# American Institute of Mathematical Sciences

• Previous Article
Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains
• DCDS Home
• This Issue
• Next Article
Instability of the soliton for the focusing, mass-critical generalized KdV equation
doi: 10.3934/dcds.2021173
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Fujita type results for quasilinear parabolic inequalities with nonlocal terms

 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 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.
Citation: Roberta Filippucci, Marius Ghergu. Fujita type results for quasilinear parabolic inequalities with nonlocal terms. Discrete & Continuous Dynamical Systems, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [13] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan. Acad., 49 (1973), 503-505.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [20] É. Mitidieri and S. I. Pokhozhaev, Fujita type theorems for quasilinear parabolic inequalities with nonlinear gradient, Doklady Mathematics, 66 (2002), 187-191.   Google Scholar [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.  Google Scholar [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.  Google Scholar [23] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar [24] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, 2007.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [13] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan. Acad., 49 (1973), 503-505.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [20] É. Mitidieri and S. I. Pokhozhaev, Fujita type theorems for quasilinear parabolic inequalities with nonlinear gradient, Doklady Mathematics, 66 (2002), 187-191.   Google Scholar [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.  Google Scholar [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.  Google Scholar [23] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar [24] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, 2007.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451 [19] Abdelkader Boucherif. Nonlocal problems for parabolic inclusions. Conference Publications, 2009, 2009 (Special) : 82-91. doi: 10.3934/proc.2009.2009.82 [20] Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048

2020 Impact Factor: 1.392