### Electronic Research Archive

2020, Issue 2: 777-794. doi: 10.3934/era.2020039

# $H^2$ blowup result for a Schrödinger equation with nonlinear source term

• Received: 01 February 2020 Revised: 01 April 2020
• Primary: 35L71; Secondary: 35B30, 35B44

• In this paper, we consider the nonlinear Schrödinger equation on $\mathbb{R}^N, N\ge1$,

$\partial_tu = i\Delta u+\lambda|u|^\alpha u ,$

with $H^2$-subcritical nonlinearities: $\alpha>0, (N-4)\alpha<4$ and Re$\lambda>0$. For any given compact set $K\subset\mathbb{R}^N$, we construct $H^2$ solutions that are defined on $(-T, 0)$ for some $T>0$, and blow up exactly on $K$ at $t = 0$. We generalize the range of the power $\alpha$ in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.

Citation: Xuan Liu, Ting Zhang. $H^2$ blowup result for a Schrödinger equation with nonlinear source term[J]. Electronic Research Archive, 2020, 28(2): 777-794. doi: 10.3934/era.2020039

### Related Papers:

• In this paper, we consider the nonlinear Schrödinger equation on $\mathbb{R}^N, N\ge1$,

$\partial_tu = i\Delta u+\lambda|u|^\alpha u ,$

with $H^2$-subcritical nonlinearities: $\alpha>0, (N-4)\alpha<4$ and Re$\lambda>0$. For any given compact set $K\subset\mathbb{R}^N$, we construct $H^2$ solutions that are defined on $(-T, 0)$ for some $T>0$, and blow up exactly on $K$ at $t = 0$. We generalize the range of the power $\alpha$ in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.

 [1] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., 17. Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2 [2] A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation. São Paulo J. Math. Sci. (2015) 9: 146-161. [3] Continuous dependence for NLS in fractional order spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire (2011) 28: 135-147. [4] Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation. Trans. Amer. Math. Soc. (2016) 368: 7911-7934. [5] T. Cazenave, Z. Han and Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, (2019), arXiv: 1906.02983. [6] Finite-time blowup for a Schröding equation with nonlinear source term. Discrete Contin. Dynam. Systems. (2019) 39: 1171-1183. [7] Solutions blowing up on any given compact set for the energy subcritical wave equation. J. Differential Equations (2020) 268: 680-706. [8] Solutions with prescribed local blow-up surface for the nonlinear wave equation. Adv. Nonlinear Stud. (2019) 19: 639-675. [9] Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete Contin. Dyn. Syst. (2019) 39: 1171-1183. [10] C. Collot, T. E. Ghouland N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity, (2018), arXiv: 1803.07826. [11] A multivariate Faa di Bruno formula with applications. Trans. Amer. Math. Soc. (1996) 348: 503-520. [12] On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. (1987) 46: 113-129. [13] S. Kawakami and S. Machihara, Blowup solutions for the nonlinear Schrödinger equation with complex coefficient, (2019), arXiv: 1905.13037. [14] The radial mass-subcritical NLS in negative order Sobolev spaces. Discrete Contin. Dyn. Syst. (2019) 39: 553-583. [15] Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. (2005) 127: 1103-1140. [16] Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. (1990) 129: 223-240. [17] O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete Contin. Dyn. (2002) 8: 435-450. [18] On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations. Comm. Math. Phys. (2015) 333: 1529-1562. [19] I. Moerdijk and G. Reyes, Models for Smooth Infinitesimal Analysis, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4757-4143-8 [20] Construction of a blow-up solution for the complex ginzburg-landau equation in a critical case. Arch. Ration. Mech. Anal. (2018) 228: 995-1058. [21] Solutions of semilinear Schrödinger equations in $H^s$. Ann. Inst. H. Poincareé Phys. Théor. (1997) 67: 259-296. [22] Compact sets in the space $L^p(0, T; B)$. Ann. Mat. Pura Appl. (1987) 146: 65-96. [23] J. Speck, Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearity, Analysis and PDE, 13 (2020), 93–146, arXiv: 1709.04778. doi: 10.2140/apde.2020.13.93 [24] Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrödinger equations. Electron. J. Differential Equations (2018) 2018: 1-52. [25] Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential. Adv. Nonlinear Anal. (2020) 9: 882-894.
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