (8). By Theorem 3, this equation has at most one Hence u(x) = p 0 (x 2,..., x n). But u is a classical solution if and only if p 0 C m. Remark 4. In Theorems 6, 7 and 8, respectively, the strict inequalities (7), (8) and (9) cannot be replaced by unstrict ones. To verify this, consider the equation u (m) = p 0 (x 2,..., x n) u, where m is odd and p 0 (x 2,..., x n) is an smooth (ω 2,..., ω n) periodic function such that p 0 (x 2,..., x n) 0, p 0 (x 2,..., x n) 0. By Theorem 2, this equation has at most one Therefore u is a solution of the equation p 0 (x 2,..., x n) u = 0. But the latter equation has a continuous solution if and only if p 0 (x 2,..., x n) > 0 for (x 2,..., x n) R n. References. A. K. Aziz and S. L. Brodsky, Periodic solutions of a class of weakly nonlinear hyperbolic partial differential equations. SIAM J. Math. Anal. 3 (972), No 2, A. K. Aziz and M. G. Horak, Periodic solutions of hyperbolic partial differential equations in the large. SIAM J. Math. Anal. 3(972), No, L. Cesari, Existence in the large of periodic solutions of hyperbolic partial differential equations. Arch. Rational Mech. Anal. 20 (965), No 2, G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities. Cambridge Univ. Press, Cambridge, I. Kiguradze and T. Kusano, On periodic solutions of higher order nonautonomous ordinary differential equations. (Russian) Differentsialnye Uravneniya 35 (999), No., 72-78; translation in Differ. Equations 35 (999), No., T. Kiguradze, On periodic in the plane solutions of second order linear hyperbolic systems. Arch. Math. 33 (997), No 4, T. Kiguradze, On doubly periodic solutions of a class of nonlinear hyperbolic equations. (Russian). Differentsialnye Uravneniya 34 (998), N 2, ; translation in Differ. Equations 34 (998), No. 2, T. Kiguradze, On periodic in the plane solutions of nonlinear hyperbolic equations. Nonlin. Anal. 39 (2000), T. Kiguradze and V. Lakshmikantham, On doubly periodic solutions of forthorder linear hyperbolic equations. Nonlinear Analysis 49 (2002), T. Kiguradze and T. Smith, On doubly periodic solutions of quasilinear hyperbolic equations of the fourth order. Proc. Confer. Differential & Difference Equations Appl. (Melbourne, Florida, 2005) (to appear). Author s address: T. Kiguradze Florida Institute of Technology Department of Mathematical Sciences 50 W. University Blvd. Melbourne, Fl 3290 USA tkigurad@fit.edu