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 & 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.

[2]

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

[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. 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. doi: 10.1016/S0362-546X(97)00542-7.

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A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem,, Indiana Univ. Math. J., 31 (1982), 167. 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), 94 (1986), 83. doi: 10.1007/BF00278244.

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

[8]

Y.W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation,, Nonlinear Anal. 31 (1998), 31 (1998), 117. 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. 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), 44 (2001), 1133. 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), 91 (1986), 231. 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), 91 (1986), 247. doi: 10.1007/BF00250744.

[13]

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

[14]

E. Yanagida, Uniqueness of rapidly decreasing solutions to the Haraux-Weissler equation,, J. Differential Equations 127 (1996), 127 (1996), 561. 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.

[2]

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

[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. 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. 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. 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), 94 (1986), 83. 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, 6 (1996), 197.

[8]

Y.W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation,, Nonlinear Anal. 31 (1998), 31 (1998), 117. 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. 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), 44 (2001), 1133. 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), 91 (1986), 231. 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), 91 (1986), 247. doi: 10.1007/BF00250744.

[13]

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

[14]

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

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