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

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  • 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.


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  • [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.


    G.I. Barenblatt, On self-similar motions of compressible fluids in porous media, Prikl. Mat. Mekh., 16 (1952), 679-698 (in Russian).


    K. Deng and H.A. Levine, The role of critical exponents in blowup theorems: The sequel, J. Math. Anal. Appl., 243 (2000), 85-126.


    P. Drábek and J. Milota, Methods of Nonlinear Analysis, Birkhäuser Advanced Texts, Birkhäuser, 2007.


    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.


    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.


    V.A. Galaktionov and H.A. Levine, A general approach to critical Fujita exponents in nonlinear parbolic problems, Nonlinear Analysis, 34 (1998), 1005-1027.


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


    Xinfeng Liu and Mingxin Wang, The critical exponent of doubly singular parabolic equations, Journal of Mathematical Analysis and Applications, 257 (2001), 170-188.


    H.A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.


    A. Matas and J. Merker, Existence of weak solutions to doubly degenerate diffusion equations, Applications of Mathematics, 57 (2012), 43-69.


    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.


    F. Otto, $L^1$-Contraction and Uniqueness for Quasilinear Elliptic-Parabolic Equations, Journal of Differential Equations, 131 (1996), 20-38.


    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.


    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.


    M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic problems, Publ. RIMS Kyoto Univ., 8 (1972/73), 211-229.


    J.L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, Oxford lecture series in mathematics and its applications 33, Oxford University Press, 2006.


    J.L. Vázquez, The porous medium equation, Oxford mathematical monographs, Oxford University Press, 2007.

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