Abstract This paper deals with the Cauchy problem for a quasilinear first-order equation that includes a possibly discontinuous hysteresis operator F : ∂ ∂ t [ u + F ( u ) ] + ∂ u ∂ x = f in R , for t > 0 . Existence of a weak solution is proved for F equal to a completed relay operator. In the case of f ≡ 0 , an entropy-type condition yields Lipschitz-continuous and monotone dependence on the initial data, hence uniqueness.
Quasilinear first-order PDEs with hysteresis
Published 2005 in Journal of Mathematical Analysis and Applications
ABSTRACT
PUBLICATION RECORD
- Publication year
2005
- Venue
Journal of Mathematical Analysis and Applications
- Publication date
2005-12-15
- Fields of study
Mathematics
- Identifiers
- External record
- Source metadata
Semantic Scholar
CITATION MAP
EXTRACTION MAP
CLAIMS
CONCEPTS
- completed relay operator
A specific relay-type hysteresis operator used as the concrete choice of F in the existence result.
Aliases: relay operator
- entropy-type condition
An additional admissibility condition imposed on solutions when f ≡ 0.
Aliases: entropy condition
- hysteresis operator
A possibly discontinuous operator F[u] that encodes hysteresis effects in the evolution equation.
Aliases: F, hysteresis
- initial data
The prescribed starting state used to determine the solution of the Cauchy problem.
Aliases: initial conditions
- lipschitz-continuous dependence
A stability property in which solution changes are bounded linearly by changes in the initial data.
Aliases: Lipschitz dependence
- monotone dependence
A dependence relation in which the ordering of initial data is preserved by the induced solutions.
Aliases: monotonic dependence
- quasilinear first-order equation
A first-order partial differential equation with nonlinear dependence on the unknown or its derivatives.
Aliases: quasilinear first-order PDE
- uniqueness
The property that the admissible solution is determined uniquely by the initial data and equation.
- weak solution
A solution concept for the PDE formulated in an integral or distributional sense.
REFERENCES
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