# American Institute of Mathematical Sciences

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.   Google Scholar [2] M.F. Bidaut-Véron, The $p$-Laplace heat equation with a source term: self-similar solutions revisited,, Advanced Nonlinear Studies, 6 (2006), 69.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.   Google Scholar [2] M.F. Bidaut-Véron, The $p$-Laplace heat equation with a source term: self-similar solutions revisited,, Advanced Nonlinear Studies, 6 (2006), 69.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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