Spectral Bounds for Tricomi Problems and Application to Semilinear Existence and Existence with Uniqueness Results
Source: Journal of Differential Equations, Volume 184, Number 1, September 2002 , pp. 139-162(24)
Publisher: Academic Press
Abstract:For the linear Tricomi problem, it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum. Exploiting the linear solvability theory and spectral information, standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems. In particular, using the Leray–Schauder principle, existence of generalized solutions with sublinear nonlinearities is established. For sublinear or asymptotically linear nonlinearities that satisfy a Lipschitz condition, the contraction mapping principle is employed to give results on existence with uniqueness. The Lipschitz constant depends on lower bounds for the spectrum of the linear problem. For certain superlinear problems, maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence. © 2002 Elsevier Science (USA).
Document Type: Research Article
Affiliations: 1: Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci, 32, Milano, 20133, Italy 2: Dipartimento di Matematica “F. Enriques”, Università di Milano, Via Saldini, 50, Milano, 20133, Italy
Publication date: September 1, 2002