On structures of L2-well-posed mixed problems for hyperbolic operators

1975 ◽  
pp. 235-254
Author(s):  
Taira Shirota
Keyword(s):  
Mixed Problems ◽  
Hyperbolic Operators ◽  
Well Posed

LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

2014 ◽  
Vol 11 (01) ◽  
pp. 185-213 ◽  
Author(s):  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Energy Estimates ◽  
Characteristic Set ◽  
Finite Speed ◽  
Hyperbolic Operators ◽  
New Energy ◽  
The Cauchy Problem ◽  
Well Posed

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

scholarly journals The Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition

2020 ◽  
Vol 11 (4) ◽  
pp. 1991-2022
Author(s):  
Annamaria Barbagallo ◽  
Vincenzo Esposito
Keyword(s):  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Existence And Uniqueness Results ◽  
Mixed Problems ◽  
Uniqueness Results ◽  
Hyperbolic Operators ◽  
Priori Estimates

Abstract The mixed Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition is investigated. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, existence and uniqueness results for the mixed problems are obtained.

scholarly journals Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity

2015 ◽  
Vol 12 (03) ◽  
pp. 535-579 ◽  
Author(s):  
Enrico Bernardi ◽  
Antonio Bove ◽  
Vesselin Petkov
Keyword(s):  
Cauchy Problem ◽  
Principal Symbol ◽  
Energy Estimates ◽  
Hyperbolic Operator ◽  
Third Order ◽  
Hyperbolic Operators ◽  
The Cauchy Problem ◽  
Loss Of Regularity ◽  
Well Posed ◽  
Lower Order Terms

We study a class of third-order hyperbolic operators P in G = {(t, x): 0 ≤ t ≤ T, x ∈ U ⋐ ℝn} with triple characteristics at ρ = (0, x0, ξ), ξ ∈ ℝn ∖{0}. We consider the case when the fundamental matrix of the principal symbol of P at ρ has a couple of non-vanishing real eigenvalues. Such operators are called effectively hyperbolic. Ivrii introduced the conjecture that every effectively hyperbolic operator is strongly hyperbolic, that is the Cauchy problem for P + Q is locally well posed for any lower order terms Q. This conjecture has been solved for operators having at most double characteristics and for operators with triple characteristics in the case when the principal symbol admits a factorization. A strongly hyperbolic operator in G could have triple characteristics in G only for t = 0 or for t = T. We prove that the operators in our class are strongly hyperbolic if T is small enough. Our proof is based on energy estimates with a loss of regularity.

scholarly journals The C∞-well posed Cauchy problem for hyperbolic operators dominated by time functions

2004 ◽  
Vol 30 (2) ◽  
pp. 283-348 ◽  
Author(s):  
Kunihiko KAJITANI ◽  
Seiichiro WAKABAYASHI ◽  
Karen YAGDJIAN
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Operators ◽  
Well Posed

scholarly journals A MIXED PROBLEM FOR THE FOUR-ORDER ONE-DIMENSIONAL HYPERBOLIC EQUATION WITH PERIODIC CONDITIONS

2018 ◽  
Vol 54 (2) ◽  
pp. 135-148
Author(s):  
V. I. Korzyuk ◽  
Nguyen Van Vinh
Keyword(s):  
Hyperbolic Equation ◽  
Classical Solution ◽  
Differential Operators ◽  
Hyperbolic Equations ◽  
One Dimensional ◽  
First Order ◽  
Mixed Problems ◽  
Uniqueness Of The Solution ◽  
Different Characteristics ◽  
Well Posed

This article considers a classical solution of the boundary problem for the four-order strictly hyperbolic equation with four different characteristics. Note that the well-posed statement of mixed problems for hyperbolic equations not only depends on the number of characteristics, but also on their location. The operator appearing in the equation involves a composition of first-order differential operators. The equation is defined in the half-strip of two independent variables. There are Cauchy’s conditions at the domain bottom and periodic conditions at other boundaries. Using the method of characteristics, the analytic solution of the considered problem is obtained. The uniqueness of the solution is proved. We have also noted that the solution in the whole given domain is a composition of the solutions obtained in some subdomains. Thus, for the obtained classical solution to possess required smoothness, the values of these piecewise solutions, as well as their derivatives up to the fourth order must coincide at the boundary of these subdomains. A classical solution is understood as a function that is defined everywhere at all closure points of a given domain and has all classical derivatives entering the equation and the conditions of the problem.

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