Periodic solutions of parabolic and telegraph equations with asymmetric nonlinearities

• Autor(a) principal: Grossinho, Maria do Rosário
• Data de Publicação: 1998
• Outros Autores: Nkashama, M. N.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.5/27656
• topic: Jumping Nonlinearities; Time-periodic Solutions; Parabolic Equations; Telegraph Equations; Fuçik Spectrum; Resonance; A Priori Estimates; Degree Theory; LP-theory; Anisotropic Sobolev-Slobodeckii Spaces

This paper is organized as follows: Introduction, In Section 2, we collect the notation and basic assumptions that we shall suppose fulfilled throughout this paper. Section 3 is devoted to second order nonlinear one-dimensional parabolic and (linearly) damped hyperbolic equations. We compare, in some sense, the nonlinearity g(x, u) with the Fuçik spectrum of the corresponding piecewise linear differential equations with homogeneous Dirichlet boundary conditions, and a resonance condition of Landesman-Lazer type with respect to the forcing term h(x, t). More specifically, we assume that (the asymptotic behavior of) u - ¹ g ( x , u) lies in a rectangle located in what we should call the Fucik -Landesman-Lazer "resolvent" set. In Section 4, we take up the case of second-order multi-dimensional equations, and we prove results on crossing at not necessarily simple (higher) eigenvalues. Finally, in Section 5 we indicate the conditions under which one can extend our results to higher-order multi-dimensional equations.