Fujita type results for quasilinear parabolic inequalities with nonlocal terms

Previous Article
On $ L^1 $ estimates of solutions of compressible viscoelastic system
DCDS Home
This Issue
Next Article
Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $

April 2022, 42(4): 1817-1833. doi: 10.3934/dcds.2021173

Fujita type results for quasilinear parabolic inequalities with nonlocal terms

Roberta Filippucci ^1, and Marius Ghergu ^2,,

1.	Dipartimento di Matematica e Informatica, Universitá degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2.	School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland, and, Institute of Mathematics Simion Stoilow of the Romanian Academy, 21 Calea Grivitei St., 010702 Bucharest, Romania

^* Corresponding author

Received June 2021 Published April 2022 Early access November 2021

In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form

$ \begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $

where

$ u_0\in L^1_{loc}({\mathbb R}^N) $

,

$ L_{\mathcal{A}} $

denotes a weakly

$ m $

-coercive operator, which includes as prototype the

$ m $

-Laplacian or the generalized mean curvature operator,

$ p,\,q>0 $

, while

$ K\ast u^p $

stands for the standard convolution operator between a weight

$ K>0 $

satisfying suitable conditions at infinity and

$ u^p $

. For problem

$ (P^-) $

we obtain a Fujita type exponent while for

$ (P^+) $

we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.

Keywords: Quasilinear parabolic inequalities, nonlocal terms, Fujita exponent.

Mathematics Subject Classification: Primary: 35K59, 35A23; Secondary: 35B33, 35B53.

Citation: Roberta Filippucci, Marius Ghergu. Fujita type results for quasilinear parabolic inequalities with nonlocal terms. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1817-1833. doi: 10.3934/dcds.2021173

References:

[1]	A. T. Duong and Q. H. Phan, Optimal Liouville-type theorems for a system of parabolic inequalities, Commun. Contemp. Math., 22 (2020), 1950043, 22 pp. doi: 10.1142/S0219199719500433.
[2]	R. Filippucci and M. Ghergu, Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms, Nonlinear Anal., 197 (2020), 111857, 22 pp. doi: 10.1016/j.na.2020.111857.
[3]	R. Filippucci and S. Lombardi, Fujita type results for parabolic inequalities with gradient terms, J. Differential Equations, 268 (2020), 1873-1910. doi: 10.1016/j.jde.2019.09.026.
[4]	H. Fujita, On the blowing up of solutions to the Cauchy problem for $u_t = \Delta u + u^{1 + \alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.
[5]	V. A. Galaktionov and H. A. Levine, A general approach to Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005-1027. doi: 10.1016/S0362-546X(97)00716-5.
[6]	M. Ghergu, P. Karageorgis and G. Singh, Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms, J. Differential Equations, 268 (2020), 6033-6066. doi: 10.1016/j.jde.2019.11.013.
[7]	M. Ghergu, P. Karageorgis and G. Singh, Quasilinear elliptic inequalities with Hardy potential and nonlocal terms, Proc. Royal Soc. Edinburgh Sect. A, 151 (2021), 1075-1093. doi: 10.1017/prm.2020.50.
[8]	M. Ghergu, Y. Miyamoto and V. Moroz, Polyharmonic inequalities with nonlocal terms, J. Differential Equations, 296 (2021), 799-821. doi: 10.1016/j.jde.2021.06.019.
[9]	M. Ghergu and S. D. Taliaferro, Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217. doi: 10.1016/j.jde.2016.03.004.
[10]	D. R. Hartree, The wave mechanics of an atom with a non-Coulomb central field, Part Ⅰ. Theory and methods, Math. Proc. Cambridge Philosophical Soc., 24 (1928), 89-110. doi: 10.1017/S0305004100011919.
[11]	D. R. Hartree, The wave mechanics of an atom with a non-Coulomb central field, Part Ⅱ. Some results and discussion, Math. Proc. Cambridge Philosophical Soc., 24 (1928), 111-132.
[12]	D. R. Hartree, The wave mechanics of an atom with a non-Coulomb central field, Part Ⅲ. Term values and intensities in series in optical spectra, Math. Proc. Cambridge Philosophical Soc., 24 (1928), 426-437. doi: 10.1017/S0305004100015954.
[13]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan. Acad., 49 (1973), 503-505.
[14]	A. G. Kartsatos and V. V. Kurta, On the critical Fujita exponents for solutions of first-order nonlinear evolution inequalities, J. Math. Anal. Appl., 269 (2002), 73-86. doi: 10.1016/S0022-247X(02)00005-7.
[15]	K. Kobayashi, T. Sirao and H. Tanaka, On the blowing up problem for semi-linear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.
[16]	H. A. Levine, G. M. Lieberman and P. Meier, On critical exponents for some quasilinear parabolic equations, Math. Methods Appl. Sci., (1990), 429-438. doi: 10.1002/mma.1670120507.
[17]	V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes, J. Differential Equations, 247 (2009), 2740-2777. doi: 10.1016/j.jde.2009.08.018.
[18]	V. Liskevich, I. I. Skrypnik and Z. Sobol, Gradient estimates for degenerate quasi-linear parabolic equations, J. London Math. Soc., 84 (2011), 446-474. doi: 10.1112/jlms/jdr020.
[19]	É. Mitidieri and S. I. Pokhozhaev, Apriori Estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
[20]	É. Mitidieri and S. I. Pokhozhaev, Fujita type theorems for quasilinear parabolic inequalities with nonlinear gradient, Doklady Mathematics, 66 (2002), 187-191.
[21]	E. Mitidieri and S. I. Pokhozhaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $\mathbb R^n$, J. Evol. Equ., 1 (2001), 189-220. doi: 10.1007/PL00001368.
[22]	V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813. doi: 10.1007/s11784-016-0373-1.
[23]	S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
[24]	P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, 2007.
[25]	A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-Up in Problems for Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.535.
[26]	C. Zhang, Estimates for parabolic equations with measure data in generalized Morrey spaces, Commun. Contemp. Math., 21 (2019), 1850044, 21 pp. doi: 10.1142/S021919971850044X.
[27]	Y. Zheng and Z. Bo Fang, New critical exponents, large time behavior, and life span for a fast diffusive $p$-Laplacian equation with nonlocal source, Z. Angew. Math. Phys., 70 (2019), Paper No. 144, 17 pp. doi: 10.1007/s00033-019-1191-2.
[28]	J. Zhou, Fujita exponent for an inhomogeneous pseudo parabolic equation, Rocky Mountain J. Math., 50 (2020), 1125-1137. doi: 10.1216/rmj.2020.50.1125.

show all references

References:

[1]	A. T. Duong and Q. H. Phan, Optimal Liouville-type theorems for a system of parabolic inequalities, Commun. Contemp. Math., 22 (2020), 1950043, 22 pp. doi: 10.1142/S0219199719500433.
[2]	R. Filippucci and M. Ghergu, Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms, Nonlinear Anal., 197 (2020), 111857, 22 pp. doi: 10.1016/j.na.2020.111857.
[3]	R. Filippucci and S. Lombardi, Fujita type results for parabolic inequalities with gradient terms, J. Differential Equations, 268 (2020), 1873-1910. doi: 10.1016/j.jde.2019.09.026.
[4]	H. Fujita, On the blowing up of solutions to the Cauchy problem for $u_t = \Delta u + u^{1 + \alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.
[5]	V. A. Galaktionov and H. A. Levine, A general approach to Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005-1027. doi: 10.1016/S0362-546X(97)00716-5.
[6]	M. Ghergu, P. Karageorgis and G. Singh, Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms, J. Differential Equations, 268 (2020), 6033-6066. doi: 10.1016/j.jde.2019.11.013.
[7]	M. Ghergu, P. Karageorgis and G. Singh, Quasilinear elliptic inequalities with Hardy potential and nonlocal terms, Proc. Royal Soc. Edinburgh Sect. A, 151 (2021), 1075-1093. doi: 10.1017/prm.2020.50.
[8]	M. Ghergu, Y. Miyamoto and V. Moroz, Polyharmonic inequalities with nonlocal terms, J. Differential Equations, 296 (2021), 799-821. doi: 10.1016/j.jde.2021.06.019.
[9]	M. Ghergu and S. D. Taliaferro, Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217. doi: 10.1016/j.jde.2016.03.004.
[10]	D. R. Hartree, The wave mechanics of an atom with a non-Coulomb central field, Part Ⅰ. Theory and methods, Math. Proc. Cambridge Philosophical Soc., 24 (1928), 89-110. doi: 10.1017/S0305004100011919.
[11]	D. R. Hartree, The wave mechanics of an atom with a non-Coulomb central field, Part Ⅱ. Some results and discussion, Math. Proc. Cambridge Philosophical Soc., 24 (1928), 111-132.
[12]	D. R. Hartree, The wave mechanics of an atom with a non-Coulomb central field, Part Ⅲ. Term values and intensities in series in optical spectra, Math. Proc. Cambridge Philosophical Soc., 24 (1928), 426-437. doi: 10.1017/S0305004100015954.
[13]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan. Acad., 49 (1973), 503-505.
[14]	A. G. Kartsatos and V. V. Kurta, On the critical Fujita exponents for solutions of first-order nonlinear evolution inequalities, J. Math. Anal. Appl., 269 (2002), 73-86. doi: 10.1016/S0022-247X(02)00005-7.
[15]	K. Kobayashi, T. Sirao and H. Tanaka, On the blowing up problem for semi-linear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.
[16]	H. A. Levine, G. M. Lieberman and P. Meier, On critical exponents for some quasilinear parabolic equations, Math. Methods Appl. Sci., (1990), 429-438. doi: 10.1002/mma.1670120507.
[17]	V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes, J. Differential Equations, 247 (2009), 2740-2777. doi: 10.1016/j.jde.2009.08.018.
[18]	V. Liskevich, I. I. Skrypnik and Z. Sobol, Gradient estimates for degenerate quasi-linear parabolic equations, J. London Math. Soc., 84 (2011), 446-474. doi: 10.1112/jlms/jdr020.
[19]	É. Mitidieri and S. I. Pokhozhaev, Apriori Estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
[20]	É. Mitidieri and S. I. Pokhozhaev, Fujita type theorems for quasilinear parabolic inequalities with nonlinear gradient, Doklady Mathematics, 66 (2002), 187-191.
[21]	E. Mitidieri and S. I. Pokhozhaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $\mathbb R^n$, J. Evol. Equ., 1 (2001), 189-220. doi: 10.1007/PL00001368.
[22]	V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813. doi: 10.1007/s11784-016-0373-1.
[23]	S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
[24]	P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, 2007.
[25]	A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-Up in Problems for Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.535.
[26]	C. Zhang, Estimates for parabolic equations with measure data in generalized Morrey spaces, Commun. Contemp. Math., 21 (2019), 1850044, 21 pp. doi: 10.1142/S021919971850044X.
[27]	Y. Zheng and Z. Bo Fang, New critical exponents, large time behavior, and life span for a fast diffusive $p$-Laplacian equation with nonlocal source, Z. Angew. Math. Phys., 70 (2019), Paper No. 144, 17 pp. doi: 10.1007/s00033-019-1191-2.
[28]	J. Zhou, Fujita exponent for an inhomogeneous pseudo parabolic equation, Rocky Mountain J. Math., 50 (2020), 1125-1137. doi: 10.1216/rmj.2020.50.1125.

[1]	Lingwei Ma, Zhong Bo Fang. A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081
[2]	Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226
[3]	Hui-Ling Li, Heng-Ling Wang, Xiao-Liu Wang. A quasilinear parabolic problem with a source term and a nonlocal absorption. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1945-1956. doi: 10.3934/cpaa.2018092
[4]	Gary Lieberman. Nonlocal problems for quasilinear parabolic equations in divergence form. Conference Publications, 2003, 2003 (Special) : 563-570. doi: 10.3934/proc.2003.2003.563
[5]	Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure and Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587
[6]	Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105
[7]	Genni Fragnelli, Paolo Nistri, Duccio Papini. Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3831-3834. doi: 10.3934/dcds.2013.33.3831
[8]	Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174
[9]	Rui Huang, Yifu Wang, Yuanyuan Ke. Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1005-1014. doi: 10.3934/dcdsb.2005.5.1005
[10]	Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541
[11]	Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35
[12]	Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[13]	Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[14]	Angelo Alvino, Roberta Volpicelli, Bruno Volzone. A remark on Hardy type inequalities with remainder terms. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 801-807. doi: 10.3934/dcdss.2011.4.801
[15]	Roberta Filippucci, Chiara Lini. Existence of solutions for quasilinear Dirichlet problems with gradient terms. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 267-286. doi: 10.3934/dcdss.2019019
[16]	Leszek Gasiński. Existence results for quasilinear hemivariational inequalities at resonance. Conference Publications, 2007, 2007 (Special) : 409-418. doi: 10.3934/proc.2007.2007.409
[17]	Siegfried Carl. Comparison results for a class of quasilinear evolutionary hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 221-229. doi: 10.3934/proc.2007.2007.221
[18]	Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451
[19]	Jean Dolbeault, An Zhang. Parabolic methods for ultraspherical interpolation inequalities. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022080
[20]	Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Roberta Filippucci Marius Ghergu
on Google Scholar
Roberta Filippucci Marius Ghergu

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Roberta Filippucci Marius Ghergu
on Google Scholar
Roberta Filippucci Marius Ghergu

[Back to Top]