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Milliy Universitet » Darslik » ASYMPTOTIC BEHAVIOR OF THE SELF-SIMILAR SOLUTIONS OF NONLINEAR KLEIN-GORDON TYPE EQUATIONS AND SYSTEMS

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ASYMPTOTIC BEHAVIOR OF THE SELF-SIMILAR SOLUTIONS OF NONLINEAR KLEIN-GORDON TYPE EQUATIONS AND SYSTEMS
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Kategoriya: Darslik

Authors: M.M. Aripov,  A.S. Matyakubov. 

Asymptotic behavior of self-similar solutions of the nonlinear Klein-Gordon type equations and systems.

Tashkent, “University” publishing house, 2011, 98 p.

e-mail: mirsaidaripov@mail.ru

 

Abstract.

 

Consider in the area $Q = \left( {\left( {t,x} \right):0 < t < \infty ,\,\,x \in R^N } \right)$  a generalised system of the N –quasilinear equations of Klein-Gordon type:

$$\frac{{\partial ^2 u_i }}{{\partial \,t^2 }} = \nabla (|x|^m |\nabla u_i^{k_i } |^{n - 1} \nabla u_i^{k_i } ) + F_i (u_1 ,u_2 ,...,u_N )\, $$

where $F_i (u_1 ,u_2 ,...,u_N ) = \prod\limits_{j = 1}^N {u_j^{p_{ij} } } ,\,\,\,\,\,p_{ij} ,\,\,\,\,k_i \,\,\,(i,j = 1,2,...,N),\,\,\,n,\,\,\,\,m\,\,\; - $ given positive real numbers, $\nabla ( \cdot ) = grad_x ( \cdot ),$ $u_i  = u_i (t,x) \ge 0 \quad (i=1, …, N) $  required solutions.

Depending on values of the parameters included into the equations and systems of equations of Klein-Gordon type, the asymptotic representations of self-similar solutions, the necessary and sufficient signs of existence of these solutions on the basis of researching the positive solutions of the ordinary differential equations, received by a method of nonlinear split, have been found.

Unlimited (blow up) solutions over the finite time, the equations and the systems of the equations of Klein-Gordon type have been investigated. Necessary conditions of their existence have been received.

Researches of the asymptotic behaviour of solutions at $t \to  + \infty ,$ $t^2  \to \sum\limits_{i = 1}^N {x_i^2 }, $  and $t \to T <  + \infty$ for the equations of Klein-Gordon type and at $t \to T \le  + \infty $.

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