American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 117-125. doi: 10.3934/proc.2011.2011.117

Existence results to a quasilinear and singular parabolic equation

Mehdi Badra 1, , Kaushik Bal 2, and  Jacques Giacomoni 2,

1. 

Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex
2. 

LMAP (UMR 5142), Bat. IPRA, Université de Pau et des Pays de l'Adour, Avenue de l'Université, 64013 cedex Pau, France, France

Received  July 2010 Revised  January 2011 Published  October 2011

We investigate the following quasilinear parabolic and singular equation, $ u_t-\Delta_p u &=\frac{1}{u^\delta}+f(t,x)\;\text{ in }\,Q_T=(0,T]\times\Omega\\ u>0 \text{ in }\, Q_T\; , u &=0\,\text{ on} \;\Gamma=[0,T]\times\partial\Omega,\\ u(0,x) &=u_0(x)\;\text{ in }\Omega $ where $\Omega$ is an open bounded domain with smooth boundary in ${\rm R}^N$, $1 < p< \infty$ and $0<\delta$, $T>0$, $f\in L^\infty(Q_T)$ and $u_0\in L^\infty(\Omega)\cap W^{1,p}_0(\Omega)$. For any $\delta\in (0,2+\frac{1}{p-1})$, $u_0$ satisfying a cone condition defined below and any $T>0$, In this paper we prove the existence and the uniqueness of a weak solution $u$ to $({\rm P_t})$.
Keywords: sub and supersolutions, singular nonlinearity, quasilinear parabolic equation.
Mathematics Subject Classification: Primary 35J65, 35J20; Secondary 35J70.
Citation: Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117
[1]

Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713
[2]

Jie Zhao. A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021193
[3]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493
[4]

Dung Le. Strong positivity of continuous supersolutions to parabolic equations with rough boundary data. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1521-1530. doi: 10.3934/dcds.2015.35.1521
[5]

Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009
[6]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[7]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335
[8]

Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399
[9]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[10]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027
[11]

Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations & Control Theory, 2021, 10 (4) : 723-732. doi: 10.3934/eect.2020088
[12]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[13]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885
[14]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[15]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[16]

Amal Attouchi, Eero Ruosteenoja. Gradient regularity for a singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5955-5972. doi: 10.3934/dcds.2020254
[17]

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
[18]

Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303
[19]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[20]

U. Biccari, V. Hernández-Santamaría, J. Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021032

 Impact Factor: 

Tools

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy