July  2013, 18(5): 1389-1414. doi: 10.3934/dcdsb.2013.18.1389

Sign-changing solutions of a quasilinear heat equation with a source term

1. 

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

2. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States

Received  June 2012 Revised  August 2012 Published  March 2013

The Cauchy problem of a heat equation with a source term $$ \psi_t=\Delta \left(|\psi|^{m-1}\psi\right)+|\psi|^{\gamma-1}\psi\ \ \mbox{in}\ \ (0, \infty)\times R^n $$ is considered, where $\gamma>m>1$. We are interested in global solutions with H$\ddot{o}$lder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time $ t > 0$. Through detailed analysis of the self-similarity problem, we prove the existence of two type of such solutions, one with compact support and the other decays to zero as $ | x| \rightarrow \infty$ with an algebraic rate determined uniquely by $ n, m$ and $\gamma$. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of $m = 1$.
Citation: Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389
References:
[1]

B. Bettioui and A. Gmira, On the radial solutions of a degenerate quasilinear elliptic equation in $R^N$, Ann. Fac. Sci. Toulouse, 8 (1999), 411-438.

[2]

M.F. Bidaut-Véron, The $p$-Laplace heat equation with a source term: self-similar solutions revisited, Advanced Nonlinear Studies, 6 (2006), 69-108.

[3]

T. Cazenave, F. Dickstein and F.B. Weissler, Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball Math. Ann., 344 (2009), 431-449. doi: 10.1007/s00208-008-0312-6.

[4]

C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation, Nonlinear Anal., 33 (1998), 51-69. doi: 10.1016/S0362-546X(97)00542-7.

[5]

A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189. doi: 10.1512/iumj.1982.31.31016.

[6]

L.A. Peletier, D. Terman and F.B. Weissler, On the equation $\Delta u+\frac {1}{2}x \nabla u +f(u)=0$, Arch. Rat. Mech. Anal. 94 (1986), 83-99. doi: 10.1007/BF00278244.

[7]

Y.W. Qi, The existence of moving boundary solution of a porous media equation with a source term, Mathematical Sciences and Applications Gakkōtosho, Tokyo, 6 (1996), 197-215.

[8]

Y.W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation, Nonlinear Anal. 31 (1998), 117-136. doi: 10.1016/S0362-546X(96)00298-2.

[9]

R. Suzuki, Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity, J. Differential Equations, 190 (2003), 150-181. doi: 10.1016/S0022-0396(02)00086-4.

[10]

M. Wang and X. Wang, Existence of positive solutions to a nonlinear initial problem, Nonlinear Anal. 44 (2001), 1133-1136. doi: 10.1016/S0362-546X(99)00344-2.

[11]

F.B. Weissler, Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation, Arch. Rat. Mech. Anal. 91 (1986), 231-245. doi: 10.1007/BF00250743.

[12]

F.B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Rat. Mech. Anal. 91 (1986), 247-266. doi: 10.1007/BF00250744.

[13]

F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation Israel J. Math. 38 (1981), 29-40. doi: 10.1007/BF02761845.

[14]

E. Yanagida, Uniqueness of rapidly decreasing solutions to the Haraux-Weissler equation, J. Differential Equations 127 (1996), 561-570. doi: 10.1006/jdeq.1996.0083.

show all references

References:
[1]

B. Bettioui and A. Gmira, On the radial solutions of a degenerate quasilinear elliptic equation in $R^N$, Ann. Fac. Sci. Toulouse, 8 (1999), 411-438.

[2]

M.F. Bidaut-Véron, The $p$-Laplace heat equation with a source term: self-similar solutions revisited, Advanced Nonlinear Studies, 6 (2006), 69-108.

[3]

T. Cazenave, F. Dickstein and F.B. Weissler, Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball Math. Ann., 344 (2009), 431-449. doi: 10.1007/s00208-008-0312-6.

[4]

C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation, Nonlinear Anal., 33 (1998), 51-69. doi: 10.1016/S0362-546X(97)00542-7.

[5]

A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189. doi: 10.1512/iumj.1982.31.31016.

[6]

L.A. Peletier, D. Terman and F.B. Weissler, On the equation $\Delta u+\frac {1}{2}x \nabla u +f(u)=0$, Arch. Rat. Mech. Anal. 94 (1986), 83-99. doi: 10.1007/BF00278244.

[7]

Y.W. Qi, The existence of moving boundary solution of a porous media equation with a source term, Mathematical Sciences and Applications Gakkōtosho, Tokyo, 6 (1996), 197-215.

[8]

Y.W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation, Nonlinear Anal. 31 (1998), 117-136. doi: 10.1016/S0362-546X(96)00298-2.

[9]

R. Suzuki, Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity, J. Differential Equations, 190 (2003), 150-181. doi: 10.1016/S0022-0396(02)00086-4.

[10]

M. Wang and X. Wang, Existence of positive solutions to a nonlinear initial problem, Nonlinear Anal. 44 (2001), 1133-1136. doi: 10.1016/S0362-546X(99)00344-2.

[11]

F.B. Weissler, Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation, Arch. Rat. Mech. Anal. 91 (1986), 231-245. doi: 10.1007/BF00250743.

[12]

F.B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Rat. Mech. Anal. 91 (1986), 247-266. doi: 10.1007/BF00250744.

[13]

F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation Israel J. Math. 38 (1981), 29-40. doi: 10.1007/BF02761845.

[14]

E. Yanagida, Uniqueness of rapidly decreasing solutions to the Haraux-Weissler equation, J. Differential Equations 127 (1996), 561-570. doi: 10.1006/jdeq.1996.0083.

[1]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454

[2]

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

[3]

Qingsong Gu, Jiaxin Hu, Sze-Man Ngai. Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates. Communications on Pure and Applied Analysis, 2020, 19 (2) : 641-676. doi: 10.3934/cpaa.2020030

[4]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801

[5]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[6]

Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499

[7]

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

[8]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313

[9]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703

[10]

Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure and Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004

[11]

Weiwei Ao, Chao Liu. Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5047-5077. doi: 10.3934/dcds.2020211

[12]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002

[13]

Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235

[14]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565

[15]

Angela Pistoia, Tonia Ricciardi. Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5651-5692. doi: 10.3934/dcds.2017245

[16]

Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003

[17]

M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057

[18]

Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013

[19]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256

[20]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]