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Article Contents

# Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations

• The aim of this short note is to study self-similiar radially symmetric solutions of the scalar doubly nonlinear reaction-diffusion equation \begin{equation*} \frac{\partial u^{m-1}}{\partial t} - \Delta_p u = \lambda u^{q-1} \end{equation*} on $\mathbb{R}^n$, where the parameters $1 < m, p,q < \infty$ and $0 < \lambda < \infty$ are fixed. Particularly, for $m < p < q < q_c := p(1+\frac{m-1}{n})$ (where $q_c$ is Fujita's critical exponent of blow-up) we show that there exist self-similar and radially symmetric solutions $u$, which do not blow up in finite time, but instantly become sign-changing for $t>0$ inside some subdomain.
Mathematics Subject Classification: Primary: 35B05, 35K92; Secondary: 35K65.

 Citation:

•  [1] D. Andreucci and A. Tedeev, A Fujita type result for a degenerate Neumann problem in domains with non compact boundary, J. Math. Anal. Appl., 231 (1999), 543-567. [2] G.I. Barenblatt, On self-similar motions of compressible fluids in porous media, Prikl. Mat. Mekh., 16 (1952), 679-698 (in Russian). [3] K. Deng and H.A. Levine, The role of critical exponents in blowup theorems: The sequel, J. Math. Anal. Appl., 243 (2000), 85-126. [4] P. Drábek and J. Milota, Methods of Nonlinear Analysis, Birkhäuser Advanced Texts, Birkhäuser, 2007. [5] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. [6] V.A. Galaktionov, Conditions for global nonexistence and localization for a class of nonlinear parabolic equations, USSR Computational Mathematics and Mathematical Physics, 23(6) (1983), 35-44. [7] V.A. Galaktionov and H.A. Levine, A general approach to critical Fujita exponents in nonlinear parbolic problems, Nonlinear Analysis, 34 (1998), 1005-1027. [8] A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189. [9] Xinfeng Liu and Mingxin Wang, The critical exponent of doubly singular parabolic equations, Journal of Mathematical Analysis and Applications, 257 (2001), 170-188. [10] H.A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288. [11] A. Matas and J. Merker, Existence of weak solutions to doubly degenerate diffusion equations, Applications of Mathematics, 57 (2012), 43-69. [12] D.J. Needham and P.G. Chamberlain, Global similarity solutions to a class of semilinear parabolic equations: existence, bifurcations and asymptotics, Proc. R. Soc. Lond. A, 454 (1998), 1933-1959. [13] F. Otto, $L^1$-Contraction and Uniqueness for Quasilinear Elliptic-Parabolic Equations, Journal of Differential Equations, 131 (1996), 20-38. [14] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in quasilinear parabolic equations, de Gruyter expositions in mathematics 19, de Gruyter, 1995. [15] P. Souplet and F.B. Weissler, Self-similar subsolutions and blowup for nonlinear parabolic equations, Journal of Mathematical Analysis and Applications, 212 (1997), 60-74. [16] M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic problems, Publ. RIMS Kyoto Univ., 8 (1972/73), 211-229. [17] J.L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, Oxford lecture series in mathematics and its applications 33, Oxford University Press, 2006. [18] J.L. Vázquez, The porous medium equation, Oxford mathematical monographs, Oxford University Press, 2007.
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