1,720,996 research outputs found
Well posedness of the Cauchy problem for nonlinear weakly hyperbolic equations
No full textThe authors study well-posedness of the Cauchy problem for several classes of nonlinear (semilinear) weakly hyperbolic equations. It is assumed that the principal part of the operator possesses real characteristic roots of constant multiplicity, and that Levi-type conditions and Levi conditions of nonlinear type [respectively, Gevrey-Levi conditions and nonlinear Gevrey-Levi-type conditions]
are satisfied in C1 [respectively, in Gevrey categories] with respect to the space variables. Local existence and uniqueness results in the time variable t are proved both in C1 and Gevrey classes
Optimal well-posedness of the Cauchy problem for evolution equations with C^N coefficients
No full textPropagation of analytic and Gevrey singularities for operators with noninvolutive characteristics.
No full textThe authors consider classical analytic pseudodifferential operators of the form . The arguments are microlocal, near a point of the symplectic characteristic manifold , generalizing results by A. Bove, J. E. Lewis and C. Parenti \ref[ Propagation of singularities for Fuchsian operators, Lecture Notes in Math., 984, Springer, Berlin, 1983; and, in the case , previous contributions by Hanges, Ivrii, and Melrose. With respect to the papers of the above-mentioned authors, here no Levi condition is needed on the lower order terms , the result being stated in terms of Gevrey classes with (if then ). In particular, propagation of singularities is proved along the half-bicharacteristics emanating from
Nonlinear hyperbolic Cauchy problems in Gevrey classes
No full textWell-posedness of the Cauchy problem in the Gevrey classes for quasi-linear equations of constant multiplicity is stated. More precisely, let be a hyperbolic equation with Hölder continuous coefficients with respect to time, and let be the largest multiplicity of the characteristics of . Then the Cauchy problem is well-posed in the Gevrey classes of index smaller than . For and for linear , the result goes back to the classical theory of perturbation of hyperbolic equations. For non-linear , this is an improvement of related results by K. Kajitani
Quasilinear weakly hyperbolic equations with Levi's conditions
No full textThis paper deals with quasilinear weakly hyperbolic equations satisfying the Levi condition. The authors introduce in the quasi-linear case the notions of weakly hyperbolic equation of constant multiplicity, Levi's condition and domain of influence. The solution of the corresponding equation is assumed to be real-valued and to belong to , while the equation depends analytically on . According to the main result of this paper, analyticity propagates across an analytic hypersurface into the domain of influence based on . This paper is a continuation of some previous investigations of the same authors
Analytic regularity for solutions to semi-linear weakly hyperbolic equations
No full textLet be a solution to a weakly hyberbolic semilinear partial differential equation with real analytic Cauchy data. The main result of the paper asserts that if is assumed to be in a Gevrey space close enough to the space of analytic functions, then it is analytic
A well-posed Cauchy problem for an evolution equation with coefficients of low regularity
No full textIn the hyperbolic Cauchy problem, the well-posedness in Sobolev spaces is strictly related to the modulus of continuity of the coefficients.
This holds true for -evolution equations with real characteristics ( hyperbolic equations, vibrating plate and Scr\"odinger type models, ...). We show that, for , a lack of regularity in can be balanced by a damping of the too fast oscillations as
the space variable . This can not happen in the hyperbolic case because of the finite speed of propagation
Analytic regularity for solutions of nonlinear weakly hyperbolic equations
No full textAnalytic regularity of real solutions of a nonlinear weakly hyperbolic equation with characteristic roots of constant multiplicity is studied in the paper. It is proved that the analytic regularity of Cauchy data propagates according to the geometry of the influence domains of the equation if its solution is "sufficiently regular". Local propagation results are achieved as an application of a theorem concerning the continuity of a class of infinite-order Fourier integral operators in Sobolev spaces with weight of exponential type
- …
