# American Institute of Mathematical Sciences

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
 $$$\label{cond_on_f}{ \limsup\limits_{y \to +\infty} \frac{f(x,y)}{y}<\lambda_1^s(\Omega)},$$$
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 and Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022
##### References:
 [1] B. Abdellaoui, M. Medina, I. 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. [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. [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. [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. [5] B. Avelin, U. 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. [6] M. Badra, K. 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. [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. [8] B. Barrios, I. De Bonis, M. 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. [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. [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. [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. [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. [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. [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. [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. [16] J. Giacomoni, T. 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. [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. [18] T. Leonori, I. Peral, A. 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. [19] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electronic Journal of Differential Equations, 54 (2016), 1-23. [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. [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. [22] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. [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. [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. [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. [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.

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##### References:
 [1] B. Abdellaoui, M. Medina, I. 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. [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. [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. [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. [5] B. Avelin, U. 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. [6] M. Badra, K. 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. [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. [8] B. Barrios, I. De Bonis, M. 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. [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. [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. [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. [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. [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. [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. [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. [16] J. Giacomoni, T. 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. [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. [18] T. Leonori, I. Peral, A. 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. [19] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electronic Journal of Differential Equations, 54 (2016), 1-23. [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. [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. [22] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. [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. [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. [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. [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.
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