Application of H-measures to evolution equations

doctoral thesis

• Title (croatian): Primjena H-mjera na evolucijske jednadžbe
• Author: Matko Grbac
• Mentor: Nenad Antonić (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Ivan Ivec (član povjerenstva)
• Committee member: Mario Bukal (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-12-19, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

In the thesis evolution equations of second order in \(t\) are studied. Some standard results regarding the existence, uniqueness, and well-posedness of the semilinear wave equation are revisited in the first part and generalised to the case where the coefficients have only bounded variation with respect to time. The second part begins with an introduction to the theory of pseudodifferential operators and microlocal analysis tools, with an emphasis on the notion of H-measures and their main properties. The theory is then applied to both linear and semilinear wave equations. The results in this part generalize the work of Francfort and Murat, allowing the coefficients of the equation to depend on the time variable as well.

Abstract (croatian)

Disertacija se bavi proučavanjem evolucijskih jednadžbi drugog reda u \(t\). U prvom dijelu disertacije revidiraju se i generaliziraju standardni rezultati o postojanju i jedinstvenosti rješenja, te dobroj postavljenosti polulinearne valne jednadžbe na slučaj kada koeficijenti imaju samo ograničenu varijaciju s obzirom na vremensku varijablu. Drugi dio disertacije započinje uvodom u teoriju pseudodiferencijalnih operatora i alata mikrolokalne analize, s naglaskom na pojam H-mjera i njihovih glavnih svojstava. Zatim se teorija primjenjuje na slučajeve linearne i polulinearne valne jednadžbe. Rezultati u ovom dijelu generaliziraju rad Francforta i Murata, omogućujući da koeficijenti jednadžbe također ovise o vremenskoj varijabli.

• Keywords: H-measures; pseudodifferential operators; semilinear wave equation
• Keywords (croatian): H-mjere; pseudodiferencijalni operatori; polulinearna valna jednadzba
• Language: english
• URN:NBN: urn:nbn:hr:217:028449
• Study programme: Title: Doctoral study; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica u području prirodnih znanosti (doktor/doktorica u području prirodnih znanosti)
• Type of resource: Text
• Extent: v, 99 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-01-31 11:19:45