Geodesic
Blow-up of solutions of Cauchy problem for nonlinear Schr\"odinger equation
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2013), pp. 159-171.

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In this work we study the effect of time finiteness of the existence of Cauchy problem for nonlinear Schrödinger equation solution. Together with the ill-posed Cauchy problem we consider its neighborhood in the space of operators, representing Cauchy problem. We explore the convergence of sequence of solutions of Cauchy problems with the operators, approximating the initial Hamiltonian.
Keywords: nonlinear Schrödinger equation, regularization, blow-up regime, blow-up of solution, viscosity solution. 
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V. Zh. Sakbaev. Blow-up of solutions of Cauchy problem for nonlinear Schr\"odinger equation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2013), pp. 159-171. http://geodesic.mathdoc.fr/item/VSGTU_2013_1_a14/

[1] A. A. Samarskiy, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhaylov, Peaking modes in problems for quasilinear parabolic equations, Nauka, Moscow, 1987, 478 pp. | MR

[2] E. Mitidieri, S. I. Pohozaev A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1–362 | MR | Zbl

[3] H. Fujita, “On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha }$”, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124 | MR | Zbl

[4] S. N. Kruzhkov, Lectures on partial differential equations, Moscow State Univ., Moscow, 1970

[5] O. A. Oleynik, E. V. Radkevich, Equations with nonnegative characteristic form, Moscow Univ., Moscow, 2010, 360 pp.

[6] V. Zh. Sakbaev, “The Cauchy problem for a linear differential equation with degeneration and averaging of its approximating regularizations”, Partial differential equations, CMFD, 43, PFUR, Moscow, 2012, 3–172

[7] R. T. Glassey, “On the blowing up of solution to the Cauchy Problem for nonlinear Schrödinger equations”, J. Math. Phys, 18:9 (1977), 1794–1797 | DOI | MR | Zbl

[8] F. Merle, Y. Tsutsumi, “$L_2$ convergence of blow-up solutions for nonlinear Shrodinger equation with critical power nonlinearity”, J. Differ. Equations, 84:2 (1990), 205–214 | DOI | MR | Zbl

[9] P. E. Zhidkov, “Korteweg-de Vries and nonlitear Schrödinger equations: qualitative theory. Lecture Notes in Math”, Lecture Notes in Mathematics, 1756, Springer-Verlag, Berlin, 2001, vi+147 pp. | MR | Zbl

[10] P. Baras, J. Goldstein, “The heat equation with a singular potential”, Trans. Amer. Math. Soc., 284:1 (1984), 121–139 | DOI | MR | Zbl

[11] V. A. Galaktionov, I. V. Kamotski, “On nonexistance of Baras-Goldstein type for higher-order parabolic eqations with singular potential”, Trans. Amer. Math. Soc., 362:8 (2010), 4117–4136 | DOI | MR | Zbl

[12] N. Mizoguchi, F. Quirós, J. L. Vázquez, “Multiple blow-up for a porous medium equation with reaction”, Math. Ann., 350:4 (2011), 801–827 | DOI | MR | Zbl

[13] V. Zh. Sakbaev, “Averaging of quantum dynamical semigroups”, Theoret. and Math. Phys., 164:3 (2010), 1215–1221 | DOI | DOI | Zbl

[14] V. A. Galaktionov, J. L. Vázquez, “Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations”, Arch. Rational Mech. Anal., 129:3 (1995), 225–244 | DOI | MR | Zbl

[15] V. Zh. Sakbaev, “On the properties of ambiguity and irreversibility of dynamical maps of initial data space of Cauchy problems”, p-Adic Numbers Ultrametric Anal. Appl., 4:4 (2012), 306–318 | DOI | MR | Zbl

[16] J. Ginibre, G. Velo, “On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case”, J. Funct. Anal., 32:1 (1979), 1–32 | DOI | MR | Zbl

[17] O. Kavian, “A remark on the blowing-up solutions to the Cauchy problem for nonlinear Schrödinger equation”, Trans. Amer. Math. Soc., 299:1 (1987), 192–203 | MR

[18] A. Pazy, “Semigroups of linear operators and applications to partial differential equations”, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983, viii+279 pp. | DOI | MR | Zbl

[19] S. M. Nikol'skiy, Approximation of functions of several variables and imbedding theorems, Nauka, Moscow, 1969, 480 pp. | MR | Zbl

[20] S. G. Kreyn, Linear differential equations in a Banach space, Nauka, Moscow, 1967, 464 pp. | MR