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

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 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.
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. GherguP. 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. GherguP. 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. GherguY. 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. KobayashiT. 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. LevineG. 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. LiskevichI. 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. GherguP. 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. GherguP. 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. GherguY. 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. KobayashiT. 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. LevineG. 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. LiskevichI. 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.

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