doi: 10.3934/dcdss.2022115
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Nonexistence for time-fractional wave inequalities on Riemannian manifolds

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

* Corresponding author

Received  March 2022 Revised  April 2022 Early access May 2022

We establish necessary conditions for the existence of global weak solutions to a class of semilinear time-fractional wave inequalities with nonlinearity of derivative type, defined on complete noncompact Riemannian manifolds. A potential function depending of both time and space, is allowed in front of the power nonlinearity. The obtained conditions depend on the parameters of the problem, the initial conditions and the geometry of the manifold. Our results are new even in the Euclidean case.

Citation: Mohamed Jleli, Bessem Samet. Nonexistence for time-fractional wave inequalities on Riemannian manifolds. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022115
References:
[1]

K. Adolfsson and M. Enelund, Fractional derivative viscoelasticity at large deformations, Nonlinear Dyn., 33 (2003), 301-321.  doi: 10.1023/A:1026003130033.

[2]

L. J. Alias, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer, 2016. doi: 10.1007/978-3-319-24337-5.

[3]

D. Bianchi and A. G. Setti, Laplacian cut-offs, porous and fast diffusion on manifolds and other applications, Calc. Var. Part. Diff. Eq., 57 (2018), Paper No. 4, 33 pp. doi: 10.1007/s00526-017-1267-9.

[4]

S. Buonocore and F. Semperlotti, Tomographic imaging of non-local media based on space-fractional diffusion models, J. Appl. Phys., 123 (2018), Article 214902. doi: 10.1063/1.5026789.

[5]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific, 2020. doi: 10.1142/10550.

[6]

M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Anal., 149 (2017), 1-40.  doi: 10.1016/j.na.2016.10.010.

[7]

T. A. Dao and A. Z. Fino, Critical exponent for semi-linear structurally damped wave equation of derivative type, Math. Methods Appl. Sci., 43 (2020), 9766-9775.  doi: 10.1002/mma.6649.

[8]

T. A. Dao and M. Reissig, $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping, Discrete Contin. Dyn. Syst., 39 (2019), 5431-5463.  doi: 10.3934/dcds.2019222.

[9]

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 Sec. 1., 13 (1966), 109-124. 

[10]

A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36 (1999), 135-249.  doi: 10.1090/S0273-0979-99-00776-4.

[11]

A. Grigor'yan and V. A. Kondratiev, On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds, In Around the Research of Vladimir Maz'ya. II, volume 12 of Int. Math. Ser. (N. Y.), pages 203–218, Springer, New York, 2010. doi: 10.1007/978-1-4419-1343-2_8.

[12]

A. Grigor'yan and Y. Sun, On non-negative solutions of the inequality $\Delta u+ u^{\sigma}\leq 0$ on Riemannian manifolds, Comm. Pure Appl. Math., 67 (2014), 1336-1352.  doi: 10.1002/cpa.21493.

[13]

A. Grigor'yan, Y. Sun and I. Verbitsky, Superlinear elliptic inequalities on manifolds, J. Funct. Anal., 278 (2020), 108444, 34 pp. doi: 10.1016/j.jfa.2019.108444.

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[15]

M. Kirane and A. Abdeljabbar, Nonexistence of global solutions of systems of time fractional differential equations posed on the Heisenberg group, Math. Methods Appl. Sci., (2022).  doi: 10.1002/mma.8243.

[16]

M. Kirane and Y. Laskri, Nonexistence of global solutions to a hyperbolic equation with a space-time fractional damping, Appl. Math. Comput., 167 (2005), 1304-1310.  doi: 10.1016/j.amc.2004.08.038.

[17]

M. KiraneY. Laskri and N.-E. Tatar, Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives, J. Math. Anal. Appl., 312 (2005), 488-501.  doi: 10.1016/j.jmaa.2005.03.054.

[18]

V. V. Kulish and J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng., 124 (2002), 803-806.  doi: 10.1115/1.1478062.

[19]

F. Mainardi, Fractional calculus in wave propagation problems, Forum der Berliner Mathematischer Gesellschaft., 19 (2011), 20-52. 

[20]

È. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems in $\mathbb{R}^N$, Tr. Mat. Inst. Steklova., 227 (1999), 192-222. 

[21]

D. D. MonticelliF. Punzo and M. Squassina, Nonexistence for hyperbolic problems on Riemannian manifolds, Asymptot. Anal., 120 (2020), 87-101.  doi: 10.3233/ASY-191580.

[22]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, New York, NY: Springer New York, 2006.

[23]

B. Samet, Blow-up phenomena for a nonlinear time fractional heat equation in an exterior domain, Comput. Math. Appl., 78 (2019), 1380-1385.  doi: 10.1016/j.camwa.2018.10.003.

[24]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach: Yverdon, Switzerland, 1993.

[25]

N.-E. Tatar, Nonexistence results for a fractional problem arising in thermal diffusion in fractal media, Chaos Solitons Fractals., 36 (2008), 1205-1214.  doi: 10.1016/j.chaos.2006.08.001.

[26]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations., 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.

[27]

Q. ZhangH.-R. Sun and Y. Li, The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance, Appl. Math. Lett., 64 (2017), 119-124.  doi: 10.1016/j.aml.2016.08.017.

[28]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.

show all references

References:
[1]

K. Adolfsson and M. Enelund, Fractional derivative viscoelasticity at large deformations, Nonlinear Dyn., 33 (2003), 301-321.  doi: 10.1023/A:1026003130033.

[2]

L. J. Alias, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer, 2016. doi: 10.1007/978-3-319-24337-5.

[3]

D. Bianchi and A. G. Setti, Laplacian cut-offs, porous and fast diffusion on manifolds and other applications, Calc. Var. Part. Diff. Eq., 57 (2018), Paper No. 4, 33 pp. doi: 10.1007/s00526-017-1267-9.

[4]

S. Buonocore and F. Semperlotti, Tomographic imaging of non-local media based on space-fractional diffusion models, J. Appl. Phys., 123 (2018), Article 214902. doi: 10.1063/1.5026789.

[5]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific, 2020. doi: 10.1142/10550.

[6]

M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Anal., 149 (2017), 1-40.  doi: 10.1016/j.na.2016.10.010.

[7]

T. A. Dao and A. Z. Fino, Critical exponent for semi-linear structurally damped wave equation of derivative type, Math. Methods Appl. Sci., 43 (2020), 9766-9775.  doi: 10.1002/mma.6649.

[8]

T. A. Dao and M. Reissig, $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping, Discrete Contin. Dyn. Syst., 39 (2019), 5431-5463.  doi: 10.3934/dcds.2019222.

[9]

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 Sec. 1., 13 (1966), 109-124. 

[10]

A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36 (1999), 135-249.  doi: 10.1090/S0273-0979-99-00776-4.

[11]

A. Grigor'yan and V. A. Kondratiev, On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds, In Around the Research of Vladimir Maz'ya. II, volume 12 of Int. Math. Ser. (N. Y.), pages 203–218, Springer, New York, 2010. doi: 10.1007/978-1-4419-1343-2_8.

[12]

A. Grigor'yan and Y. Sun, On non-negative solutions of the inequality $\Delta u+ u^{\sigma}\leq 0$ on Riemannian manifolds, Comm. Pure Appl. Math., 67 (2014), 1336-1352.  doi: 10.1002/cpa.21493.

[13]

A. Grigor'yan, Y. Sun and I. Verbitsky, Superlinear elliptic inequalities on manifolds, J. Funct. Anal., 278 (2020), 108444, 34 pp. doi: 10.1016/j.jfa.2019.108444.

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[15]

M. Kirane and A. Abdeljabbar, Nonexistence of global solutions of systems of time fractional differential equations posed on the Heisenberg group, Math. Methods Appl. Sci., (2022).  doi: 10.1002/mma.8243.

[16]

M. Kirane and Y. Laskri, Nonexistence of global solutions to a hyperbolic equation with a space-time fractional damping, Appl. Math. Comput., 167 (2005), 1304-1310.  doi: 10.1016/j.amc.2004.08.038.

[17]

M. KiraneY. Laskri and N.-E. Tatar, Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives, J. Math. Anal. Appl., 312 (2005), 488-501.  doi: 10.1016/j.jmaa.2005.03.054.

[18]

V. V. Kulish and J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng., 124 (2002), 803-806.  doi: 10.1115/1.1478062.

[19]

F. Mainardi, Fractional calculus in wave propagation problems, Forum der Berliner Mathematischer Gesellschaft., 19 (2011), 20-52. 

[20]

È. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems in $\mathbb{R}^N$, Tr. Mat. Inst. Steklova., 227 (1999), 192-222. 

[21]

D. D. MonticelliF. Punzo and M. Squassina, Nonexistence for hyperbolic problems on Riemannian manifolds, Asymptot. Anal., 120 (2020), 87-101.  doi: 10.3233/ASY-191580.

[22]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, New York, NY: Springer New York, 2006.

[23]

B. Samet, Blow-up phenomena for a nonlinear time fractional heat equation in an exterior domain, Comput. Math. Appl., 78 (2019), 1380-1385.  doi: 10.1016/j.camwa.2018.10.003.

[24]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach: Yverdon, Switzerland, 1993.

[25]

N.-E. Tatar, Nonexistence results for a fractional problem arising in thermal diffusion in fractal media, Chaos Solitons Fractals., 36 (2008), 1205-1214.  doi: 10.1016/j.chaos.2006.08.001.

[26]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations., 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.

[27]

Q. ZhangH.-R. Sun and Y. Li, The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance, Appl. Math. Lett., 64 (2017), 119-124.  doi: 10.1016/j.aml.2016.08.017.

[28]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.

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