پیوندهای مفید
آمار
| تعداد نشریات | 21 |
| تعداد شمارهها | 687 |
| تعداد مقالات | 9,985 |
| تعداد مشاهده مقاله | 70,957,067 |
| تعداد دریافت فایل اصل مقاله | 62,422,382 |
Global solutions for a nonlinear degenerate nonlocal problem | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 2، دوره 14، شماره 10، دی 2023، صفحه 9-17 اصل مقاله (387.08 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2023.26510.3338 | ||
| نویسنده | ||
| Eugenio Cabanillas Lapa* | ||
| Instituto de Investigacion, FCM-UNMSM, Av. Venezuela S/N, Lima, Peru | ||
| تاریخ دریافت: 02 فروردین 1401، تاریخ بازنگری: 21 مرداد 1402، تاریخ پذیرش: 24 مرداد 1402 | ||
| چکیده | ||
| In this paper, we consider the existence and asymptotic behavior of solutions to the following new nonlocal problem $$ u_{tt}- M\Big(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\Big)\triangle u + \delta u_{t}= |u|^{\rho-2}u\hspace{1.0cm} \text{in}\ \Omega \times ]0,\infty[, $$ where \begin{equation*} M(s)=\begin{cases} a-bs &\text{for } \ \, s \in [0,\frac{a}{b}[,\\ 0, &\text{for } s \in [\frac{a}{b}, +\infty[. \end{cases} \end{equation*} We first state a local existence theorem. Next, if the initial energy is appropriately small, by using Tartar's method and the decay rate of the energy, we derive the global existence theorem. As a biproduct, we also obtain the exponential decay property of the global solution. | ||
| کلیدواژهها | ||
| global solutions؛ degenerate nonlocal problem؛ asymptotic behavior | ||
| مراجع | ||
|
[1] M. Aassila and A. Benaissa, Existence globale et comportement asymptotique des solutions des equations de Kirchhoff moyennement degenerees avec un terme nonlinear dissipatif, Funkcial. Ekvac. 44 (2001), 309–333. [2] F.D. Araruna, F.O. Matias, M.L. Oliveira, and S.M.S.Souza, Well-posedness and asymptotic behavior for a nonlinear wave equation, Recent Advances in PDEs: Analysis, Numerics and Control, Springer, 17 (2018), 17–32. [3] A. Arosio and S. Garavaldi, On the mildly degenerate Kirchhoff string, Math. Meth. Appl. Sci. 14 (1991), 177–195 [4] A. Benaissa and L. Rahmani, Global existence and energy decay of solutions for Kirchhoff-Carrier equations with weakly nonlinear dissipation, Bull. Belg. Math. Soc. 11 (2004), 547–574. [5] M.M. Chaharlang and A. Razani Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28 (2021), no. 3, 429–438. [6] M. Chipot and B. Lovat, Some remarks on non local elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619-–4627. [7] M. Ghisi and M. Gobbino, Global existence and asymptotic behaviour for a mildly degenerate dissipative hyperbolic equation of Kirchhoff type, Asymptot. Anal. 40 (2004), 25–36. [8] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn, Springer, Berlin, 1983. [9] M.K. Hamdani, N.T. Chung, and D.D. Repovs, New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions, Adv. Nonlinear Anal. 10 (2021), 1117–1131. [10] G. Kirchhoff, Vorlesungen ¨uber Mechanik, Leipzig, Teubner, 1883. [11] J.E. Lions, On some questions in boundary value problems of mathematical physics, Proc. Int. Symp. Continuum Mech. Partial Diff. Equ., North Holland, Amsterdam, The Netherlands, 1978. [12] J.L. Lions, Some Remarks on the Optimal Control on Singular Distributed Systems, INRIA, France, 1984. [13] B. Lovat, Etudes de quelques probl`ems paraboliques non locaux, Th`ese, Universite de Metz, 1995. [14] M. Milla Miranda, A.T. Louredo, and L.A. Medeiros, Nonlinear perturbations of the Kirchhof equation, Electron J. Diff. Equ. 77 (2017), 1–21. [15] Z. Musbah and A. Razani, A class of biharmonic nonlocal quasilinear systems consisting of Leray-Lions type operators with Hardy potential, Bound. Value Probl. 2022 (2022), no. 1, 88. [16] M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. 30 (1976), 257–265. [17] K. Ono, Global existence and decay properties of solutions for coupled degenerate dissipative hyperbolic systems of Kirchhoff type, Funkcial. Ekvac. 57 (2014), 319–337. [18] K. Ono, Global existence and decay properties of solutions of some mildly degenerate nonlinear dissipative Kirchhofi strings, Funkcial. Ekvac. 40 (1997), 255–270. [19] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation, Math. Meth. Appl. Sci. 20 (1997), 151–177. [20] X. Qian, Multiplicity of positive solutions for a class of nonlocal problem involving critical exponent, Electron. J. Qual. Theory Differ. Equ. 57 (2021), 1–14. [21] M. Ghergu and V. Radulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Heidelberg, Springer Verlag, 2012. [22] A. Razani, Two Weak Solutions for Fully Nonlinear Kirchhoff-Type Problem, Filomat 35 (2021), no. 10, 3267–3278. [23] A. Razani and G. Figuieredo, Weak solution by sub-super solution method for a nonlocal elliptic system involving Lebesgue generalized spaces, Electron J. Diff. Equ. 2022 (2022), no. 36, 1–18. 36, 1-18. [24] A. Razani and F. Behboudi, Weak solutions for some fractional singular (p, q)-Laplacian nonlocal problems with Hardy potential, Rend. Circ. Mate. Palermo Ser. 2 72 (2023), 1639–1654. [25] L. Tartar, Topics in Nonlinear Analysis, Publications Math´ematiques d’Orsay, Uni.Paris Sud. Dep. Math., Orsay, France, 1978. [26] Y. Ye, Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation, J. Inequal. Appl. 2013 (2013), no. 1, 1–10. [27] G.S. Yin and J.S. Liu, Existence and multiplicity of nontrivial solutions for a nonlocal problem, Bound. Value Probl. 2015 (2015), no. 26, 1—7. [28] Y. Wang and X. Yang, Infinitely many solutions for a new Kirchhoff-type equation with subcritical exponent, Appl. Anal. 101 (2022), no. 3, 1038–1051. | ||
|
آمار تعداد مشاهده مقاله: 16,635 تعداد دریافت فایل اصل مقاله: 10,341 |
||