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Advisor(s)
Abstract(s)
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.
Description
Keywords
Jumping Nonlinearities Time-periodic Solutions Parabolic Equations Telegraph Equations Fuçik Spectrum Resonance A Priori Estimates Degree Theory LP-theory Anisotropic Sobolev-Slobodeckii Spaces
Pedagogical Context
Citation
Grossinho, Maria do Rosário and M.N. Nkashama .(1998). “Periodic solutions of parabolic and telegraph equations with asymmetric nonlinearities”. Nonlinear Analysis, Theory, Methods & Applications, Vol. 33, No. 2: pp. 187-210. (Search PDF in 2023).