Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

Previous Article
Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents
CPAA Home
This Issue
Next Article
Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

September 2019, 18(5): 2717-2733. doi: 10.3934/cpaa.2019121

Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

Xumin Wang ^,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

^* Corresponding author

Received October 2018 Revised January 2019 Published April 2019

Fund Project: The author was partly supported by grant from the NNSF of China (No.11371056)

We present the singular Hardy-Trudinger-Moser inequality and the existence of their extremal functions on the unit disc

$ B $

$ \mathbb{R}^2 $

. As our first main result, we show that for any

$ 0<t<2 $

and

$ u \in C_0^\infty({B}) $

satisfying

$ \int_{{B}}|\nabla u|^2 dx- \int_{{B}}\frac{u^2}{(1-|x|^2)^2}dx\leq1, $

there exists a constant

$ C_{0}>0 $

such that the following inequality holds

$ \int_{{B}}\frac{e^{4\pi(1-t/2)u^2}}{|x|^t} dx\leq C_{0}. $

Furthermore, by the method of blow-up analysis, we establish the existence of extremal functions in a suitable function space. Our results extend those in Wang and Ye [36] from the non-singular case

$ t = 0 $

to the singular case for

$ 0<t<2 $

Keywords: Singular Hardy-Trudinger-Moser inequality, rearrangement, blow-up analysis.

Mathematics Subject Classification: Primary: 35J50; Secondary: 46E30, 46E35.

Citation: Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121

References:

[1]	D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 128 (1988), 385-398. doi: 10.2307/1971445. Google Scholar
[2]	Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, Nolinear Differential Equations Application, 13 (2007), 585-603.Google Scholar
[3]	Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not., 13 (2010), 2394-2426. Google Scholar
[4]	L. Carleson and A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127. Google Scholar
[5]	L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in $R^4$, Adv. Nonlinear Stud., 18 (2018), 429-452. doi: 10.1515/ans-2018-2020. Google Scholar
[6]	M. Dong, N. Lam and G. Lu, Sharp weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities and their extremal functions, Nonlinear Anal., 173 (2018), 75-98. doi: 10.1016/j.na.2018.03.006. Google Scholar
[7]	M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger-Moser inequalities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 88, 26 pp. doi: 10.1007/s00526-016-1014-7. Google Scholar
[8]	Y. Dong and Q. Yang, An interpolation of Hardy inequality and Moser-Trudinger inequality on Riemannian manifolds with negative curvature, Acta. Mathematica Sinica., English Series, 32 (2016), 856-866. doi: 10.1007/s10114-016-5129-8. Google Scholar
[9]	M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514. Google Scholar
[10]	N. Lam, Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal., 16 (2017), 973-997. doi: 10.3934/cpaa.2017047. Google Scholar
[11]	N. Lam and G. Lu, Sharp constants and optimizers for a class of Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlinear Stud., 17 (2017), 457-480. doi: 10.1515/ans-2017-0012. Google Scholar
[12]	N. Lam and G. Lu, Sharp singular Trudinger-Moser-Adams type inequalities with exact growth, Geometric methods in PDE's, 43–80, Springer INdAM Ser., 13, Springer, Cham, 2015. Google Scholar
[13]	N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument, J. Differential Equations, 255 (2013), 298-325. doi: 10.1016/j.jde.2013.04.005. Google Scholar
[14]	N. Lam and G. Lu, Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math., 231 (2012), 3259-3287. doi: 10.1016/j.aim.2012.09.004. Google Scholar
[15]	J. Li, G. Lu and Q. Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math., 33 (2018), 350-385. doi: 10.1016/j.aim.2018.05.035. Google Scholar
[16]	J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations, (2018), 57-84. doi: 10.1007/s00526-018-1352-8. Google Scholar
[17]	Y. Li, Trudinger-Moser inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192. Google Scholar
[18]	Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A., 48 (2005), 618-648. doi: 10.1360/04ys0050. Google Scholar
[19]	Y. Li, Remarks on the extremal functions for the Moser-Trudinger inequality, Acta Math. Sin. (Engl. Ser.), 22 (2006), 545-550. doi: 10.1007/s10114-005-0568-7. Google Scholar
[20]	Y. Li and C. Ndiaye, Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds, J. Geom. Anal., 17 (2007), 669-699. doi: 10.1007/BF02937433. Google Scholar
[21]	Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^n$, Indiana Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137. Google Scholar
[22]	K. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3. Google Scholar
[23]	G. Lu and H. Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud., 13 (2013), 1035-1052. doi: 10.1515/ans-2013-0415. Google Scholar
[24]	G. Lu and H. Tang, Sharp Moser-Trudinger inequalities on hyperbolic spaces with exact growth condition, J. Geom. Anal., 26 (2016), 837-857. doi: 10.1007/s12220-015-9573-y. Google Scholar
[25]	G. Lu and Q. Yang, A sharp Trudinger-Moser inequality on any bounded and convex plannar domain, Calc. Var. Partial Differential Equations, 55 (2016). doi: 10.1007/s00526-016-1077-5. Google Scholar
[26]	G. Lu and Q. Yang, Sharp Hardy-Adams inequalities for bi-laplacian on hyperbolic space of dimension four, Advances in Mathematics, 319 (2017), 567-598. doi: 10.1016/j.aim.2017.08.014. Google Scholar
[27]	G. Lu and Q. Yang, Paneitz operators on hyperbolic spaces and higher order Hardy-Sobolev-Maz'ya inequalities on half spaces, Amer. J. Math., to appear.Google Scholar
[28]	G. Lu and Y. Yang, Adams' inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170. doi: 10.1016/j.aim.2008.10.011. Google Scholar
[29]	G. Lu and Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979. doi: 10.3934/dcds.2009.25.963. Google Scholar
[30]	G. Lu and M. Zhu, A sharp Trudinger-Moser type inequality involving $L ^n$ norm in the entire space $\mathbb{R}^n$.Google Scholar
[31]	G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commum. Contemp. Math., 12 (2010), 1055-1068. doi: 10.1142/S0219199710004111. Google Scholar
[32]	G. Mancini, K. Sandeep and K. Tintarev, Trudinger-Moser inequality in the hyperbolic spaces $\mathbb{H}^N$, Adv. Nonlinear Anal., 2 (2013), 309-324. doi: 10.1515/anona-2013-0001. Google Scholar
[33]	J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101. Google Scholar
[34]	S. I. Pohozaev, The Sobolev embedding in the case pl = n, Proceeding of the Technical Scientific Conference on Advances of Scientific Research, 1964–1965. Mathematics Section, Moskov. Energet. Inst., (1965), 158–170.Google Scholar
[35]	N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484. doi: 10.1512/iumj.1968.17.17028. Google Scholar
[36]	G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320. doi: 10.1016/j.aim.2011.12.001. Google Scholar
[37]	Q. Yang, D. Su and Y. Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Annali di Matematica Pura ed Applicata, 195 (2016), 459-471. doi: 10.1007/s10231-015-0472-4. Google Scholar
[38]	V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749. Google Scholar
[39]	C. Zhang and L. Chen, Concentration-compactness principle of singular Trudinger-Moser inequalities in $R^n$ and n-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585. doi: 10.1515/ans-2017-6041. Google Scholar

show all references

References:

[1]	D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 128 (1988), 385-398. doi: 10.2307/1971445. Google Scholar
[2]	Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, Nolinear Differential Equations Application, 13 (2007), 585-603.Google Scholar
[3]	Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not., 13 (2010), 2394-2426. Google Scholar
[4]	L. Carleson and A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127. Google Scholar
[5]	L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in $R^4$, Adv. Nonlinear Stud., 18 (2018), 429-452. doi: 10.1515/ans-2018-2020. Google Scholar
[6]	M. Dong, N. Lam and G. Lu, Sharp weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities and their extremal functions, Nonlinear Anal., 173 (2018), 75-98. doi: 10.1016/j.na.2018.03.006. Google Scholar
[7]	M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger-Moser inequalities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 88, 26 pp. doi: 10.1007/s00526-016-1014-7. Google Scholar
[8]	Y. Dong and Q. Yang, An interpolation of Hardy inequality and Moser-Trudinger inequality on Riemannian manifolds with negative curvature, Acta. Mathematica Sinica., English Series, 32 (2016), 856-866. doi: 10.1007/s10114-016-5129-8. Google Scholar
[9]	M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514. Google Scholar
[10]	N. Lam, Equivalence of sharp Trudinger-Moser-Adams inequalities, Commun. Pure Appl. Anal., 16 (2017), 973-997. doi: 10.3934/cpaa.2017047. Google Scholar
[11]	N. Lam and G. Lu, Sharp constants and optimizers for a class of Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlinear Stud., 17 (2017), 457-480. doi: 10.1515/ans-2017-0012. Google Scholar
[12]	N. Lam and G. Lu, Sharp singular Trudinger-Moser-Adams type inequalities with exact growth, Geometric methods in PDE's, 43–80, Springer INdAM Ser., 13, Springer, Cham, 2015. Google Scholar
[13]	N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument, J. Differential Equations, 255 (2013), 298-325. doi: 10.1016/j.jde.2013.04.005. Google Scholar
[14]	N. Lam and G. Lu, Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math., 231 (2012), 3259-3287. doi: 10.1016/j.aim.2012.09.004. Google Scholar
[15]	J. Li, G. Lu and Q. Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math., 33 (2018), 350-385. doi: 10.1016/j.aim.2018.05.035. Google Scholar
[16]	J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations, (2018), 57-84. doi: 10.1007/s00526-018-1352-8. Google Scholar
[17]	Y. Li, Trudinger-Moser inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192. Google Scholar
[18]	Y. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A., 48 (2005), 618-648. doi: 10.1360/04ys0050. Google Scholar
[19]	Y. Li, Remarks on the extremal functions for the Moser-Trudinger inequality, Acta Math. Sin. (Engl. Ser.), 22 (2006), 545-550. doi: 10.1007/s10114-005-0568-7. Google Scholar
[20]	Y. Li and C. Ndiaye, Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds, J. Geom. Anal., 17 (2007), 669-699. doi: 10.1007/BF02937433. Google Scholar
[21]	Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^n$, Indiana Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137. Google Scholar
[22]	K. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3. Google Scholar
[23]	G. Lu and H. Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud., 13 (2013), 1035-1052. doi: 10.1515/ans-2013-0415. Google Scholar
[24]	G. Lu and H. Tang, Sharp Moser-Trudinger inequalities on hyperbolic spaces with exact growth condition, J. Geom. Anal., 26 (2016), 837-857. doi: 10.1007/s12220-015-9573-y. Google Scholar
[25]	G. Lu and Q. Yang, A sharp Trudinger-Moser inequality on any bounded and convex plannar domain, Calc. Var. Partial Differential Equations, 55 (2016). doi: 10.1007/s00526-016-1077-5. Google Scholar
[26]	G. Lu and Q. Yang, Sharp Hardy-Adams inequalities for bi-laplacian on hyperbolic space of dimension four, Advances in Mathematics, 319 (2017), 567-598. doi: 10.1016/j.aim.2017.08.014. Google Scholar
[27]	G. Lu and Q. Yang, Paneitz operators on hyperbolic spaces and higher order Hardy-Sobolev-Maz'ya inequalities on half spaces, Amer. J. Math., to appear.Google Scholar
[28]	G. Lu and Y. Yang, Adams' inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170. doi: 10.1016/j.aim.2008.10.011. Google Scholar
[29]	G. Lu and Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979. doi: 10.3934/dcds.2009.25.963. Google Scholar
[30]	G. Lu and M. Zhu, A sharp Trudinger-Moser type inequality involving $L ^n$ norm in the entire space $\mathbb{R}^n$.Google Scholar
[31]	G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commum. Contemp. Math., 12 (2010), 1055-1068. doi: 10.1142/S0219199710004111. Google Scholar
[32]	G. Mancini, K. Sandeep and K. Tintarev, Trudinger-Moser inequality in the hyperbolic spaces $\mathbb{H}^N$, Adv. Nonlinear Anal., 2 (2013), 309-324. doi: 10.1515/anona-2013-0001. Google Scholar
[33]	J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101. Google Scholar
[34]	S. I. Pohozaev, The Sobolev embedding in the case pl = n, Proceeding of the Technical Scientific Conference on Advances of Scientific Research, 1964–1965. Mathematics Section, Moskov. Energet. Inst., (1965), 158–170.Google Scholar
[35]	N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484. doi: 10.1512/iumj.1968.17.17028. Google Scholar
[36]	G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320. doi: 10.1016/j.aim.2011.12.001. Google Scholar
[37]	Q. Yang, D. Su and Y. Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Annali di Matematica Pura ed Applicata, 195 (2016), 459-471. doi: 10.1007/s10231-015-0472-4. Google Scholar
[38]	V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749. Google Scholar
[39]	C. Zhang and L. Chen, Concentration-compactness principle of singular Trudinger-Moser inequalities in $R^n$ and n-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585. doi: 10.1515/ans-2017-6041. Google Scholar

[1]	Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
[2]	Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
[3]	Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212
[4]	Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505
[5]	Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
[6]	Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963
[7]	Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521
[8]	Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[9]	Philippe Souplet, Juan-Luis Vázquez. Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 221-234. doi: 10.3934/dcds.2006.14.221
[10]	Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697
[11]	C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88
[12]	Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443
[13]	Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[14]	Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155
[15]	Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[16]	W. Edward Olmstead, Colleen M. Kirk, Catherine A. Roberts. Blow-up in a subdiffusive medium with advection. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1655-1667. doi: 10.3934/dcds.2010.28.1655
[17]	Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691
[18]	Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[19]	Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
[20]	José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43

American Institute of Mathematical Sciences