On an initial-value problem for second order partial differential equations with self-reference

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1 Note di Matematica ISSN , e-issn Note Mat no. 1, doi:1.185/i15993v35n1p75 On an initial-value problem for second order partial differential equations with self-reference Nguyen T.T. Lan Faculty of Mathematics and Applications, Saigon University, 73 An Duong Vuong Str., Ward 3, district 5, Ho Chi Minh City, Viet Nam. Received: ; accepted: Abstract. In this paper, we study the local existence and uniqueness of the solution to an initial-value problem for a second-order partial differential equation with self-reference. Keywords: Cauchy problem, second-order partial differential equation, self-reference MSC classification: primary 35R9, secondary 35F55 45G15 1 Introduction In [1], Eder obtained the existence, uniqueness, analyticity and analytic dependence of solutions to the following equation of an one-variable unknown function u : I R R : u t = u ut. 1.1 This is so-called a differential equation with self-reference, since the right-hand side is the composition of the unknown and itself. This equation has attracted much attention. As a more general case than 1.1, Si and Cheng [4] investigated the functional-differential equation u t = u at + but, 1. where a 1 and b are complex numbers; the unknown u : C C is a complex function. By using the power series method, analytic solutions of this equation are obtained. By generalizing 1., in [9] Cheng, Si and Wang considered the equation αt + βu t = u at + bu t, c 15 Università del Salento

2 76 Nguyen T.T. Lan where α and β are complex numbers. Existence theorems are established for the analytic solutions, and systematic methods for deriving explicit solutions are also given. In [11], Stanek studied maximal solutions of the functional-differential equation utu t = ku ut 1.3 with < k < 1. Here u : I R R is a real unknown. This author showed that properties of maximal solutions depend on the sign of the parameter k for two separate cases k 1, and k, 1. For earlier work of Stanek than 1.3, see [16] [1]. For a more general model than the above, in [6], Miranda and Pascali studied the existence and uniqueness of a local solution to the following initial-valued problem for a partial differential equation with self-reference and heredity t ux, t = u ux, sds, t, x R, a.e. t >, t ux, = u x, x R, 1.4 by assuming that u is a bounded, Lipschitz continuous function. With suitable weaker conditions on u, namely u is a non-negative, non-decreasing, bounded, lower semi-continuous real function, in [3], Pascali and Le obtained the existence of a global solution of 1.4. In [], T. Nguyen and L. Nguyen, generalizing [7], studied the system of partial differential equations with self-reference and heredity t ux, t = u αvx, t + v ux, sds, t t, t t 1.5 vx, t = v βux, t + u vx, sds, t t, t associated with initial conditions ux, = u x, vx, = v x, 1.6 where α and β are non-negative coefficients. By the boundedness and Lipschitz continuity of u and v, we obtained the existence and uniqueness of a local solution to this system. We also proved that this system has a global solution, provided u and v are non-negative, non-decreasing, bounded and lower semicontinuous functions.

3 On an initial-value problem for second order partial differential equations 77 In [5], Pascali and Miranda considered an initial-valued problem for a secondorder partial differential equation with self-reference as follows: t ux, t = k 1u t ux, t + k ux, t, t, ux, = αx, 1.7 ux, = βx. t These authors proved that if αx and βx are bounded and Lipschitz continuous functions, k 1 and k are given real numbers, this problem has a unique local solution. It is noted that this result still holds when k i k i x, t, i = 1,, are real functions satisfying some technical conditions. Motivated from problem 1.7 and related questions in [5], in this paper we establish the existence and uniqueness of a local solution to the following Cauchy problem of an partial differential equation with self-reference: t ux, t = µ 1u t ux, t + µ u t ux, t + µ 3ux, t, t, t, ux, = px 1.8 ux, = qx, t where p and q are given functions, µ i, i = 1,, 3, given real numbers x R and t [, T ] for some T >. It is clear that this problem is a non-trivial generalization of 1.7. Let us specify some reasons as follows: The operator t ux, t + µ u t ux, t + µ 3ux, t, t is actually a doubly self-reference form, which is more complicated than that of 1.7; If k = µ =, problem 1.8 coincides with problem 1.7. This is the only coincidence of these two problems. This means that the problem we study in this paper is not a natural generalization of 1.7, not including 1.7 as a special case. Finally we present the problem 1.8 in the case that px = p and qx = q, where p and q are two given constants and we remark a particular strange situation.

4 78 Nguyen T.T. Lan Existence and uniqueness of a local solution By integrating the partial differential equation in 1.8, we obtain the following integral equation: ux, t = u x, t+ µ 1 u s ux, s + µ u s ux, s + µ 3ux, s, s, s dsdτ,.9 where u x, t = px + tqx and x R and t [, T ]. The following theorem is so clear that its proof is omitted. Theorem.1. If u is a continuous solution of problem.9, then it is also a solution of problem 1.8. This theorem allows us to consider problem.9 only in the rest of this paper. For simplicity, we assume that µ 1 = µ = µ 3 = 1. Now we state our main result. Theorem.. Assume that p and q are bounded and Lipschitz continuous on R. Let σ be the lipschitz constant of p and assume that σ < 1. Then there exists a positive constant T such that problem.9 has a unique solution, denoted by u x, t, in R [, T ]. Moreover, the function u x, t is also bounded and Lipschitz continuous with respect to each of variables x R and t [, T ]. Proof. To prove this theorem, we use an iterative algorithm. The proof includes some steps as below. Step 1:An iterate sequence of functions. We define the following sequence of real functions u n n defined for x R, t [.T ] for T > : u x, t = px + tqx, u 1 x, t = u x, t + µ 1 u µ u µ 3 u x, s, s, s dsdτ, u n+1 x, t = u x, t + µ 1 u n s u nx, s + µ u n s u nx, s + µ 3 u n x, s, s, s dsdτ..1 Step : Proof of the boundedness of u n. With simple calculations, taking into account the boundedness of p and q, we get u x, t px + t qx p L + t q L,

5 On an initial-value problem for second order partial differential equations 79 Moreover, u 1 x, t u x, t + p L + t q L + = u x, t u x, t t p L +! p L + t q L + = µ 1 u µ u µ 3 u x, s, s, s t + t3 3! dsdτ p L + s q L dsdτ q L. µ 1u 1 s u 1x, s + µ u 1 s u 1x, s + µ 3 u 1 x, s, s, s dsdτ 1 + t! + t4 p L + 4! By induction on n we find 1 + s! t + t3 3! + t5 5! u n x, t e T p L + q L p L + q L. s + s3 q L dsdτ 3!, n N, t [, T ]..11 Step 3: Every u n is lipschitz with respect to the first variable. From the Lipschitz continuity of p and q px py σ x y, x, y R, qx qy ω x y, x, y R..1 where < σ, ω are real numbers with σ < 1 as in the hypotheses. Using.1, we derive u x, t u y, t px py + t qx qy σ + tω x y := L t x y,.13 where L t := σ + tω.

6 8 Nguyen T.T. Lan In addition, u 1 x, t u 1 y, t L t x y + where L 1 t := L t + t Moreover L t + := L 1 t x y, µ 1 u µ u µ 3 u x, s, s, s µ 1 u µ u µ 3 u y, s, s, s dsdτ L 3 sdsdτ x y τ C sdsdτ, with C t := L 3 t..14 t u 1x, t t u 1y, t L t µ u µ 3 u x, t, t µ u µ 3 u y, t, t L t µ 3 u x, t µ 3 u y, t L 3 t x y := C t x y. Similarly, we have u x, t u y, t t τ L t x y + L 1 s s u 1x, s s u 1y, s + µ u 1 s u 1x, s + µ 3 u 1 x, s, s µ u 1 s u 1y, s + µ 3 u 1 y, s, s dsdτ L 1 s L 3 s x y L t x y + + L 1 s L 3 s x y + L 1 s x y dsdτ L t + L 1 s + L 1s C s + L 3 1s dsdτ := L t + C 1 sdsdτ x y := L t x y, x y.15

7 On an initial-value problem for second order partial differential equations 81 where L t := L t + C 1 sdsdτ, C 1 t := L 1 t + L 1t C t + L 3 1t. Moreover t u x, t t u y, t L 1 t t u 1x, t t u 1y, t + µ u 1 t u 1x, t + µ 3 u 1 x, t, t µ u 1 t u 1y, t + µ 3 u 1 y, t, t L 1 t L 3 t x y + L 1 t t u 1x, t t u 1y, t + µ 3 u 1 x, t µ 3 u 1 y, t L 1 t + L 1t L 3 t + L 3 1t x y = L 1 t + L 1t C t + L 3 1t x y := C 1 t x y. Repeating the previous calculation for u 3 we get u 3 x, t u 3 y, t L t x y + L s s u x, s s u y, s + µ u s u x, s + µ 3 u x, s, s µ u s u y, s + µ 3 u y, s, s dsdτ L t + L s + L s C 1 s + Lsdsdτ 3 x y := L t + C sdsdτ x y := L 3 t x y,.16

8 8 Nguyen T.T. Lan where and We have also L 3 t := L t + C t := C sdsdτ, L t + L t C 1 t + L 3 t. t u 3x, t t u 3y, t C t x y. Next, we proceed by induction. Let L t := σ + tω and C t := L 3 t, C n t := L n t + L nt C n 1 t + L 3 nt L n t := L t + From.13.16, by induction on n, we obtain C n 1 sdsdτ, n u n+1 x, t u n+1 y, t L n+1 t x y,.18 t u n+1x, t t u n+1y, t C n t x y..19 We introduce a definition. We call v n a stationary sequence in x if v n+1 x, t v n x, t f n t, where f n is a non-negative sequence of real function defined on [, T ]. If f n = f for all n, we say that v n is uniformly stationary sequence in x. Step 4: u n and u t n are stationary sequence in x. Direct calculations show that u 1 x, t u x, t = µ 1 u µ u µ 3 u x, s, s, s dsdτ p L + t q L dsdτ = t p L + t3 6 q L := A 1t.. t u 1x, t t u x, t = µ 1 u µ u µ 3 u x, s, s, s p L + t q L := B 1 t..1

9 On an initial-value problem for second order partial differential equations 83 From. and.1, we deduce A 1 t := B 1 sdsdτ.. u x, t u 1 x, t A 1 s + L s s u 1x, s s u x, s + µ u s u 1x, s + µ 3 u 1 x, s, s µ u s u x, s + µ 3 u x, s, s dsdτ 1 + L s + L s A 1 s + L s + L s B 1 s dsdτ := A t. t u x, t t u 1x, t A 1 t + L t t u 1x, t t u x, t + µ u t u 1x, t + µ 3 u 1 x, t, t µ u t u x, t + µ 3 u x, t, t A 1 t 1 + L t + L t + L t + L t B 1 t := B t..3.4 Combining.3 and.4 gives A t := B sdsdτ..5 From. and.3, by inducting on n, we derive u n+1 x, t u n x, t A n+1 t.6 and t u n+1x, t t u nx, t B n+1t,.7

10 84 Nguyen T.T. Lan where A n+1 t := B n+1 sdsdτ, B n+1 t := 1 + L n 1 t + L n 1t A n t + L n 1 t + L n 1t B n t, n 1..8 In the following step, we select T for which we prove also that u n and u t n are uniformly stationary sequences. Step 5: Existence of a local solution. Because σ < 1, we can find T >, < M < 1, < h < 1 such that for t [, T ], we have σ + tω + M t M < M < h; M + M 1; M M t < h..9 From.9 we obtain L 1 t σ + tω + L t σ + tω + L t = σ + tω M, M 3 dsdτ = σ + tω + M 3 t M, M 3 + M 4 + M 5 dsdτ σ + tω + M t M, C t = L 3 t M 3 M, C 1 t M + M M + M 3 = MM + M M, C t M + M M + M 3 = MM + M M. Now, by induction on n, we conclude that.3 C n t M, L n+1 t σ + tω + M t M..31

11 On an initial-value problem for second order partial differential equations 85 Hence we derive B t A 1 t1 + M + M + B 1 tm + M 1 + M + M B 1 sdsdτ + B 1 tm + M t B 1 L 1 + M + M + B 1 L M + M t B 1 L 1 + M + M B 1 L h..3 From.3 we obtain B L B 1 L h..33 By a similar argument, we get t B 3 t B L 1 + M + M + B L M + M t B L 1 + M + M B L h..34 So B 3 L B L h..35 From.33 and.35, by induction on n, we conclude that B n+1 L B n L h..36 In addition, from.8 we deduce T A n+1 L B n+1 L..37 Due to.36, we see the series B n+1 t converges absolutely and uniformly, hence by.7 there exists φ such that uniformly in R [, T ]. t u n φ.38

12 86 Nguyen T.T. Lan Similarly, from.6 and.37, we conclude that A n+1 t converges absolutely and uniformly and there exists u such that u n u.39 uniformly in R [, T ]. We remark that u x, t u y, t M x y. Now we are proving that u x, t is a solution of.9. It is clear that µ 1u n t u nx, t + µ u n t u nx, t + µ 3 u n x, t, t, t µ 1 u φ x, t + µ u φ x, t + µ 3 u x, t, t, t u n u L + M t u nx, t φ x, t + µ u n t u nx, t + µ 3 u n x, t, t µ u φ x, t + µ 3 u x, t, t.4 u n u L 1 + M + M + t u L n φ M + M as n. From.4, we deduce that u x, t = u x, t + µ 1 u φ x, s + µ u φ x, s + µ 3 u x, s, s, s dsdτ..41

13 On an initial-value problem for second order partial differential equations 87 Moreover, we have φ x, t t u x, t φ t u n + L t u nx, t t u x, t φ t u n + u n 1 u L L + M t u n 1x, t t u x, t + µ u n 1 t u n 1x, t + µ 3 u n 1 x, t, t µ u t u x, t + µ 3 u x, t, t φ t u n + u n 1 u L L 1 + M + M as n..4 Hence, u x, t = u x, t + µ 1 u s u x, s + µ u s u x, s + µ 3 u x, s, s, s dsdτ,.43 for all x R, t [, T ]. Then u is a solution of.9 in R [, T ]. Step 6: Uniqueness of the local solution u. We assume that there exists another

14 88 Nguyen T.T. Lan lipschitz solution u x, t of.9. Then u x, t u x, t t τ u t u x, s + u t u x, s + u x, s, s, s u t u x, s + u t u x, s + u x, s, s, s + u t u x, s + u t u x, s + u x, s, s, s u t u x, s + u t u x, s + u x, s, s, s dsdτ t τ u u L + M s u x, s s u x, s + u t u x, s + u x, s, s u t u x, s + u x, s, s dsdτ t τ u u L + M s u x, s s u x, s + u t u x, s + u x, s, s u t u x, s + u x, s, s + u t u x, s + u x, s, s u t u x, s + u x, s, s dsdτ t τ u u L + M s u s u + u u L L + M s u s u + u u L dsdτ L 1 + M + M u u L + M + M s u s u dsdτ. L.44

15 On an initial-value problem for second order partial differential equations 89 Additionally, t u x, t t u x, t = u t u x, t + u t u x, t + u x, t, t, t u t u x, t + u t u x, t + u x, t, t, t u t u x, t + u t u x, t + u x, t, t, t u t u x, t + u t u x, t + u x, t, t, t + u t u x, t + u t u x, t + u x, t, t, t u t u x, t + u t u x, t + u x, t, t, t u u L + M t u x, t t u x, t + u t u x, t + u x, t, t u t u x, t + u x, t, t u u L + M t u t u L + u t u x, t + u x, t, t u t u x, t + u x, t, t + u t u x, t + u x, t, t u t u x, t + u x, t, t u u L + M t u t u + u u L L + M t u x, t t u x, t + u x, t u x, t u u L 1 + M + M + t u t u M + M L 1 + M u u L + M t u t u. L.45

16 9 Nguyen T.T. Lan From.45, we deduce t u t u 1 + M L 1 M u u L..46 From.11,.44 and.46, we deduce u x, t u x, t 1 + M T 1 M u u L..47 This shows that u u and the proof is complete. QED Remark.1. It is clear that a trivial example for problem 1.8 is that ux, t = is a solution for px = qx =. Remark.. In this paper, we only consider the existence and uniqueness of a local solution to problem 1.8. It is of course interesting to investigate the behavior of this solution for some special cases of the initial more regular data p and q. We do not think that such problems are trivial. Remark.3. A numerical algorithm for problem 1.8 is still open. We believe that various specific differential equations with self-reference of the general form Aux, t = u Bux, t, t, where A : X R and B : X R are two functionals, X is a function space, u = ux, t, x, t R [, + is an unknown function, can be solved numerically. 3 A remark for particular initial data Now we present a particular situation that show as the initial value are very important in the study the iterative procedure considered in previous section, in particular if we assume px = p, qx = q ; p and q are two given real constants. Now, suppose px = p and qx = q, where p and q are two given real constants. We consider, as in previous section u x, t = p + tq, 3.48

17 On an initial-value problem for second order partial differential equations 91 and remark that u 1 x, t = u x, t + = u x, t + = p + tq + u u u x, s, s, sdsdτ u u p + sq, s, sdsdτ p + sq dsdτ 3.49 Therefore In addition, we get t = p + tq + p + q t 3 6 = p 1 + t + q t + t3.! 3! t u 1x, t = p + tq = u x, t. 3.5 u x, t = u x, t + u 1 s u 1x, s + u 1 s u 1x, s + u 1 x, s, s, s dsdτ = u x, t + p 1 + s + q s + s3 dsdτ! 3! = p 1 + t! + t4 + q t + t3 4! 3! + t5. 5! Now, by induction on k we obtain We deduce u k+1 x, t = u x, t + = p 1 + u k x, t = p k i= From we obtain u n x, t = p k i= p t i i! + q k i= k i= s i i! + q t i+ + q t + i +! n i= t i i! + q k i= n i= t i+1 i + 1!. k i= t i+1 dsdτ i + 1! t i+3. i + 3! t i+1 i + 1!, 3.53

18 9 Nguyen T.T. Lan t u n+1x, t = u n 1 x, t Letting n go to infinity, for all t [, T ], T >, we get u x, t = { Ce t, p = q = C p n= tn n! + q n= tn+1 n+1! = p cosh t + q sinh t, p q But it is easy to prove that u are solution of 1.8. The functions u are solution of the ordinary differential equation üt = ut. Hence we have the following situation. The problem.9 generated an integral equation; starting from non-constant initial condition so that almost one of p, q depend explicitely on x the iteration procedure give a local solution of problem.9. But starting from constant initial condition p and q togheter constant the same iteration procedure give a solution of a different problem. This seem to be a very interesting situation. References [1] Eder. E.: The functional-differential equation x t = xxt, J. Differ. Equ. 54, [] W. T. Li, S. Zhang: Classification and existence of positive solutions of higer order nonlinear iterative functional differential equations, J. Comput. Appl. Math.,139, [3] U. V. Lê and E. Pascali: An existence theorem for self-referred and hereditary differential equations, Adv. Differential Equations Control Process. 1, [4] J. G. Si and S. S. Cheng: Analytic solutions of a functional-differential equation with state dependent argument, Taiwanese J. Math. 4, [5] M. Miranda and E. Pascali: On a class of differential equations with self-reference, Rend. Mat., serie VII, 5, Roma [6] M. Miranda and E. Pascali: On a type of evolution of self-referred and hereditary phenomena, Aequationes Math. 71, [7] E. Pascali: Existence of solutions to a self-referred and hereditary system of differential equations, Electron. J. Diff. Eqns. Vol. 6 No. 7, pp [8] X. P. Wang, J. G. Si: Smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients, J. Math. Anal. Appl. 6, [9] X. Wang, J. G. Si and S. S. Cheng: Analytic solutions of a functional differential equation with state derivative dependent delay, Aequationes Math. 1, [1] U. V. Lê and L. T. T. Nguyen: Existence of solutions for systems of self-referred and hereditary differential equations, Electron. J. Diff. Eqns. Vol. 8 51, pp [11] Stanek, Svatoslav: Global properties of solutions of the functional differential equation utu t = kx xt, < k < 1, Funct. Differ. Equ. 9,

19 On an initial-value problem for second order partial differential equations 93 [1] Slenak, Benat: On the smooth parameter-dependence of the solutions of abstract functional differential equations with state-dependent delay, Funct. Differ. Equa., 17, [13] A. Domoshnitsky; A. Drakhlin; E. Litsyn: On equations with delay depending on solution, Nonlinear Anal., 49, [14] A. Domoshnitsky; A. Drakhlin; E. Litsyn: Nonocillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments., J. Differential Equations, 88 no. 1, [15] F. Hartung: Linearized stability in periodic functional differential equations with statedependent delays, J. Comput. Anal. Math., 174 no., [16] Stanek, Svatoslav: On global properties of solutions of the equation u t = au t but, Hokkaido Math. J. 3, [17] Stanek, Svatoslav: Global properties of decreasing solutions for the equation u t = u ut but, b, 1, Soochow J. Math. 6, [18] Stanek, Svatoslav: Global properties of increasing solutions for the equation u t = u ut but, b, 1, Soochow J. Math. 6, [19] Stanek, Svatoslav: Global properties of solutions of iterative-differential equations, Funct. Differ. Equ. 5, [] Stanek, Svatoslav: Global properties of decreasing solutions of the equation u t = u ut + ut, Funct. Differ. Equ. 4, [1] Stanek, Svatoslav: On global properties of solutions of functional differential equation u t = u ut + ut, Dynamical Systems and Appl. 4, [] N. M. Tuan, N. T. T. Lan: On solutions of a system of hereditary and self-referred partial-differential equations, Numer. Algor., DOI 1.17/s [3] P. K. Anh, N. T. T. Lan, N. M. Tuan: Solutions to systems of partial differential equations with weighted self-reference and heredity Electron. J. Diff. Eqns. Vol. 11, No. 117, pp ISSN: [4] V. Volterra: Opere Matematiche: Memorie e note, Vol. V, , Accad. Naz. Lincei. Roma 196.

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