| Title:
|
Results of nonexistence of solutions for some nonlinear evolution problems (English) |
| Author:
|
Djilali, Medjahed |
| Author:
|
Hakem, Ali |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
269-284 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), $$ u_{tt} +f(x)u_t +(-\Delta)^{\alpha/2}(u^m)= h(t,x) |u|^{p}, $$ posed in $(0,T)\times \mathbb{R}^{N},$ where $(-\Delta)^{{\alpha}/{2}}, 0<\alpha \leq 2$ is ${\alpha}/{2}$-fractional power of $\,-\Delta.$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times2$ system of the same type. (English) |
| Keyword:
|
nonexistence |
| Keyword:
|
test functions |
| Keyword:
|
global weak solution |
| Keyword:
|
fractional Laplacian |
| Keyword:
|
critical exponent |
| MSC:
|
35A01 |
| MSC:
|
35D30 |
| MSC:
|
47J35 |
| idZBL:
|
Zbl 07144893 |
| idMR:
|
MR3982472 |
| DOI:
|
10.14712/1213-7243.2019.001 |
| . |
| Date available:
|
2019-08-05T09:52:31Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147814 |
| . |
| Reference:
|
[1] Cazenave T., Haraux A.: Introduction aux problèmes d'évolution semi-linéaires.Mathématiques & Applications, 1. Ellipses, Paris, 1990 (French). MR 1299976 |
| Reference:
|
[2] Fino A. Z., Ibrahim H., Wehbe A.: A blow-up result for a nonlinear damped wave equation in exterior domain: the critical case.Comput. Math. Appl. 73 (2017), no. 11, 2415–2420. MR 3648022, 10.1016/j.camwa.2017.03.030 |
| Reference:
|
[3] Fujita H.: On the blowing up of solutions of the problem for $ u_t=\Delta u + u^{1+\alpha}$.J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124. MR 0214914 |
| Reference:
|
[4] Guedda M., Kirane M.: A note on nonexistence of global solutions to a nonlinear integral equation.Bull. Belg. Math. Soc. Simon Stevin 6 (1999), no. 4, 491–497. MR 1732885, 10.36045/bbms/1103055577 |
| Reference:
|
[5] Guedda M., Kirane M.: Local and global nonexistence of solutions to semilinear evolution equations.Proc. of the 2002 Fez Conf. on Partial Differential Equations, Electron. J. Differ. Equ. Conf. 09 (2002), 149–160. MR 1976692 |
| Reference:
|
[6] Hakem A.: Nonexistence of weak solutions for evolution problems on $\mathbb{R}^{N}$.Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 1, 73–82. MR 2134858, 10.36045/bbms/1113318131 |
| Reference:
|
[7] Hakem A., Berbiche M.: On the blow-up behavior of solutions to semi-linear wave models with fractional damping.IAENG Int. J. Appl. Math. 41 (2011), no. 3, 206–212. MR 2954229 |
| Reference:
|
[8] Ju N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations.Comm. Math. Phys. 255 (2005), 161–181. MR 2123380, 10.1007/s00220-004-1256-7 |
| Reference:
|
[9] Ogawa T., Takeda H.: Non-existence of weak solutions to nonlinear damped wave equations in exterior domains.Nonlinear Anal. 70 (2009), no. 10, 3696–3701. MR 2504456, 10.1016/j.na.2008.07.025 |
| Reference:
|
[10] Pozrikidis C.: The Fractional Laplacian.CRC Press, Boca Raton, 2016. MR 3470013 |
| Reference:
|
[11] Sun F., Wang M.: Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping.Nonlinear Anal. 66 (2007), no. 12, 2889–2910. MR 2311644, 10.1016/j.na.2006.04.012 |
| Reference:
|
[12] Takeda H.: Global existence and nonexistence of solutions for a system of nonlinear damped wave equations.J. Math. Anal. Appl. 360 (2009), no. 2, 631–650. MR 2561260, 10.1016/j.jmaa.2009.06.072 |
| Reference:
|
[13] Todorova G., Yordanov B.: Critical exponent for a nonlinear wave equation with damping.C. R. Acad. Sci. Paris Sèrie I Math. 330 (2000), no. 7, 557–562. MR 1760438, 10.1016/S0764-4442(00)00228-7 |
| Reference:
|
[14] Zhang Q. S.: A blow up result for a nonlinear wave equation with damping: the critical case.C. R. Acad. Sci. Paris Sèrie I Math. 333 (2001), no. 2, 109–114. MR 1847355, 10.1016/S0764-4442(01)01999-1 |
| Reference:
|
[15] Zheng S.: Nonlinear Evolution Equations.Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 113, Chapman & Hall/CRC Press, Boca Raton, 2004. MR 2088362 |
| . |
