Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes

Kajitani, Kunihiko

doi:10.14492/hokmj/1525852966

Cited by 38 publications

(28 citation statements)

References 2 publications

(5 reference statements)

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“…For example see the article of Bernstein [2] and of Pohozaev [9]. When M{r]} = 1, Theorem 1,1 is proved in [3]. In [7] we proved that Thorem 1,1 holds for (1.1) and (1.4) under the assumption s ^ d < 4/3.…”

Section: 2)mentioning

confidence: 82%

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Propagation of analyticity of solutions to the Cauchy problem for Kirchhoff type equations

Kajitani¹

2000

Journées Équations Aux Dérivées Partielles

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Section: 2)mentioning

confidence: 82%

“…The proof of this lemma is given in Lemma 1.2 of [3]. We can prove the following lemma by use of the above lemma.…”

Section: A Priori Estimates For Linearized Equationsmentioning

confidence: 96%

Propagation of analyticity of solutions to the Cauchy problem for Kirchhoff type equations

Kajitani¹

2000

Journées Équations Aux Dérivées Partielles

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“…is hyperbolic in z, that is, the imaginary part of its roots Th (t, x,~) vanishes identically, we can construct a solution of (0.1) by solving the Cauchy problem with zero initial data (see [2] and [5]). Therefore, we assume that p is not hyperbolic.…”

Section: Remark 1 (Borderline Cases)mentioning

confidence: 99%

Strong Gevrey Solvability for a System of Linear Partial Differential Equations

Kajitani

Spagnolo

2003

Partial Differential Equations and Mathematical Physics

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We consider a class of linear systems whose principal symbol satisfies a certain condition of semi-hyperbolicity, and we prove the local sUljectivity in suitable Gevrey spaces. o IntroductionWe investigate the local solvability for m x m systems of typeHere u(t, x) and f(t, x) are em-valued functions on R l+n, while the A j (t, x)'s are m x m matrix functions, uniformly analytic in R l+n in the sense that there is some Co > 0 for which la:a~Aj(t, x)1~C6+We denote by t"1 (t , x,~), ... , t"m (t, x,~) the eigenvalues of the matrixrepeated following their multiplicities, and defineWe do not assume t"h 1= t"k for h 1= k, hence we cannot apply the classical theory of Coo-solvability of Hormander, Nirenberg and Treves for equations of principal type. Consequently, we expect solvability only in suitable Gevrey classes.Our main assumption is that, for each fixed~E R n , the imaginary parts of all the t"h'S keep the same sign when (t, x) runs in R l+n, that is

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“…[5]). An interesting problem should be to prove Gevrey-well posedness and to find the critical Gevrey index depending on s. For s ∈ [1, 2] Levi conditions don't appear ( [6]). …”

Section: Remarkmentioning

confidence: 99%

“…If we restrict ourselves to Gevrey classes of order ≤ 2, then we can use ideas of [6] to prove a local existence result for…”

Section: Introductionmentioning

confidence: 99%

Weakly hyperbolic equations with time degeneracy in Sobolev spaces

Reissig

1997

Abstract and Applied Analysis

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Abstract. The theory of nonlinear weakly hyperbolic equations was developed during the last decade in an astonishing way. Today we have a good overview about assumptions which guarantee local well posedness in spaces of smooth functions (C ∞ , Gevrey). But the situation is completely unclear in the case of Sobolev spaces. Examples from the linear theory show that in opposite to the strictly hyperbolic case we have in general no solutions valued in Sobolev spaces. If so-called Levi conditions are satisfied, then the situation will be better. Using sharp Levi conditions of C ∞ -type leads to an interesting effect. The linear weakly hyperbolic Cauchy problem has a Sobolev solution if the data are sufficiently smooth. The loss of derivatives will be determined in essential by special lower order terms. In the present paper we show that it is even possible to prove the existence of Sobolev solutions in the quasilinear case although one has the finite loss of derivatives for the linear case. Some of the tools are a reduction process to problems with special asymptotical behaviour, a Gronwall type lemma for differential inequalities with a singular coefficient, energy estimates and a fixed point argument.

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Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes

Cited by 38 publications

References 2 publications

Propagation of analyticity of solutions to the Cauchy problem for Kirchhoff type equations

Propagation of analyticity of solutions to the Cauchy problem for Kirchhoff type equations

Strong Gevrey Solvability for a System of Linear Partial Differential Equations

Weakly hyperbolic equations with time degeneracy in Sobolev spaces

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