Existence and nonexistence results for higher order evolution inequalities posed outside the half-ball

Manal Alfulaij; Mohamed Jleli

doi:10.3934/dcdss.2024070

Article Contents

2024, Volume 17, Issue 10: 3046-3063. Doi: 10.3934/dcdss.2024070

This issue Previous Article Local well-posedness of the fractional inhomogeneous NLS in Sobolev spaces Next Article On mathematical model of infectious disease by using fractals fractional analysis

Existence and nonexistence results for higher order evolution inequalities posed outside the half-ball

Manal Alfulaij^1, and
Mohamed Jleli^1, ,

1.
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

^*Corresponding author: Mohamed Jleli
^*Corresponding author: Mohamed Jleli

Received: March 2024

Revised: May 2024

Early access: May 13, 2024

Published online: May 13, 2024

The second author is supported by Researchers Supporting Project number (RSP2024R57), King Saud University, Riyadh, Saudi Arabia.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

This article focuses on existence and nonexistence of weak solutions to semilinear higher order (in time) evolution inequalities with Hardy potential and posed outside the half-ball of $ \mathbb{R}^N $ ($ N\geq 2) $. We impose inhomogeneous Dirichlet-type boundary conditions, and show the dividing line with respect to existence or nonexistence is given by a Fujita-type critical exponent which depends on a suitable parameter $ \lambda $, but does not depend on the order of the time derivative. We conclude naturally optimal nonexistence results for the corresponding elliptic inequality.

Keywords:

Mathematics Subject Classification: Primary: 35R45, 35A01; Secondary: 35B44.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	B. Abdellaoui, A. Attar, R. Bentifour and I. Peral, On fractional $p$-Laplacian parabolic problem with general data, Ann. Mat. Pura Appl., 197 (2018), 329-356. doi: 10.1007/s10231-017-0682-z.
[2]	B. Abdellaoui, M. Medina, I. Peral and A. Primo, Optimal results for the fractional heat equation involving the Hardy potential, Nonlinear Anal., 140 (2016), 166-207. doi: 10.1016/j.na.2016.03.013.
[3]	B. Abdellaoui, S. E. Miri, I. Peral and T. M. Touaoula, Some remarks on quasilinear parabolic problems with singular potential and a reaction term, Nonlinear Differ. Equ. Appl., 21 (2014), 453-490. doi: 10.1007/s00030-013-0253-y.
[4]	B. Abdellaoui, I. Peral and A. Primo, Influence of the Hardy potential in a semi-linear heat equation, Proc. R. Soc. Edinburgh. Sect. A., 139 (2009), 897-926. doi: 10.1017/S0308210508000152.
[5]	B. Abdellaoui, I. Peral and A. Primo, A note on the Fujita exponent in fractional heat equation involving the Hardy potential, Mathematics in Engineering., 2 (2020), 639-656. doi: 10.3934/mine.2020029.
[6]	C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (1989), 595-622. doi: 10.1090/S0002-9947-1989-0937878-9.
[7]	M. B. Borikhanov and B. T. Torebek, On an inhomogeneous exterior Robin problems with critical nonlinearities, J. Differential Equations, 380 (2024), 1-23. doi: 10.1016/j.jde.2023.10.020.
[8]	H. Chen and L. Véron, Schrödinger operators with Leray-Hardy potential singular on the boundary, J. Diff. Equ., 269 (2020), 2091-2131. doi: 10.1016/j.jde.2020.01.029.
[9]	M. D'Abbicco, R. Ikehata and H. Takeda, Critical exponent for semi-linear wave equations with double damping terms in exterior domains, Nonlinear Differ. Equ. Appl., 26 (2019), Paper No. 56, 25 pp. doi: 10.1007/s00030-019-0603-5.
[10]	A. El Hamidi and G. G. Laptev, Existence and nonexistence results for higher-order semilinear evolution inequalities with critical potential, J. Math. Anal. Appl., 304 (2005), 451-463. doi: 10.1016/j.jmaa.2004.09.019.
[11]	A. Z. Fino, Finite time blow-up for wave equations with strong damping in an exterior domain, Mediterranean Journal of Mathematics., 17 (2020), Paper No. 174, 21 pp. doi: 10.1007/s00009-020-01607-2.
[12]	J. A. Goldstein, D. Hauer and A. Rhandi, Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential, Nonlinear Anal., 131 (2016), 121-154. doi: 10.1016/j.na.2015.07.016.
[13]	J. W. He, Nonexistence of global solutions for time fractional wave equations in an exterior domain, Journal of Integral Equations and Applications, 35 (2023), 11-26. doi: 10.1216/jie.2023.35.11.
[14]	R. Ikehata, Small data global existence of solutions for dissipative wave equations in an exterior domain, Funkcial. Ekvac., 45 (2002), 259-269.
[15]	M. Jleli and B. Samet, Nonexistence for higher order evolution inequalities with Hardy potential in $\mathbb{R}^N$, J. Math. Anal. Appl., 531 (2024), Paper No. 127755, 11 pp. doi: 10.1016/j.jmaa.2023.127755.
[16]	M. Jleli, B. Samet and C. Vetro, On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space, J. Math. Anal. Appl., 526 (2023), Paper No. 127325, 24 pp. doi: 10.1016/j.jmaa.2023.127325.
[17]	M. Jleli, B. Samet and D. Ye, Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains, J. Differential Equations, 268 (2020), 3035-3056. doi: 10.1016/j.jde.2019.09.051.
[18]	T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Commun. Pure Appl.Math., 33 (1980), 501-505. doi: 10.1002/cpa.3160330403.
[19]	H. A. Levine and Q. S. Zhang, The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 591-602. doi: 10.1017/S0308210500000317.
[20]	X. F. Li and G. X. Wang, Blow up of solutions to nonlinear wave equations in 2D exterior domains, Arch. Math., 98 (2012), 265-275. doi: 10.1007/s00013-012-0366-2.
[21]	E. Mitidieri and S. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162. doi: 10.1007/s00032-004-0032-7.
[22]	T. Ogawa and H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal., 70 (2009), 3696-3701. doi: 10.1016/j.na.2008.07.025.
[23]	S. Pohozaev and L. Véron, Blow-up results for nonlinear hyperbolic inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 393-420.
[24]	A. B. Sultan, M. Jleli and B. Samet, A system of wave inequalities with inverse-square potentials in an exterior domain, Applicable Analysis., 103 (2024), 843-857. doi: 10.1080/00036811.2023.2210154.
[25]	Y. Sun, Nonexistence results for systems of elliptic and parabolic differential inequalities in exterior domains of $\mathbb{R}^n$, Pacific J. Math., 293 (2018), 245-256. doi: 10.2140/pjm.2018.293.245.
[26]	Q. S. Zhang, A general blow-up result on nonlinear boundary-value problems on exterior domains, Proc. R. Soc. Edinb., 131 (2001), 451-475. doi: 10.1017/S0308210500000950.