On the critical behavior for a Sobolev-type inequality with Hardy potential

Jleli, Mohamed; Samet, Bessem

Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 87-97

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

Classification : 35R45, 35A01, 35B33
Keywords: Sobolev-type inequality, Hardy potential, bounded domain, existence, nonexistence, critical exponent

[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., Volume 21 (2014) no. 4, pp. 453-490 | Zbl | DOI | MR

[2] Abdellaoui, Boumediene; Peral, Ireneo Some results for semilinear elliptic equations with critical potential, Proc. R. Soc. Edinb., Sect. A, Math., Volume 132 (2002) no. 1, pp. 1-24 | DOI | MR

[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., Volume 139 (2009) no. 5, pp. 897-926 | DOI | Zbl

[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., Volume 258 (2010) no. 4, pp. 1247-1272 | DOI | MR | Zbl

[5] Alsaedi, Ahmed; Alhothuali, Mohammed S.; Ahmad, Bashir; Kerbal, Sebti; Kirane, Mokhtar Nonlinear fractional differential equations of Sobolev type, Math. Methods Appl. Sci., Volume 37 (2014) no. 13, pp. 2009-2016 | DOI | MR | Zbl

[6] Al’shin, Alexander B.; Korpusov, Maksim O.; Sveshnikov, Alekseĭ G. Blow-up in nonlinear Sobolev type equations, de Gruyter Series in Nonlinear Analysis and Applications, 15, Walter de Gruyter, 2011, xii+648 pages

[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., Volume 46 (2010) no. 9, pp. 1354-1358 | MR | Zbl

[8] Aristov, Anatoliĭ I. On the Cauchy problem for a Sobolev type equation with a quadratic nonlinearity, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 75 (2011) no. 5, pp. 3-18 | MR

[9] Aristov, Anatoliĭ I. On the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient, Math. Notes, Volume 91 (2012) no. 5, pp. 603-612 | DOI | MR | Zbl

[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., Volume 24 (1960), pp. 1286-1303 | DOI | Zbl

[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., Volume 59 (2019) no. 2, pp. 175-192 | DOI | MR | Zbl

[12] Brill, Heinz A semilinear Sobolev evolution equation in a Banach space, J. Differ. Equations, Volume 24 (1977), pp. 412-425 | DOI | MR | Zbl

[13] Cao, Yang; Nie, Yuanyuan Blow-up of solutions of the nonlinear Sobolev equation, Appl. Math. Lett., Volume 28 (2014), pp. 1-6 | MR | Zbl

[14] Colton, David; Wimp, Jet Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., Volume 69 (1979), pp. 411-418 | DOI | MR | Zbl

[15] Dzektser, E. S. A generalization of the equations of motion of subterranean water with free surface, Dokl. Akad. Nauk SSSR, Volume 202 (1972), pp. 1031-1033

[16] El Hamidi, Abdallah; Laptev, Gennady G. Existence and nonexistence results for higher-order semilinear evolution inequalities with critical potential, J. Math. Anal. Appl., Volume 304 (2005) no. 2, pp. 451-463 | DOI | MR | Zbl

[17] Fedorov, Vladimir E.; Urazaeva, A. V. An inverse problem for linear Sobolev type equations, J. Inverse Ill-Posed Probl., Volume 12 (2004) no. 4, pp. 387-395 | Zbl | DOI | MR

[18] Guezane-Lakoud, Assia; Belakroum, D. Time-discretization schema for an integrodifferential Sobolev type equation with integral conditions, Appl. Math. Comput., Volume 218 (2012) no. 9, pp. 4695-4702 | MR | Zbl

[19] Hoff, Nicholas J. Creep buckling, Aeron. Quart., Volume 7 (1956) no. 1, pp. 1-20 | DOI

[20] Jleli, Mohamed; Samet, Bessem Instantaneous blow-up for a fractional in time equation of Sobolev type, Math. Methods Appl. Sci., Volume 43 (2020) no. 8, pp. 5645-5652 | MR | Zbl | DOI

[21] Jleli, Mohamed; Samet, Bessem Instantaneous blow-up for nonlinear Sobolev type equations with potentials on Riemannian manifolds, Commun. Pure Appl. Anal., Volume 21 (2022) no. 6, pp. 2065-2078 | DOI | MR | Zbl

[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., Volume 10 (2021), pp. 1267-1283 | DOI | MR | Zbl

[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., Volume 442 (2016) no. 2, pp. 451-468 | DOI | MR | Zbl

[24] Korpusov, Maksim O.; Sveshnikov, Alekseĭ G. Blowup of solutions to initial value problems for nonlinear operator-differential equations, Dokl. Math., Volume 71 (2005) no. 2, pp. 168-171 | Zbl

[25] Korpusov, Maksim O.; Sveshnikov, Alekseĭ G. Application of the nonlinear capacity method to differential inequalities of Sobolev type, Differ. Equ., Volume 45 (2009) no. 7, pp. 951-959 | DOI | Zbl

[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, Volume 31 (2014) no. 1, pp. 1-22 | Zbl | MR | Numdam

[27] Mukhartova, Yu. V.; Panin, Aleksandr A. Blow-up of the solution of an inhomogeneous system of Sobolev-type equations, Math. Notes, Volume 91 (2012) no. 2, pp. 217-230 | DOI | Zbl | MR

[28] Sviridyuk, Georgiĭ A.; Fedorov, Vladimir E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Inverse and Ill-Posed Problems Series, VSP, 2003, viii+216 pages | Zbl | DOI

[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., Volume 54 (2010) no. 5, pp. 47-55 | DOI

[30] Zamyshlyaeva, Alena A.; Lut, Aleksandr Inverse problem for the Sobolev type equation of higher order, Mathematics, Volume 9 (2021) no. 14, 1647 | DOI

[31] Zamyshlyaeva, Alena A.; Surovtsev, S V. Numerical investigation of one Sobolev type mathematical model, J. Comput. Eng. Math., Volume 2 (2015) no. 3, pp. 72-80 | DOI | Zbl