Mathematical Analysis of Nonlinear Nonlocal Partial Differential
Equations
ABSTRACT
This project aims to obtain some rigorous results on mathematical analysis of nonlinear partial differential equations
arising in nonlocal elastic models of solids. The cornerstones of the project are local and global existence of solutions of
initial-value problems, finite time blow-up of solutions, global existence of small amplitude solutions and existence and
orbital stability of travelling waves.
Arts and Sciences
Various generalizations of the conventional (local) theory of elastic solids have been proposed in the literature. One of these
generalized theories is called non-local elasticity theory in which the equations of motion are integro-differential equations.
It is very important to suggest an applicable nonlinear model within the context of nonlocal elasticity theory. In recent two
studies (N. Duruk, A. Erkip and H.A. Erbay, IMA Journal of Applied Mathematics vol. 74, page 97-106, 2009 and N. Duruk,
H.A. Erbay, A. Erkip, Nonlinearity vol. 22, page 1-12, 2010), one such model has been proposed in one-dimensional case
and the corresponding initial-value problem has been studied. The model involves a convolution integral operator with a
general kernel function whose Fourier transform is nonnegative. For suitable choices of the kernel function, the equation of
motion reduces to some well-known examples of nonlinear wave equations, such as Boussinesq-type equations. In Duruk
et al (2010), assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term
it has been shown that the Cauchy problem is globally well-posed in suitable Sobolev spaces. Furthermore, conditions for
finite time blow-up have been provided.
Prof. Dr.
Hüsnü Ata Erbay
DEPARTMENT
Natural and Mathematical
Sciences
CONTACT
[email protected]
This project aims to study various extensions of the model proposed in Duruk et al (2009, 2010). The five fundamental
problems to be studied are:
FUNDING SCHEME
TÜBİTAK 1001
1. The aim of the first problem is to extend the analysis presented in Duruk et al (2010) about the global existence and finite
time blow-up of solutions to the case of two coupled nonlocal nonlinear partial differential equations.
START DATE
1/1/11
2. The aim of the problem is to prove the global existence of small amplitude solutions of the Cauchy problem introduced
in Duruk et al (2010).
2011 National Grants
3. This problem aims to do what has been done in Duruk et al (2010) in one-dimensional case for the case of two-dimensional
wave propagation.
4. The aim of the problem is to prove existence and orbital stability of travelling wave solutions of the Cauchy problem
introduced in Duruk et al (2010).
5. The aim of this problem is to discuss local existence, global existence and finite time blow-up of solutions of the Cauchy
problem arising in the peridynamic formulation of one dimensional elasticity theory. The peridynamic model proposed
by Silling (S.A. Silling, journal of the Mechanics and Physics of Solids vol. 48, page 175-209, 2000) has also nonlocal
character but it has a different flavor.
24
DURATION
24 months
OZU BUDGET
94,496.00 TL