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September  2021, 41(9): 4125-4144. doi: 10.3934/dcds.2021031

Existence of solution for a class of heat equation in whole $ \mathbb{R}^N $

1. 

Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB, Brazil

2. 

Department of Mathematics, Faculty of Exact Sciences, Lab. of Applied Mathematics, University of Bejaia, Bejaia, 06000, Algeria

* Corresponding author: Tahir Boudjeriou

Received  March 2020 Revised  December 2020 Published  September 2021 Early access  February 2021

Fund Project: C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7

In this paper we study the local and global existence of solutions for a class of heat equation in whole $ \mathbb{R}^{N} $ where the nonlinearity has a critical growth for $ N \geq 2 $. In order to prove the global existence, we will use the potential well theory combined with the Nehari manifold, and also with the Pohozaev manifold that is a novelty for this type of problem. Moreover, the blow-up phenomena of local solutions is investigated by combining the subdifferential approach with the concavity method.

Citation: Claudianor O. Alves, Tahir Boudjeriou. Existence of solution for a class of heat equation in whole $ \mathbb{R}^N $. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4125-4144. doi: 10.3934/dcds.2021031
References:
[1]

C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differential Equations, 34 (2009), 377-411.  doi: 10.1007/s00526-008-0188-z.

[2]

C. O. Alves, Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity ith exponential critical growth in $\mathbb{R}^2$, Milan J. Math, 84 (2016), 1-22.  doi: 10.1007/s00032-015-0247-9.

[3]

C. O. AlvesJ. M. B. do Ó and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbb{R}^2$ involving critical growth, Nonlinear Anal, 56 (2004), 781-791.  doi: 10.1016/j.na.2003.06.003.

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973.

[5]

G. BarlesS. Biton and O. Ley, Uniqueness for parabolic equations without growth condition and applications to the mean curvature flow in $\mathbb{R}^{2}$, J. Differential Equations, 187 (2003), 456-472.  doi: 10.1016/S0022-0396(02)00071-2.

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Archive for Rational Mechanics and Analysis, 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[7]

T. Boudjeriou, Global existence and blow-up for the fractional p-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 17 (2020), Paper No. 162, 24 pp. doi: 10.1007/s00009-020-01584-6.

[8]

T. Boudjeriou, Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-Laplacian, Applicable Analysis, (2020). doi: 10.1080/00036811.2020.1829601.

[9]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Comm. Partial Differential Equation, 17 (1992), 407-435.  doi: 10.1080/03605309208820848.

[10]

X. D. Cao and Z. Zhang, Differential Harnack estimates for parabolic equations, Complex and Differential Geometry, 8 (2011), 87-98.  doi: 10.1007/978-3-642-20300-8_5.

[11]

M. M. CavalcantiV. N. Domingos Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158.  doi: 10.1016/j.jde.2004.04.011.

[12]

M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, J. Math. Anal. Appl, 291 (2004), 109-127.  doi: 10.1016/j.jmaa.2003.10.020.

[13]

H. ChenP. Luo and G. Liu, Global solution and below-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl, 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[14]

H. Ding and J. Zhou, Global existence and blow-up for a parabolic problem of Kirchhoff type with logarithmic nonlinearity, J. Appl. Math. Optim., (2019). doi: 10.1007/s00245-019-09603-z.

[15]

J. A. Esquivel-Avila, A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations, Nonlinear Anal., 52 (2003), 1111-1127.  doi: 10.1016/S0362-546X(02)00155-4.

[16] J. C. Robinson, Infinite-dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. 
[17]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 1998.

[18]

D. G. FigueiredoO. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearity in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003.

[19]

H. Fujita, On the blowing up solutions of the Cauchy problem for $u_{t} = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[20]

Y. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qualitative Theory Differ. Equ, 70 (2016), 1-17.  doi: 10.14232/ejqtde.2016.1.70.

[21]

V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal, 34 (1998), 1005-1027.  doi: 10.1016/S0362-546X(97)00716-5.

[22]

V. A. Galaktionov and J. L. V$\acute{\text{a}}$zquez, Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24 (1993), 1254-1276.  doi: 10.1137/0524071.

[23]

V. A. Galaktionov and J. L. V$\acute{\text{a}}$zquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst, 8 (2002), 399-433.  doi: 10.3934/dcds.2002.8.399.

[24]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differ Integral Equ, 18 (2005), 961-990. 

[25]

C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ, 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4.

[26]

H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differential Equations, 26 (1977), 291-319.  doi: 10.1016/0022-0396(77)90196-6.

[27]

R. Jiang and J. Zhou, Blow-up and global existence of solutions to a parabolic equation associated with fractional $p$-Laplacian, Com on Pure. Appl Anal, 18 (2019), 1205-1226.  doi: 10.3934/cpaa.2019058.

[28]

Y. Liu and J. Zhao, On the potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal, 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.

[29]

X. MingqiD. V. R$\breve{\text{a}}$dulescu and B. L. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.

[30]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel. J. Math, 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[31]

S. I. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. 

[32]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Math, 30 (1968), 148-172.  doi: 10.1007/BF00250942.

[33]

R. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Q. Appl. Math, 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.

[34]

S. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL. 2004. doi: 10.1201/9780203492222.

show all references

References:
[1]

C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differential Equations, 34 (2009), 377-411.  doi: 10.1007/s00526-008-0188-z.

[2]

C. O. Alves, Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity ith exponential critical growth in $\mathbb{R}^2$, Milan J. Math, 84 (2016), 1-22.  doi: 10.1007/s00032-015-0247-9.

[3]

C. O. AlvesJ. M. B. do Ó and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbb{R}^2$ involving critical growth, Nonlinear Anal, 56 (2004), 781-791.  doi: 10.1016/j.na.2003.06.003.

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973.

[5]

G. BarlesS. Biton and O. Ley, Uniqueness for parabolic equations without growth condition and applications to the mean curvature flow in $\mathbb{R}^{2}$, J. Differential Equations, 187 (2003), 456-472.  doi: 10.1016/S0022-0396(02)00071-2.

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Archive for Rational Mechanics and Analysis, 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[7]

T. Boudjeriou, Global existence and blow-up for the fractional p-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 17 (2020), Paper No. 162, 24 pp. doi: 10.1007/s00009-020-01584-6.

[8]

T. Boudjeriou, Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-Laplacian, Applicable Analysis, (2020). doi: 10.1080/00036811.2020.1829601.

[9]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Comm. Partial Differential Equation, 17 (1992), 407-435.  doi: 10.1080/03605309208820848.

[10]

X. D. Cao and Z. Zhang, Differential Harnack estimates for parabolic equations, Complex and Differential Geometry, 8 (2011), 87-98.  doi: 10.1007/978-3-642-20300-8_5.

[11]

M. M. CavalcantiV. N. Domingos Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158.  doi: 10.1016/j.jde.2004.04.011.

[12]

M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, J. Math. Anal. Appl, 291 (2004), 109-127.  doi: 10.1016/j.jmaa.2003.10.020.

[13]

H. ChenP. Luo and G. Liu, Global solution and below-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl, 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[14]

H. Ding and J. Zhou, Global existence and blow-up for a parabolic problem of Kirchhoff type with logarithmic nonlinearity, J. Appl. Math. Optim., (2019). doi: 10.1007/s00245-019-09603-z.

[15]

J. A. Esquivel-Avila, A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations, Nonlinear Anal., 52 (2003), 1111-1127.  doi: 10.1016/S0362-546X(02)00155-4.

[16] J. C. Robinson, Infinite-dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. 
[17]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 1998.

[18]

D. G. FigueiredoO. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearity in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003.

[19]

H. Fujita, On the blowing up solutions of the Cauchy problem for $u_{t} = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[20]

Y. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qualitative Theory Differ. Equ, 70 (2016), 1-17.  doi: 10.14232/ejqtde.2016.1.70.

[21]

V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal, 34 (1998), 1005-1027.  doi: 10.1016/S0362-546X(97)00716-5.

[22]

V. A. Galaktionov and J. L. V$\acute{\text{a}}$zquez, Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24 (1993), 1254-1276.  doi: 10.1137/0524071.

[23]

V. A. Galaktionov and J. L. V$\acute{\text{a}}$zquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst, 8 (2002), 399-433.  doi: 10.3934/dcds.2002.8.399.

[24]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differ Integral Equ, 18 (2005), 961-990. 

[25]

C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ, 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4.

[26]

H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differential Equations, 26 (1977), 291-319.  doi: 10.1016/0022-0396(77)90196-6.

[27]

R. Jiang and J. Zhou, Blow-up and global existence of solutions to a parabolic equation associated with fractional $p$-Laplacian, Com on Pure. Appl Anal, 18 (2019), 1205-1226.  doi: 10.3934/cpaa.2019058.

[28]

Y. Liu and J. Zhao, On the potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal, 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.

[29]

X. MingqiD. V. R$\breve{\text{a}}$dulescu and B. L. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.

[30]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel. J. Math, 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[31]

S. I. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. 

[32]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Math, 30 (1968), 148-172.  doi: 10.1007/BF00250942.

[33]

R. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Q. Appl. Math, 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.

[34]

S. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL. 2004. doi: 10.1201/9780203492222.

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