April  2019, 12(2): 311-337. doi: 10.3934/dcdss.2019022

Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

1. 

Université de Pau et des Pays de l'Adour, CNRS, E2S, LMAP UMR 5142, avenue de l'université, 64013 Pau cedex, France

2. 

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India

* Corresponding author: Jacques Giacomoni

Received  April 2017 Revised  January 2018 Published  August 2018

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity
$\begin{equation*} \quad (P_{t}^s) \left\{\begin{split} \quad u_t + (-\Delta)^s u & = u^{-q} + f(x,u), \;u >0\; \text{in}\;(0,T) \times \Omega, \\ u & = 0 \; \mbox{in}\; (0,T) \times (\mathbb{R}^n \setminus \Omega ),\\ \quad \quad \quad \quad u(0,x)& = u_0(x) \; \mbox{in} \; {\mathbb{R}^n},\end{split}\quad \right.\end{equation*}$
where
$\Omega $
is a bounded domain in
$\mathbb{R}^n$
with smooth boundary
$\partial \Omega $
,
$n> 2s, \;s ∈ (0,1)$
,
$q>0$
,
${q(2s-1)<(2s+1)}$
,
$u_0 ∈ L^∞(\Omega )\cap X_0(\Omega )$
and
$T>0$
. We suppose that the map
$(x,y)∈ \Omega × \mathbb{R}^+ \mapsto f(x,y)$
is a bounded from below Carathéodary function, locally Lipschitz with respect to the second variable and uniformly for
$x ∈ \Omega $
and it satisfies
$ \begin{equation}\label{cond_on_f}{ \limsup\limits_{y \to +\infty} \frac{f(x,y)}{y}<\lambda_1^s(\Omega)}, \end{equation}$
where
$\lambda_1^s(\Omega )$
is the first eigenvalue of
$(-\Delta )^s$
in
$\Omega $
with homogeneous Dirichlet boundary condition in
$\mathbb{R}^n \setminus \Omega $
. We prove the existence and uniqueness of a weak solution to
$(P_t^s)$
on assuming
$u_0$
satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results.We also show additional regularity on the solution of
$(P_t^s)$
when we regularize our initial function
$u_0$
.
Citation: Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022
References:
[1]

B. AbdellaouiM. MedinaI. Peral and A. Primo, Optimal results for the fractional heat equation involving the hardy potential, Nonlinear Anal., 140 (2016), 166-207.  doi: 10.1016/j.na.2016.03.013.  Google Scholar

[2]

Adimurthi, J. Giacomoni and S. Santra, Positive solutions to a fractional equation with singular nonlinearity, J. Differential Equations, 265 (2018), 1191-1226, arXiv: 1706.01965 doi: 10.1016/j.jde.2018.03.023.  Google Scholar

[3]

N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data, Transactions of the American Mathematical Society, 361 (2009), 2527-2566.  doi: 10.1090/S0002-9947-08-04758-2.  Google Scholar

[4]

S. Amghibech, On the discrete version of picone's identity, Discrete Applied Mathematics, 156 (2008), 1-10.  doi: 10.1016/j.dam.2007.05.013.  Google Scholar

[5]

B. AvelinU. Gianazza and S. Salsa, Boundary estimates for certain degenerate and singular parabolic equations, Journal of the European Mathematical Society, 18 (2016), 381-424.  doi: 10.4171/JEMS/593.  Google Scholar

[6]

M. BadraK. Bal and J. Giacomoni, A singular parabolic equation: Existence, stabilization, J. Differential Equations, 252 (2012), 5042-5075.  doi: 10.1016/j.jde.2012.01.035.  Google Scholar

[7]

V. Barbu, Nonlinear Differential Equations of Monotone types in Banach Spaces, $1^{st}$ edition, Springer Monogr. Math., Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[8]

B. BarriosI. De BonisM. Medina and I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407.  doi: 10.1515/math-2015-0038.  Google Scholar

[9]

B. Bougherara and J. Giacomoni, Existence of mild solutions for a singular parabolic equation and stabilization, Adv. Nonlinear Anal., 4 (2015), 123-134.  doi: 10.1515/anona-2015-0002.  Google Scholar

[10]

L. Cafarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, Journal für die reine und angewandte Mathematik (Crelles Journal), 680 (2013), 191-233.  doi: 10.1515/crelle.2012.036.  Google Scholar

[11]

J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation, Transactions of the American Mathematical Society, 357 (2005), 1801-1828.  doi: 10.1090/S0002-9947-04-03811-5.  Google Scholar

[12]

L. M. Del Pezzo and A. J. Quaas, Non-resonant fredholm alternative and anti-maximum principle for the fractional $p$-Laplacian, Journal of Fixed Point Theory and Applications, 19 (2017), 939-958.  doi: 10.1007/s11784-017-0405-5.  Google Scholar

[13]

A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional laplacian, Monatshefte für Mathematik, 160 (2010), 375-384.  doi: 10.1007/s00605-009-0093-3.  Google Scholar

[14]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[15]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp hardy inequalities, Journal of Functional Analysis, 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.  Google Scholar

[16]

J. GiacomoniT. Mukherjee and K. Sreenadh, Positive solutions of fractional elliptic equation with critical and singular nonlinearity, Adv. Nonlinear Anal., 6 (2016), 327-354.  doi: 10.1515/anona-2016-0113.  Google Scholar

[17]

S. Kim and K.-A. Lee, Hölder estimates for singular non-local parabolic equations, Journal of Functional Analysis, 261 (2011), 3482-3518.  doi: 10.1016/j.jfa.2011.08.010.  Google Scholar

[18]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar

[19]

T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electronic Journal of Differential Equations, 54 (2016), 1-23.   Google Scholar

[20]

X. Ros-Oton and J. Serra, The dirichlet problem for the fractional laplacian: Regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[21]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional laplacian, Transactions of the American Mathematical Society, 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[22]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[23]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[24]

J. Simon, Compact sets in the space $L^p(0, T ; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[25]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Holden, Helge and Karlsen, Kenneth H. eds., Springer, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[26]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional laplacian operators, Discrete and Continuous Dynamical Systems - Series S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

show all references

References:
[1]

B. AbdellaouiM. MedinaI. Peral and A. Primo, Optimal results for the fractional heat equation involving the hardy potential, Nonlinear Anal., 140 (2016), 166-207.  doi: 10.1016/j.na.2016.03.013.  Google Scholar

[2]

Adimurthi, J. Giacomoni and S. Santra, Positive solutions to a fractional equation with singular nonlinearity, J. Differential Equations, 265 (2018), 1191-1226, arXiv: 1706.01965 doi: 10.1016/j.jde.2018.03.023.  Google Scholar

[3]

N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data, Transactions of the American Mathematical Society, 361 (2009), 2527-2566.  doi: 10.1090/S0002-9947-08-04758-2.  Google Scholar

[4]

S. Amghibech, On the discrete version of picone's identity, Discrete Applied Mathematics, 156 (2008), 1-10.  doi: 10.1016/j.dam.2007.05.013.  Google Scholar

[5]

B. AvelinU. Gianazza and S. Salsa, Boundary estimates for certain degenerate and singular parabolic equations, Journal of the European Mathematical Society, 18 (2016), 381-424.  doi: 10.4171/JEMS/593.  Google Scholar

[6]

M. BadraK. Bal and J. Giacomoni, A singular parabolic equation: Existence, stabilization, J. Differential Equations, 252 (2012), 5042-5075.  doi: 10.1016/j.jde.2012.01.035.  Google Scholar

[7]

V. Barbu, Nonlinear Differential Equations of Monotone types in Banach Spaces, $1^{st}$ edition, Springer Monogr. Math., Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[8]

B. BarriosI. De BonisM. Medina and I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407.  doi: 10.1515/math-2015-0038.  Google Scholar

[9]

B. Bougherara and J. Giacomoni, Existence of mild solutions for a singular parabolic equation and stabilization, Adv. Nonlinear Anal., 4 (2015), 123-134.  doi: 10.1515/anona-2015-0002.  Google Scholar

[10]

L. Cafarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, Journal für die reine und angewandte Mathematik (Crelles Journal), 680 (2013), 191-233.  doi: 10.1515/crelle.2012.036.  Google Scholar

[11]

J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation, Transactions of the American Mathematical Society, 357 (2005), 1801-1828.  doi: 10.1090/S0002-9947-04-03811-5.  Google Scholar

[12]

L. M. Del Pezzo and A. J. Quaas, Non-resonant fredholm alternative and anti-maximum principle for the fractional $p$-Laplacian, Journal of Fixed Point Theory and Applications, 19 (2017), 939-958.  doi: 10.1007/s11784-017-0405-5.  Google Scholar

[13]

A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional laplacian, Monatshefte für Mathematik, 160 (2010), 375-384.  doi: 10.1007/s00605-009-0093-3.  Google Scholar

[14]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[15]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp hardy inequalities, Journal of Functional Analysis, 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.  Google Scholar

[16]

J. GiacomoniT. Mukherjee and K. Sreenadh, Positive solutions of fractional elliptic equation with critical and singular nonlinearity, Adv. Nonlinear Anal., 6 (2016), 327-354.  doi: 10.1515/anona-2016-0113.  Google Scholar

[17]

S. Kim and K.-A. Lee, Hölder estimates for singular non-local parabolic equations, Journal of Functional Analysis, 261 (2011), 3482-3518.  doi: 10.1016/j.jfa.2011.08.010.  Google Scholar

[18]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar

[19]

T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electronic Journal of Differential Equations, 54 (2016), 1-23.   Google Scholar

[20]

X. Ros-Oton and J. Serra, The dirichlet problem for the fractional laplacian: Regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[21]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional laplacian, Transactions of the American Mathematical Society, 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[22]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[23]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[24]

J. Simon, Compact sets in the space $L^p(0, T ; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[25]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Holden, Helge and Karlsen, Kenneth H. eds., Springer, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[26]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional laplacian operators, Discrete and Continuous Dynamical Systems - Series S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

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