Global existence‎, ‎stability results and compact invariant sets‎ ‎for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$

نویسندگان

  • A. Pappas Civil Engineering Department, School of Technological Applications, Technological Educational Institution (TEI) of Piraeus, GR 11244, Egaleo, Athens, Greece.
  • N.L. Matiadou Department of Electronics Engineering, School of Technological Applications, Technological Educational Institution (TEI) of Piraeus, GR 11244, Egaleo, Athens, Greece
  • P. Papadopoulos adepartment of electronics engineering, school of technological applications, technological educational institution (tei) of piraeus, gr 11244, egaleo, athens, greece
چکیده مقاله:

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type [ u_{tt}-phi (x)||nabla u(t)||^{2}Delta u+delta u_{t}=|u|^{a}u,, x in mathbb{R}^{N} ,,tgeq 0;,]with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N geq 3, ; delta geq 0$ and $(phi (x))^{-1} =g (x)$  is a positive function lying in $L^{N/2}(mathbb{R}^{N})cap L^{infty}(mathbb{R}^{N})$. It is proved that, when the initial energy $ E(u_{0},u_{1})$, which corresponds to the problem, is non-negative and small, there exists a unique global solution in time in the space ${cal{X}}_{0}=:D(A) times {cal{D}}^{1,2}(mathbb{R}^{N})$. When the initial energy $E(u_{0},u_{1})$ is negative, the solution blows-up in finite time. For the proofs, a combination of the modified potential well method and the concavity method is used. Also, the existence of an absorbing set in the space ${cal{X}}_{1}=:{cal{D}}^{1,2}(mathbb{R}^{N}) times L^{2}_{g}(mathbb{R}^{N})$ is proved and that the dynamical system generated by the problem possess an invariant compact set ${cal {A}}$ in the same space.Finally, for the generalized dissipative Kirchhoff's String problem [ u_{tt}=-||A^{1/2}u||^{2}_{H} Au-delta Au_{t}+f(u) ,; ; x in mathbb{R}^{N}, ;; t geq 0;,]with the same hypotheses as above, we study the stability of the trivial solution $uequiv 0$. It is proved that if $f'(0)>0$, then the solution is unstable for the initial Kirchhoff's system, while if $f'(0)

برای دانلود باید عضویت طلایی داشته باشید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

global existence‎, ‎stability results and compact invariant sets‎ ‎for a quasilinear nonlocal wave equation on $mathbb{r}^{n}$

we discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of kirchhoff type [ u_{tt}-phi (x)||nabla u(t)||^{2}delta u+delta u_{t}=|u|^{a}u,, x in mathbb{r}^{n} ,,tgeq 0;,]with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $n geq 3, ; delta geq 0$ and $(phi (x))^{-1} =g (x)$  is a positive function lying in $l^{n/2}(mathb...

متن کامل

Strong Global Attractor for a Quasilinear Nonlocal Wave Equation on R

We study the long time behavior of solutions to the nonlocal quasilinear dissipative wave equation utt − φ(x)‖∇u(t)‖∆u+ δut + |u|u = 0, in RN , t ≥ 0, with initial conditions u(x, 0) = u0(x) and ut(x, 0) = u1(x). We consider the case N ≥ 3, δ > 0, and (φ(x))−1 a positive function in LN/2(RN ) ∩ L∞(RN ). The existence of a global attractor is proved in the strong topology of the space D1,2(RN )×...

متن کامل

Existence Results for a Dirichlet Quasilinear Elliptic Problem

In this paper, existence results of positive classical solutions for a class of second-order differential equations with the nonlinearity dependent on the derivative are established. The approach is based on variational methods.

متن کامل

global results on some nonlinear partial differential equations for direct and inverse problems

در این رساله به بررسی رفتار جواب های رده ای از معادلات دیفرانسیل با مشتقات جزیی در دامنه های کراندار می پردازیم . این معادلات به فرم نیم-خطی و غیر خطی برای مسایل مستقیم و معکوس مورد مطالعه قرار می گیرند . به ویژه، تاثیر شرایط مختلف فیزیکی را در مساله، نظیر وجود موانع و منابع، پراکندگی و چسبندگی در معادلات موج و گرما بررسی می کنیم و به دنبال شرایطی می گردیم که متضمن وجود سراسری یا عدم وجود سراسر...

Global Existence for a Quasilinear Wave Equation outside of Star-shaped Domains

which satisfy the so-called null condition [15]. The global existence for such equations in the absence of obstacles was established by Christodoulou [2] and Klainerman [11] using different techniques. We begin by describing our assumptions in more detail. We let u denote a N -tuple of functions, u = (u, u, . . . , u). We assume that K is smooth and strictly star-shaped with respect to the orig...

متن کامل

Local Existence And Stability Results For A Wave Equation With Damping On All R

We discuss the local existence results of the solutions for the nonlocal hyperbolic problem 2 ( ) || ( ) || ( ) 0, tt t u x u t u u        , N x R  0, t  with initial conditions 0 ( ,0) ( ) u x u x  and 1 ( ,0) ( ), t u x u x  in the case where 3, 0 N    and 1 ( ( )) ( ) x g x    is a positive function lying in / ( ) ( ). N N N L R L R  When the initial energy 0 1 ( , ) E u u...

متن کامل

منابع من

با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ذخیره در منابع من قبلا به منابع من ذحیره شده

{@ msg_add @}


عنوان ژورنال

دوره 6  شماره 1

صفحات  85- 95

تاریخ انتشار 2015-04-13

با دنبال کردن یک ژورنال هنگامی که شماره جدید این ژورنال منتشر می شود به شما از طریق ایمیل اطلاع داده می شود.

میزبانی شده توسط پلتفرم ابری doprax.com

copyright © 2015-2023