68 research outputs found
WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR p-EVOLUTION SYSTEMS OF PSEUDO-DIFFERENTIAL OPERATORS
No full textWe study p-evolution pseudo-differential systems of the first order with coefficients in (t,x) and real characteristics. We find sufficient conditions for the well-posedness of
the Cauchy problem in H∞. These conditions involve the behavior as x → ∞ of the coefficients, requiring some decay estimates to be satisfied
The Cauchy problem for quasilinear hyperbolic equations with non-absolutely continuous coefficients in the time variable
No full textWe consider the Cauchy problem for a quasilinear weakly hyperbolic operator with coefficients having the first time derivative with singular behavior of the type t^{−q}, q > 1, as t → 0.
We show that for t ≤ T_0 , T_0 sufficiently small, given Cauchy data in
a Gevrey class there exists a unique solution u in Gevrey classes of order σ < (qr)/(qr−1) where r denotes the largest multiplicity of the characteristic roots
A Degenerate Hyperbolic Equation under Levi conditions
No full textWe consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part's coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order.
Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C^\infty well posedness of the Cauchy problem
The Cauchy Problem for a Class of p-evolution Equations
No full textWe study p-evolution equations of order m greater than 2 with coefficients depending both on t and x, not regular with respect to the time variable. We find the sharp regularity in time for the coefficients in order to have well posedness in H^1 of the related Cauchy Problem. We consider the case of continuous in time coefficients having the first time-derivative that breaks down at a point t_0, say t_0 = 0. The case m = 2 has already been studied by Cicognani and Colombini; here we generalize the result to operators of higher order
Equazioni differenziali a derivate parziali di ecvoluzione e stocastiche (Progetto Gnampa 2015)
No full textProgetto Gnampa 2015 finanziato dal gruppo GNAMPA dell'INDAM. L'obiettivo principale è stato lo studio delle PDEs di evoluzione, sia deterministiche che stocastiche, con i metodi e gli strumenti dell'analisi microlocale
Well posedness under Levi conditions for a degenerate second order Cauchy problem
No full textWe consider the Cauchy problem for a second order equation of hyperbolic type which degenerates both in the sense that it is weakly hyperbolic and it has non Lipschitz continuous in time coefficients. Intersections between the roots of the equation are of a finite order k, and the first time derivative of the principal part's coefficients present a blow-up phenomenon at the time t=0, behaving as t^{-q}, q greater or equal 1.
The mixture of these two situations gives, under an appropriate Levi condition, C^\infty or Gevrey well posedness of the Cauchy problem, depending on the dominant between the two behaviors
FFABR2017
No full textBeneficiaria del Fondo per il finanziamento delle attività base di ricerca (FFABR) in quanto in possesso dei requisiti previsti da Anvur (art. 1, commi 295 e seguenti, legge 11 dicembre 2016, n.232), € 3.000,0
"Equazioni di evoluzione anisotrope con coefficienti variabili" FIRD 2022
No full textIntendo studiare equazioni di evoluzione che descrivano fenomeni fisici quali l'evoluzione dello stato di un sistema in meccanica quantistica o la propagazione delle onde in acque basse. Più precisamente intendo studiare:
- equazioni di tipo Schrödinger (2-evoluzione) stocastiche su spazi-tempo curvi
- equazioni di 3-evoluzione, deterministiche, non lineari.
Intendo capire sotto quali ipotesi si possa affermare l'esistenza ed unicità della soluzione in opportuni spazi funzionali, descrivendone le principali proprietà
Analisi microlocale ed applicazioni: PDEs stocastiche e di evoluzione, analisi tempo- frequenza, varietà
No full textProgetto Gnampa 202
The cauchy problem for a weakly hyperbolic equation with unbounded and non Lipschitz continuous coefficients
No full textWe consider a second order weakly hyperbolic equation with
time and space depending coefficients. We suppose the coefficients to
have globally a Hoelder type behavior and locally a blow up of the first
derivative at some time. We show that the Cauchy problem for such an
equation is well posed in Gevrey classes; the upper bound for the
Gevrey index σ depends only on the dominant between the local and
the global condition
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