Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity

We study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples...

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Bibliographic Details
Published in:Nonlinearity 2010-01, Vol.23 (1), p.107-118
Main Authors: Duruk, N, Erbay, H A, Erkip, A
Format: Article
Language:English
Subjects:
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematical methods in physics
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Other topics in mathematical methods in physics
Partial differential equations
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Physics
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided.
ISSN:0951-7715
1361-6544
DOI:10.1088/0951-7715/23/1/006