ISSN:
1572-9036
Keywords:
Primary: 35A05
;
35D05
;
46F10
;
secondary: 35L45
;
35L60
;
35R25
;
algebras of generalized functions
;
generalized derivatives
;
linear and nonlinear partial differential equations of evolution type
;
existence and uniqueness of solutions
;
symmetric hyperbolic systems
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This paper deals with a new solution concept for partial differential equations in algebras of generalized functions. Introducing regularized derivatives for generalized functions, we show that the Cauchy problem is wellposed backward and forward in time for every system of linear partial differential equations of evolution type in this sense. We obtain existence and uniqueness of generalized solutions in situations where there is no distributional solution or where even smooth solutions are nonunique. In the case of symmetric hyperbolic systems, the generalized solution has the classical weak solution as ‘macroscopic aspect’. Two extensions to nonlinear systems are given: global solutions to quasilinear evolution equations with bounded nonlinearities and local solutions to quasilinear symmetric hyperbolic systems.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00047123
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