The Cauchy Problem in for the Logarithmic Schrödinger Equation
DOI:
https://doi.org/10.65405/bp963352الكلمات المفتاحية:
Cauchy problem, Logarithmic Schrödinger equation, Damping, Sobolev spaces, Local well-posedness, Banach fixed-point theorem.الملخص
This work studies the Cauchy problem for the non-linear logarithmic Schrödinger equation (LogSE) incorporating damping and an external force:
i + Δu + u ln(|u|²) + i γ u = f
in the Sobolev space H¹(ℝ) with the initial condition u(0) = u₀ in H¹(ℝ). We investigate the local well-posedness (in the sense of Hadamard) of the Cauchy problem. To handle the logarithmic nonlinearity u ln(|u|²), which presents a singularity at the origin, a rigorous functional framework based on Sobolev space properties is established. The analytical approach relies on classical fixed-point theory (Banach fixed-point theorem) to establish the existence and uniqueness of solutions.
التنزيلات
المراجع
[1] M. Abounouh and O. Goubet, Attractor for damped cubic Schrödinger equation on a two
dimonsional thin domain, Differential Integral Equations, vol. 13, no. 1-3, pp. 311-340, 20000.
[2] H. Brezis, Analyse fonctionnelle, Édition Masson, Paris, 1983.
[3] T. Cazenave, An introduction to nonliear Schrödinger equations, Textos de Métodos
Matematicos, 26, 1993.
[4] T. Cazenave and A. Haraux, Introduction aux problèmes d’évolution semi-linéaires, Ellipses,
Paris, 1990.
[5] A. Dabaa, Comportement asymptotique des solutions d’un système d’équations de
Schrödinger-Poisson sur un domaine borné de , Annales Mathématiques Blaise Pascal, vol.
17, no. 1, pp. 199–232, 2010.
[6] J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré, vol. 5, pp. 365–405, 1988.
[7] O. Goubet, Regularité of the attracteurs for a weakly damped non linear Schrödinger equation, Analysis, vol. 60, pp. 99–119, 1995.
[8] P. Laurençot, Long time behaviour for weak damped driven non linear Schrödinger equation in , NoDEA, 1995.
[9] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation: Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.
[10] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Appl. Math. Sc. 68, Springer-Verlag, New York, Second Edition, 1997.
[11] I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Annals of Physics, vol. 100,
no. 1-2, pp. 62–93, 1976.
[12] R. Carles and I. Gallagher, Universal dynamics for the logarithmic Schrödinger equation, Duke Mathematical Journal, vol. 167, no. 16, pp. 3105–3141, 2018.
[13] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Analysis:
The-ory, Methods & Applications, vol. 7, no. 10, pp. 1127–1140, 1983.
[14] H. Hajaiej, On the regularized logarithmic Schrödinger equation, Communications on Pure
and Applied Analysis, vol. 12, no. 5, pp. 2003–2015, 2013.
[15] C. Ji and V. Rădulescu, Multi-peak solutions for logarithmic Schrödinger equations,
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 149, no. 6, pp.
1453–1468, 2019.
