×

On the critical behavior for a Sobolev-type inequality with Hardy potential. (English. French summary) Zbl 1535.35223

Summary: We investigate the existence and nonexistence of weak solutions to the Sobolev-type inequality \(-\partial_t(\Delta u)-\Delta u+\frac{\sigma}{|x|^2}u\geq |x|^{\mu}|u|^p\) in \((0,\infty)\times B\), under the inhomogeneous Dirichlet-type boundary condition \(u(t,x)=f(x)\) on \((0,\infty)\times\partial B\), where \(B\) is the unit open ball of \(\mathbb{R}^N\), \(N\geq 2\), \(\sigma>-\bigl(\frac{N-2}{2}\bigr)^2\), \(\mu\in\mathbb{R}\) and \(p>1\). In particular, when \(\sigma\neq 0\), we show that the dividing line with respect to existence and nonexistence is given by a critical exponent that depends on \(N\), \(\sigma\) and \(\mu\).

MSC:

35R45 Partial differential inequalities and systems of partial differential inequalities
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B33 Critical exponents in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.

References:

[1] Abdellaoui, Boumediene; Miri, Sofiane E. H.; Peral, Ireneo; Touaoula, Tarik M., Some remarks on quasilinear parabolic problems with singular potential and a reaction term, NoDEA, Nonlinear Differ. Equ. Appl., 21, 4, 453-490, 2014 · Zbl 1297.35118 · doi:10.1007/s00030-013-0253-y
[2] Abdellaoui, Boumediene; Peral, Ireneo, Some results for semilinear elliptic equations with critical potential, Proc. R. Soc. Edinb., Sect. A, Math., 132, 1, 1-24, 2002 · Zbl 1014.35023 · doi:10.1017/S0308210500001505
[3] Abdellaoui, Boumediene; Peral, Ireneo; Primo, Ana, Influence of the Hardy potential in a semi-linear heat equation, Proc. R. Soc. Edinb., Sect. A, Math., 139, 5, 897-926, 2009 · Zbl 1191.35157 · doi:10.1017/S0308210508000152
[4] Abdellaoui, Boumediene; Peral, Ireneo; Primo, Ana, Strong regularizing effect of a gradient term in the heat equation with the Hardy potential, J. Funct. Anal., 258, 4, 1247-1272, 2010 · Zbl 1191.35162 · doi:10.1016/j.jfa.2009.11.008
[5] Alsaedi, Ahmed; Alhothuali, Mohammed S.; Ahmad, Bashir; Kerbal, Sebti; Kirane, Mokhtar, Nonlinear fractional differential equations of Sobolev type, Math. Methods Appl. Sci., 37, 13, 2009-2016, 2014 · Zbl 1302.35396 · doi:10.1002/mma.2954
[6] Al’shin, Alexander B.; Korpusov, Maksim O.; Sveshnikov, Alekseĭ G., Blow-up in nonlinear Sobolev type equations, 15, xii+648 p. pp., 2011, Walter de Gruyter · Zbl 1259.35002
[7] Aristov, Anatoliĭ I., Large-time asymptotics of the solution of the Cauchy problem for a Sobolev type equation with a cubic nonlinearity, Differ. Uravn., 46, 9, 1354-1358, 2010 · Zbl 1213.35093
[8] Aristov, Anatoliĭ I., On the Cauchy problem for a Sobolev type equation with a quadratic nonlinearity, Izv. Ross. Akad. Nauk, Ser. Mat., 75, 5, 3-18, 2011 · Zbl 1230.35017
[9] Aristov, Anatoliĭ I., On the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient, Math. Notes, 91, 5, 603-612, 2012 · Zbl 1284.35241 · doi:10.1134/S000143461205001X
[10] Barenblatt, Grigory I.; Zheltov, Yu. P.; Kochina, I. N., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, PMM, J. Appl. Math. Mech., 24, 1286-1303, 1960 · Zbl 0104.21702 · doi:10.1016/0021-8928(60)90107-6
[11] Beshtokov, Murat Kh., Numerical analysis of initial-boundary value problem for a Sobolev-type equation with a fractional-order time derivative, Comput. Math. Math. Phys., 59, 2, 175-192, 2019 · Zbl 1447.35347 · doi:10.1134/S0965542519020052
[12] Brill, Heinz, A semilinear Sobolev evolution equation in a Banach space, J. Differ. Equations, 24, 412-425, 1977 · Zbl 0346.34046 · doi:10.1016/0022-0396(77)90009-2
[13] Cao, Yang; Nie, Yuanyuan, Blow-up of solutions of the nonlinear Sobolev equation, Appl. Math. Lett., 28, 1-6, 2014 · Zbl 1311.35063
[14] Colton, David; Wimp, Jet, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69, 411-418, 1979 · Zbl 0409.35050 · doi:10.1016/0022-247X(79)90152-5
[15] Dzektser, E. S., A generalization of the equations of motion of subterranean water with free surface, Dokl. Akad. Nauk SSSR, 202, 1031-1033, 1972
[16] El Hamidi, Abdallah; Laptev, Gennady G., Existence and nonexistence results for higher-order semilinear evolution inequalities with critical potential, J. Math. Anal. Appl., 304, 2, 451-463, 2005 · Zbl 1067.35159 · doi:10.1016/j.jmaa.2004.09.019
[17] Fedorov, Vladimir E.; Urazaeva, A. V., An inverse problem for linear Sobolev type equations, J. Inverse Ill-Posed Probl., 12, 4, 387-395, 2004 · Zbl 1081.35138 · doi:10.1515/1569394042248210
[18] Guezane-Lakoud, Assia; Belakroum, D., Time-discretization schema for an integrodifferential Sobolev type equation with integral conditions, Appl. Math. Comput., 218, 9, 4695-4702, 2012 · Zbl 1246.65251
[19] Hoff, Nicholas J., Creep buckling, Aeron. Quart., 7, 1, 1-20, 1956 · doi:10.1017/S0001925900010106
[20] Jleli, Mohamed; Samet, Bessem, Instantaneous blow-up for a fractional in time equation of Sobolev type, Math. Methods Appl. Sci., 43, 8, 5645-5652, 2020 · Zbl 1445.35085 · doi:10.1002/mma.6290
[21] Jleli, Mohamed; Samet, Bessem, Instantaneous blow-up for nonlinear Sobolev type equations with potentials on Riemannian manifolds, Commun. Pure Appl. Anal., 21, 6, 2065-2078, 2022 · Zbl 1487.35120 · doi:10.3934/cpaa.2022036
[22] Jleli, Mohamed; Samet, Bessem; Vetro, Calogero, On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain, Adv. Nonlinear Anal., 10, 1267-1283, 2021 · Zbl 1470.35443 · doi:10.1515/anona-2020-0181
[23] Korpusov, Maksim O.; Lukyanenko, Dmitrii V.; Panin, Aleksandr A.; Yushkov, Egor V., Blow-up for one Sobolev problem: theoretical approach and numerical analysis, J. Math. Anal. Appl., 442, 2, 451-468, 2016 · Zbl 1339.35061 · doi:10.1016/j.jmaa.2016.04.069
[24] Korpusov, Maksim O.; Sveshnikov, Alekseĭ G., Blowup of solutions to initial value problems for nonlinear operator-differential equations, Dokl. Math., 71, 2, 168-171, 2005 · Zbl 1272.34080
[25] Korpusov, Maksim O.; Sveshnikov, Alekseĭ G., Application of the nonlinear capacity method to differential inequalities of Sobolev type, Differ. Equ., 45, 7, 951-959, 2009 · Zbl 1181.35350 · doi:10.1134/S0012266109070027
[26] Merchán, Susana; Montoro, Luigi; Peral, Ireneo; Sciunzi, Berardino, Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy-Leray potential, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 31, 1, 1-22, 2014 · Zbl 1291.35082
[27] Mukhartova, Yu. V.; Panin, Aleksandr A., Blow-up of the solution of an inhomogeneous system of Sobolev-type equations, Math. Notes, 91, 2, 217-230, 2012 · Zbl 1381.35019 · doi:10.1134/S0001434612010233
[28] Sviridyuk, Georgiĭ A.; Fedorov, Vladimir E., Linear Sobolev Type Equations and Degenerate Semigroups of Operators, viii+216 p. pp., 2003, VSP · Zbl 1102.47061 · doi:10.1515/9783110915501
[29] Urazaeva, A. V., A mapping of a point spectrum and the uniqueness of a solution to the inverse problem for a Sobolev-type equation, Russ. Math., 54, 5, 47-55, 2010 · Zbl 1210.47066 · doi:10.3103/S1066369X10050075
[30] Zamyshlyaeva, Alena A.; Lut, Aleksandr, Inverse problem for the Sobolev type equation of higher order, Mathematics, 9, 14, 2021 · Zbl 1475.35419 · doi:10.3390/math9141647
[31] Zamyshlyaeva, Alena A.; Surovtsev, S. V., Numerical investigation of one Sobolev type mathematical model, J. Comput. Eng. Math., 2, 3, 72-80, 2015 · Zbl 1359.65166 · doi:10.14529/jcem150308
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.