1,720,970 research outputs found
Piecewise smooth solutions for 1D wave equations with discontinuous velocity and singular source term
No full textHypoellipticity and Solvability in Gelfand-Shilov spcaes for twisted laplacian type operators
No full textThe main goal of the paper is to investigate the hypoellipticity and the solvability in the Schwartz space S(R2) and the Gelfand–Shilov spaces Sμμ(R2), μ ≥ 1/2, of classes of second order Shubin type operators which generalize the twisted Laplacian. Our approach is based on the reduction to global normal forms by means of Fourier integral operators with quadratic phase functions. We describe completely the spectral properties of the original operators. The regularity/solvability results are shown by using the discrete representation of the action of Shubin operators and the characterization of S(R2) and Sμμ(R2) by means of eigenfunction expansions
Super-exponential decay and holomorphic extensions for semilinar equations with polynomial coefficients
No full textWe show that all eigenfunctions
of linear partial differential operators in with polynomial coefficients of Shubin type are extended to entire functions in \C^n of finite exponential type
and decay like for in conic neighbourhoods of the form
. We also show that under semilinear polynomial perturbations all nonzero
homoclinics %(if exist)
keep the super-exponential decay of the above type, whereas a loss of the
holomorphicity occurs, namely we show holomorphic extension into
a strip \{z\in \C^n: \, |\Im z| \leq T\} for some .
The proofs are based on geometrical and perturbative methods in
Gelfand--Shilov spaces.
The results apply in particular to semilinear Schr\"{o}dinger equations of the form
\begin{equation}
\label{schro}
-\Delta u + |x|^2u -\lambda u = F(x,u, \nabla u).
\end{equation}
Our estimates on homoclinics are sharp. In fact, we
exhibit examples of solutions of \eqref{schro} with super-exponential decay, which are
meromorphic functions, the key point of our argument being the celebrated great
Picard theorem in complex analysis
Exponential decay and regularity for SG-elliptic operators with polynomial coefficients
No full textWe study the exponential decay and the regularity for solutions of elliptic partial differential equations ,
globally defined in In particular, we consider linear operators with
polynomial coefficients which are SG-elliptic at infinity. Starting from in the so-called Gelfand-Shilov spaces, the solutions of the equation are proved to belong to the same classes.
Proofs are based on a priori estimates and arguments on the Newton polyhedron associated to the operator
Semilinear pseudo-differential equations and travelling waves
No full textWe consider semilinear perturbations of the SG-elliptic pseudodifferential equations. We prove for them a theorem on the regularity of the solutions in the functional frame of the Gelfand-Shilov classes. Applications are given to the study of travelling solitary waves
Gelfand-Shilov spaces, pseudo-differential operators and localization operators
No full textWe present new results concerning pseudo-differential operators in the function spaces of Gelfand and Shilov. In particular we discuss -regularity of solutions to SG-elliptic pseudo-differential equations, allowing lower order semilinear perturbations.
The results apply to SG-elliptic partial differential equations with polynomial coefficients. We also study the action of Weyl operators and localization operators on $S^{\mu}_{\mu}(\R^n).
Parabolic Equations With Conservative Nonlinear Term And Singular Initial Data
No full text[No abstract available]30424892496Brezis, H., Friedman, A., Nonlinear parabolic equations involving measures as initial conditions (1983) J. Math. Pures Appl., 62, pp. 73-97Andreucci, D., Degenerate parabolic equations with initial data measures (1994) Dip. di Metodi Matematici e Modelli per le, Scienze Applicate, Università di Roma "La Sapienza", , PreprintKato, T., The Navier-Stokes equation for an incompressible fluid in ℝ 2 with a measure as the initial vorticity (1994) Differential & Integral Equations, 7, pp. 949-966Kato, T., Ponce, G., The Navier-Stokes equation with weak initial data (1994) International Mathematics Research Notes, 10, pp. 1-10Kozono, H., Yamazaki, M., Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data (1994) Comm. Partial Differential Equations, 19, pp. 959-1014Biagioni, H.A., Gramchev, T., On the 2D Navier-Stokes equation with singular initial data and forcing term (1995) Proc. IV Workshop on Partial Differential Equations, , to appear IMPA, Rio de Janeiro, Mat. ContempBiagioni, H.A., Cadeddu, L., Gramchev, T., (1996) Semilinear Parabolic Equations with Singular Initial Data in Anisotropic Weighted Spaces, , preprintDemengel, F., Serre, D., Nonvanishing singular parts of measure valued solutions for scalar hyperbolic equations (1991) Comm. in Partial Differential Equations, 16, pp. 221-254Dix, D.B., Nonuniqueness and Uniqueness in the Initial Value Problem for Burgers' Equation, , preprintBiagioni, H., Oberguggenberger, M., Generalized solutions to Burgers' equation (1992) J. Differential Equations, 97, pp. 263-287Oberguggenberger, M., Wang, Y.-G., Generalized solutions to conservation laws (1994) J. for Analysis and Its Applications, 13, pp. 7-1
Evolution Pde With Elliptic Dissipative Terms: Critical Index For Singular Initial Data, Self-similar Solutions And Analytic Regularity [edp D'évolution Avec Dissipation Elliptique : L'indice Critique Pour Des Données Initiales Singulières, Solutions Auto-similaires Et Régularité Analytique]
No full textWe investigate the influence of the elliptic dissipative terms of evolution equations in ℝn and double-struck T signn on the critical Lp, 1 ≤ p ≤ ∞, index of the singularity of the initial data, the analytic regularity for positive time and the existence of self-similar solutions. © Académie des Sciences/Elsevier, Paris.32714146Bekhiranov, D., The initial value problem for the generalized Burgers' equation (1996) Differ. Integ. Eq., 9, pp. 1253-1265Biagioni, H.A., Cadeddu, L., Gramchev, T., Parabolic equations with conservative nonlinear term and singular initial data, Proc. 2nd World Congress of Nonlinear Analysts (1997) Nonlin. Anal. TMA, 30, pp. 2489-2496. , Athens, Greece, July 10-17, 1996Brézis, H., Friedman, H., Nonlinear parabolic equations involving measures as initial conditions (1983) J. Math. Pures Appl., 62, pp. 73-97Brézis, H., Cazenave, T., Nonlinear heat equation with singular initial data (1996) J. Anal. Math., 68, pp. 276-304Cannone, M., Planchon, F., Self-similar solutions for Navier-Stokes equations in ℝ3, Commun (1996) Partial Differ, Eq., 21, pp. 179-193Chemin, J.-Y., Fluides parfaits incompressibles (1995) Astérisque, 230, pp. 1-177Dix, D., Nonuniqueness and uniqueness in thé initial value problem for Burgers' equation (1996) SIAM J. Math. Anal., 27, pp. 709-724Ferrari, A., Titi, E., Gevrey regularity for nonlinear analytic parabolic equations (1998) Commun. Partial Differ. Eq., 23, pp. 1-16Foias, C., Temam, R., Gevrey class regularity for the solutions of the Navier-Stokes equations (1989) J. Funct. Anal., 87, pp. 359-369Kozono, H., Yamazaki, M., Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data (1994) Commun. Partial Differ. Eq., 19, pp. 959-1014Levermore, D., Oliver, M., Distribution-valued initial data for the complex Ginzburg-Landau equation (1997) Commun. Partial Differ. Eq., 22, pp. 39-49Planchon, F., Convérgence des solutions des équations de Navier-Stokes vers des solutions auto-similaires (1996) Séminaire X-EDP, 95-96Ribaud, F., (1996) Analyse de Littlewood-Paley Pour la Résolution d'Équations Paraboliques Semi-linéaires, , Thèse de Docteur en Sciences, OrsayBollerman, P., Doelman, A., Van Harten, A., Titi, E., Analyticity for essentially bounded solutions to strongly parabolic semilinear systems (1996) SIAM J. Math. Anal., 27, pp. 424-44
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