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MIXED PROBLEM FOR SOME SEMILINEAR WAVE EQUATION INVOLVING BESSEL'S OPERATOR

MIXED PROBLEM FOR SOME SEMILINEAR WAVE EQUATION INVOLVING BESSEL'S OPERATOR DEMONSTRATIO MATHEMATICAVol. XXXIINo 11999Nguyen Thanh Long, Alain Pham Ngoc Dinh,Duong Thi Thanh BinhMIXED PROBLEM FOR SOME SEMILINEARWAVE EQUATION INVOLVING BESSEL'S OPERATOR1. IntroductionWe will consider the following initial and boundary value problem:(1.1) utt - a(t)^urr+= f(r,t,u,ut),(1.2)| lim V n v ( r , i ) | < +oo,(1.3)u(r,0) = uQ(r),r—>0+ur(l,t)0<r<l,0<t<T,+ h(t)u(l,t)= g(t),ut(r, 0) = ui(r),where a, h, f , g, UQ, UI are given functions satisfying conditions specifiedlater.The equation (1.1) generalizes the following bidimensional nonlinearwave equationti/tt - Aw = f(y/x2+ y2,t,w,wt),x2 + y2 < 1, 0 < t < T,describing nonlinear vibrations of the unit membrane S : x2 + y2 < 1, tohave the form where the displacement is assumedw(x,y,t)= u( \/x2 + y2, t)depending only o n r = y/x2 + y2 and time t. The conditions (1.2) on theboundary of S(r — 1) and at center of S describe the elastic constraints,where the functions h(t), g(t) have a mechanical signification.In case of equation (1.1) not involving the term £u r and let a(t) — 1 wehave(1.4)utt ~ Urr = f(r, t, u, ut),0 < r < 1, 0<t<T.78Nguyen T h a n h Long, Alain P h a m Ngoc Dinh, Duong Thi T h a n http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

MIXED PROBLEM FOR SOME SEMILINEAR WAVE EQUATION INVOLVING BESSEL'S OPERATOR

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Publisher
de Gruyter
Copyright
© by Nguyen Thanh Long
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1999-0109
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXIINo 11999Nguyen Thanh Long, Alain Pham Ngoc Dinh,Duong Thi Thanh BinhMIXED PROBLEM FOR SOME SEMILINEARWAVE EQUATION INVOLVING BESSEL'S OPERATOR1. IntroductionWe will consider the following initial and boundary value problem:(1.1) utt - a(t)^urr+= f(r,t,u,ut),(1.2)| lim V n v ( r , i ) | < +oo,(1.3)u(r,0) = uQ(r),r—>0+ur(l,t)0<r<l,0<t<T,+ h(t)u(l,t)= g(t),ut(r, 0) = ui(r),where a, h, f , g, UQ, UI are given functions satisfying conditions specifiedlater.The equation (1.1) generalizes the following bidimensional nonlinearwave equationti/tt - Aw = f(y/x2+ y2,t,w,wt),x2 + y2 < 1, 0 < t < T,describing nonlinear vibrations of the unit membrane S : x2 + y2 < 1, tohave the form where the displacement is assumedw(x,y,t)= u( \/x2 + y2, t)depending only o n r = y/x2 + y2 and time t. The conditions (1.2) on theboundary of S(r — 1) and at center of S describe the elastic constraints,where the functions h(t), g(t) have a mechanical signification.In case of equation (1.1) not involving the term £u r and let a(t) — 1 wehave(1.4)utt ~ Urr = f(r, t, u, ut),0 < r < 1, 0<t<T.78Nguyen T h a n h Long, Alain P h a m Ngoc Dinh, Duong Thi T h a n

Journal

Demonstratio Mathematica – de Gruyter

Published: Jan 1, 1999

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