Counications on Stochastic Analysis Vol. 6, No. 3 (1) 43-47 Serials Publications www.serialspublications.co AN ESTIMATE FOR BOUNDED SOLUTIONS OF THE HERMITE HEAT EQUATION BISHNU PRASAD DHUNGANA Abstract. An estiate result on the partial derivatives of the Mehler kernel E(x, ξ, t) for t > is first established. Particularly for < t < 1, it extends the estiate result given by S. Thangavelu in his onograph A lecture notes on Herite and Laguerre expansions on the order of the partial derivative of the Mehler kernel with respect to the space variable. Furtherore, for each N, a growth estiate on the partial derivative U(x,t) of all bounded solutions U(x, t) of the Cauchy Dirichlet proble for the Herite heat equation is established. 1. Introduction As introduced in [1], we denote by E(x, ξ, t) the Mehler kernel defined by { E(x, ξ, t) = k= e (k+1)t h k (x)h k (ξ), t >,, t, where h k s are L noralized Herite functions defined by h k (x) = ( 1)k e x / k k! π Moreover the explicit for of E(x, ξ, t) for t > is d k dx k e x, x R. 1+e 1 e (x ξ) 1 e t 1+e t xξ E(x, ξ, t) = e t e 1 π(1 e ) 1 We note that for each ξ R, E(x, ξ, t) satisfies the Herite heat equation. In (Theore 3.1, []), we proved that {E(x, ξ, t) E(x, ξ, t)} φ(ξ)dξ (1.1) is a unique bounded solution of the following Cauchy Dirichlet proble for the Herite heat equation ( t + x ), x >, t >, U(x, ) = φ(x), x >, U(, t) =, t >,. (1.) Received 1-1-1; Counicated by K. Saitô. Matheatics Subject Classification. Priary 33C45; Secondary 35K15. Key words and phrases. Herite functions, Mehler kernel, Herite heat equation. 43
44 BISHNU PRASAD DHUNGANA where φ is a continuous and bounded function on [, ) with φ() =. It is not necessary that every bounded solution of the Herite heat equation should satisfy a fixed growth behavior on its th partial derivative with respect to the space variable. However, since the solution U(x, t) in (1.1) is a unique solution of (1.), it is natural to ake an effort for obtaining a fixed growth estiate. But it is not as easy as we anticipate. To find a growth estiate, we require first to obtain an estiate on E(x,ξ,t). Note that an estiate on the partial derivatives of the heat kernel { E(x, t) = (4πt) 1 e x 4t, t >,, t, on U(x,t) on U(x,t) with respect to the space variable has been given in [3]: E(x, t) C t (1+)! 1 e ax 4t, t >, where C is soe constant and a can be taken as close as desired to 1 such that < a < 1. Though the estiates of the following types on the Mehler kernel for < t < 1 and B independent of x, ξ and t E(x, ξ, t) Ct 1 e B t x ξ, (1.3) E(x, ξ, t) ξ Ct 3 e B t x ξ, are provided in [4], the estiate on the partial derivatives of the Mehler kernel of all order with respect to the space variable is yet to be established. Lea.1 that gives an estiate on E(x,ξ,t) for each nonnegative integer, is therefore a novelty of this paper which as an application yields t e t M in [, ) [, ) for soe constant M, the ain objective and the final part of this paper.. Main Results Lea.1. Let E(x, ξ, t) be the Mehler kernel and N. Then for soe constants a with < a < 1 and A := A(a) >! e (A+1) π 1+ e t t +1 e ae t (x ξ) 1 e.
AN ESTIMATE FOR BOUNDED SOLUTIONS OF THE HERMITE HEAT EQUATION 45 Proof. By the Cauchy integral forula, we have =! E(ζ, ξ, t) dζ πi Γ R (ζ x) +1 =! π 3 i Γ R e t e 1 1+e 1 e (ζ ξ) 1 e t 1+e t ζξ (ζ x) +1 (1 e ) 1 where Γ R is a cirle of radius R in the coplex plane C with center at x. With ζ = x + Re iθ, we have!e t π π 3 R e 1 1+e 1 e (x ξ+reiθ ) 1 e t 1+e t (x+reiθ)ξ dθ. 1 e Then, writing S for x + R cos θ, we have! e t π 3 R 1 e Let P = 1 1+e 1 e using the inequality dζ, [ π e 1 1+e ] 1 e {ξ S} + 1 e t 1+e t ξs dθ. e 1 1+e 1 e R and Q = 1 e t 1+e t. Then P > and Q > since t is positive. Now P {ξ (x + R cos θ)} + Qξ(x + R cos θ) we have! e t π 3 R 1 e! e t πr 1 e ( P Q ) {ξ (x + R cos θ)}, π e e t 1 e (x ξ+r cos θ) + 1 1+e 1 e R dθ e t e 1 e x + 1 1+e 1 e R, where x = x ξ R or or x ξ + R. Since the ratio exp 1+e iniu at R = b where b = 1 1 e, we have! e t ( e 1 + e π 1 e 1 e But with < a < 1 and β 1 e e t where A = 1 e (x ξ+βr) = e ae t a 1 a. Then clearly ( 1 1+e 1 e R) R ) e e t attains its 1 e x. (.1) 1 e (x ξ) e e t 1 e [(1 a)(x ξ) +(x ξ)βr+β R ] [ = e ae t 1 e (x ξ) e (1 a)e t 1 e (x ξ+ βr 1 a ) aβ R ] (1 a) e ae t 1 e (x ξ) e Ae t 1 e R, e e t 1 e x e ae t 1 e (x ξ) e Ae t 1 e R.
46 BISHNU PRASAD DHUNGANA Using R = (1 e ) e 1+e and the inequalities t 1+e 1, 1+e 1 e t >, (.1) reduces to Furtherore, since e t 1 e 1 t This copletes the proof.! e (A+1) π e t 1 e e t t for every t > we obtain! e (A+1) π 1+ e t t +1 Reark.. For < t < 1, in view of (.3) and e t that! e (A+3) π 1+ t +1 et t for every e ae t (x ξ) 1 e. (.) e ae t (x ξ) 1 e. (.3) 1 e e a(x ξ) 8t 1 8t it is easy to see which extends the estiate result (1.3) on the order > 1 of the partial derivative of E(x, ξ, t) with respect to the variable x. Theore.3. Every bounded solution of the Cauchy Dirichlet proble for the Herite heat equation ( t + x ) x >, t >, U(x, ) = φ(x) x >, U(, t) = t >, satisfies the following growth estiate t e t M, where N and M is soe constant. in [, ) [, ), (.4) Proof. Fro (Theore 3.1, []), every bounded solution of the Cauchy Dirichlet proble (.4) for the Herite heat equation is of the for {E(x, ξ, t) E(x, ξ, t)} φ(ξ)dξ, where φ is a continuous and bounded function on [, ) with φ() = and E(x, ξ, t), the Mehler kernel. We write where = R h(ξ) = {E(x, ξ, t) E(x, ξ, t)} φ(ξ)dξ E(x, ξ, t)h(ξ)dξ, { φ(ξ), ξ, φ( ξ), ξ <.
AN ESTIMATE FOR BOUNDED SOLUTIONS OF THE HERMITE HEAT EQUATION 47 Fro (.), we have R h(ξ) dξ h!e (A+1) e ( 1)t π 1 e (t) R e ae t (x ξ) 1 e dξ. Under the change of variable a e t 1 e (ξ x) = s and integrating, we have Clearly if we take M = h h!e (A+1) a e t. t t e t M in [, ) [, )!e (A+1) a. References 1. Dhungana, B. P.: An exaple of nonuniqueness of the Cauchy proble for the Herite heat equation, Proc. Japan. Acad. 81, Ser. A, no. 3 (5), 37 39.. Dhungana, B. P. and Matsuzawa, T.: An existence result of the Cauchy Dirichlet proble for the Herite heat equation, Proc. Japan. Acad. 86, Ser. A, no. (1), 45 47. 3. Matsuzawa, T.: A calculus approach to hyperfunctions. II, Trans. Aer. Math. Soc. 313, No. (1989), 619 654. 4. Thangavelu, S.: Lectures on Herite and Laguerre Expansions, Princeton University Press, Princeton, 1993. Bishnu Prasad Dhungana: Departent of Matheatics, Mahendra Ratna Capus, Tribhuvan University, Kathandu Nepal E-ail address: bishnupd1@yahoo.co