First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities. (English) Zbl 1492.35294
Summary: We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation
\[
iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in{\mathbb{R}}^N,
\] where \(N\ge 1, p,q>1, \lambda,\mu \in{\mathbb{C}}\), \(\lambda \ne 0\), and \(u(0,\cdot)\), \(w\in L^1_{\text{loc}}({\mathbb{R}}^N,{\mathbb{C}})\). We consider both the cases where \(\mu =0\) and \(\mu \ne 0\), respectively. We establish existence/nonexistence of global weak solutions. In each studied case, we compute the critical exponents in the sense of Fujita, and Lee and Ni. When \(\mu \ne 0\), we show that the nonlinearity \(|\nabla u|^q\) induces an interesting phenomenon of discontinuity of the Fujita critical exponent.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B33 | Critical exponents in context of PDEs |
| 35B44 | Blow-up in context of PDEs |
References:
| [1] | Cazenave, T., Semilinear Schrödinger Equations (2003), New York: American Mathematical Society, New York · Zbl 1055.35003 · doi:10.1090/cln/010 |
| [2] | Fino, AZ; Dannawi, I.; Kirane, M., Blow-up of solutions for semilinear fractional Schrödinger equations, J. Integr. Equ. Appl., 30, 67-80 (2018) · Zbl 1393.35220 · doi:10.1216/JIE-2018-30-1-67 |
| [3] | Fujita, H., 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, 109-124 (1966) · Zbl 0163.34002 |
| [4] | Ikeda, M.; Inui, T., Small data blow-up of \(L^2\) or \(H^1\)-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15, 571-581 (2015) · Zbl 1327.35353 · doi:10.1007/s00028-015-0273-7 |
| [5] | Ikeda, M.; Inui, T., Some non-existence results for the semilinear Schrödinger equation without gauge invariance, J. Math. Anal. Appl., 425, 758-773 (2015) · Zbl 1311.35288 · doi:10.1016/j.jmaa.2015.01.003 |
| [6] | Ikeda, M.; Wakasugi, Y., Small data blow-up of \(L^2\)-solution for the nonlinear Schrödinger equation without gauge invariance, Differ. Integr. Equ., 26, 1275-1285 (2013) · Zbl 1313.35324 |
| [7] | Inui, T., Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity, Proc. Am. Math. Soc., 144, 2901-2909 (2016) · Zbl 1342.35339 · doi:10.1090/proc/12938 |
| [8] | Jleli, M.; Samet, B., On the critical exponent for nonlinear Schrödinger equations without gauge invariance in exterior domains, J. Math. Anal. Appl., 469, 188-201 (2019) · Zbl 1404.35417 · doi:10.1016/j.jmaa.2018.09.009 |
| [9] | Jleli, M.; Samet, B.; Vetro, C., On a fractional in time nonlinear Schrödinger equation with dispersion parameter and absorption coefficient, Symmetry, 12, 1197 (2020) · doi:10.3390/sym12071197 |
| [10] | Kenig, C.; Ponce, G.; Vega, L., Quadratic forms for the 1-D semilinear Schrödinger equation, Trans. Am. Math. Soc., 348, 3323-3353 (1996) · Zbl 0862.35111 · doi:10.1090/S0002-9947-96-01645-5 |
| [11] | Kirane, M.; Nabti, A., Life span of solutions to a nonlocal in time nonlinear fractional Schrödinger equation, Z. Angew. Math. Phys., 66, 1473-1482 (2015) · Zbl 1322.35127 · doi:10.1007/s00033-014-0473-y |
| [12] | Lee, TY; Ni, WM, Global existence, large time behavior and life span on solution of a semilinear parabolic Cauchy problem, Trans. Am. Math. Soc., 333, 365-378 (1992) · Zbl 0785.35011 · doi:10.1090/S0002-9947-1992-1057781-6 |
| [13] | Mitidieri, E.; Pokhozhaev, SI, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234, 1-384 (2001) · Zbl 0987.35002 |
| [14] | Nabti, A., Nonexistence of non-trivial global weak solutions for higher-order nonlinear Schrödinger equations, Electron. J. Differ. Equ., 312, 1-10 (2015) · Zbl 1329.35287 |
| [15] | Pinsky, RG, Existence and nonexistence of global solutions for \(u_t= \Delta u+a(x)u^p\) in \({\mathbb{R}}^d\), J. Differ. Equ., 133, 152-177 (1997) · Zbl 0876.35048 · doi:10.1006/jdeq.1996.3196 |
| [16] | Strauss, WA, Nonlinear scattering theory at low energy, J. Funct. Anal., 41, 110-133 (1981) · Zbl 0466.47006 · doi:10.1016/0022-1236(81)90063-X |
| [17] | Tsutsumi, Y., \(L^2\)-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial Ekvac., 30, 115-125 (1987) · Zbl 0638.35021 |
| [18] | Zhang, QS, A new critical phenomenon for semilinear parabolic problems, J. Math. Anal. Appl., 219, 125-139 (1998) · Zbl 0962.35087 · doi:10.1006/jmaa.1997.5825 |
| [19] | Zhang, Q.; Sun, HR; Li, Y., The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance, Appl. Math. Lett., 64, 119-124 (2017) · Zbl 1349.35415 · doi:10.1016/j.aml.2016.08.017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
