Article Contents
Article Contents

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

• 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 $m = 1$.
Mathematics Subject Classification: 35K05, 35B40.

 Citation:

•  [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.

• on this site

/