A class of nonlinear hyperbolic problems with global solutions

D’Ancona, Piero; Spagnolo, Sergio

doi:10.1007/bf00953066

Cited by 51 publications

(47 citation statements)

References 12 publications

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“…Our condition is di erent from the conditions by D'AnconaSpagnolo [3,4,8] in the sense that we put the assumption of the smallness of the initial data of the usual Sobolev norm whereas D'Ancona-Spagnolo put that of weighted Sobolev norm, and neither implies the each other.…”

Section: Theoremmentioning

confidence: 70%

“…Greenberg-Hu [2] ÿrst showed the unique global solvability for small initial data with some decay condition in case n = 1, by introducing a transformation from the solution into a pair of unknown functions. D'Ancona and Spagnolo [3] improved the result of Greenberg-Hu [2] for arbitrary n and more general m. D'Ancona and Spagnolo [4] proved the global solvability of (1)- (2) 1=2 and D s f = F −1 ( sf ). In case n = 1, Rzymowski [5] improved [2].…”

Section: Introductionmentioning

confidence: 93%

“…D'Ancona-Spagnolo [3,4] improved their transformation, and they used the transformation expressed by | |F(u). We use the similar transformation expressed by using A 1=2 instead of the Fourier transform.…”

Section: Theoremmentioning

confidence: 99%

“…That is, we obtain su cient conditions on the usual Sobolev norm of the initial data, not on the weighted Sobolev norms of the initial data for the whole space (as in References [2][3][4]8]), nor on the weighted Sobolev norms of the generalized Fourier transform of the initial data for exterior domains (as in References [6,7,9]). We prove the above solvability for quasilinear hyperbolic equations of Kirchho type by showing that this solvability follows from a decay estimate with respect to the time valuable t of the solution of the linear wave equation, instead of using the (generalized) Fourier transform.…”

Section: Introductionmentioning

confidence: 99%

“…They applied the method by Greenberg-Hu [2], D'AnconaSpagnolo [3,4,8] to the exterior problem by applying the generalized Fourier transform instead of the Fourier transform used in References [2][3][4]8], and obtained a smallness condition on the generalized Fourier transform of the initial data su cient for the unique global solvability. Here, the generalized Fourier transform F S with respect to the selfadjoint operator A is a unitary mapping F S from L and j(x) is a cut-o function, vanishing in a neighbourhood of R n \ and being identically 1 for large |x|.…”

Section: Introductionmentioning

confidence: 99%

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Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

Yamazaki

2004

Math Methods in App Sciences

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SUMMARYWe consider the unique global solvability of initial (boundary)value problem for the Kirchho equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following su cient condition is known: initial data is su ciently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is su ciently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give su cient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation.

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Section: Theoremmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 93%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

Yamazaki

2004

Math Methods in App Sciences

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Nonlinear perturbations of the kirchhoff equation

D’Ancona

Spagnolo

1994

Comm Pure Appl Math

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The fundamental tool in the proof of Lemma 2.1 will be the following oscillating integrals associated to the functions v?, 4, 4: for any real T , and for t, s 2 0, we define J + ( T , t ) = J e -2 "~i~~( r , < ) I < I 2~~ .We now prove some decay estimates for these functionals. First of all, we observe that from inequality (18) and the assumption UO, u1 E C r , applying Lemma A.l in the Appendix, it follows that with MI = C(A,k) (11~0112,2 + llul111,2 + M ) 2 .

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A Kirchhoff type equation in a nonlinear model of shell vibration

Peradze

2016

Z Angew Math Mech

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The paper deals with the study of the initial boundary value problem for the Donnell-Mushtari-Vlasov nonlinear system of differential equations, which describes large deflections of a rectangular shallow shell. The sought functions u i = u i (x 1 , x 2 , t), i = 1, 2, and w = w(x 1 , x 2 , t) are respectively longitudinal and transverse displacements of points of the shell midsurface. It is assumed that Dirichlet and Neumann type homogeneous conditions are fulfilled for functions u 1 and u 2 . After expressing u 1 and u 2 through w, i.e. after constructing the relations u i = ψ i (w), i = 1, 2, with a help of Fourier series method, we come to a nonlinear integro-differential hyperbolic equation for the transverse displacement function w, which by its structure is analogous to Kirchhoff string, Woinowsky-Krieger beam and Berger plate equations. These equations are united under the common name -Kirchhoff type equations. The advantage of the described technique consists in splitting the initial boundary value problem at the end into two subproblems. From the integro-differential equation we first find w and using the explicit relations u i = ψ i (w), i = 1, 2, we construct u 1 and u 2 . The issue of the use of this approach in the general case not restricted by the rectangular domain and Dirichlet and Neumann conditions on the boundary is discussed. It is shown that in order to obtain formulas u i = ψ i (w), i = 1, 2, we have to solve a linear plane problem of elasticity with respect to the functions u 1 and u 2 .

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A class of nonlinear hyperbolic problems with global solutions

Cited by 51 publications

References 12 publications

Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

Nonlinear perturbations of the kirchhoff equation

A Kirchhoff type equation in a nonlinear model of shell vibration

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