Semilinear formulation of a hyperbolic system of partial differential equations

Carles Barril*, Àngel Calsina

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review


In this paper, we solve the Cauchy problem for a hyperbolic system of first-order PDEs defined on a certain Banach space X. The system has a special semilinear structure because, on the one hand, the evolution law can be expressed as the sum of a linear unbounded operator and a nonlinear Lipschitz function but, on the other hand, the nonlinear perturbation takes values not in X but on a larger space Y which is related to X. In order to deal with this situation we use the theory of dual semigroups. Stability results around steady states are also given when the nonlinear perturbation is Fréchet differentiable. These results are based on two propositions: one relating the local dynamics of the nonlinear semiflow with the linearised semigroup around the equilibrium, and a second relating the dynamical properties of the linearised semigroup with the spectral values of its generator. The later is proven by showing that the Spectral Mapping Theorem always applies to the semigroups one obtains when the semiflow is linearised.

Original languageEnglish
Article number58
Number of pages33
JournalJournal of Evolution Equations
Issue number3
Publication statusPublished - Sept 2022


  • Linearization principle
  • Semilinear hyperbolic systems
  • Spatially structured populations
  • Spectral mapping theorem
  • Sun dual formulation


Dive into the research topics of 'Semilinear formulation of a hyperbolic system of partial differential equations'. Together they form a unique fingerprint.

Cite this