2015, 2015(special): 817-825. doi: 10.3934/proc.2015.0817

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

1. 

HTWK Leipzig University of Applied Sciences, Gustav-Freytag-Straße 42a, D - 04277 Leipzig, Germany

2. 

Katedra matematiky - Západočeská univerzita v Plzni, Univerzitní 22, CZ - 306 14 Plzeň, Czech Republic

Received  July 2014 Revised  February 2015 Published  November 2015

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.
Citation: Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817
References:
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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. Google Scholar

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V.A. Galaktionov, Conditions for global nonexistence and localization for a class of nonlinear parabolic equations,, USSR Computational Mathematics and Mathematical Physics, 23 (1983), 35. Google Scholar

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

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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. Google Scholar

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

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

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

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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. Google Scholar

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

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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, 19 (1995). Google Scholar

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P. Souplet and F.B. Weissler, Self-similar subsolutions and blowup for nonlinear parabolic equations,, Journal of Mathematical Analysis and Applications, 212 (1997), 60. Google Scholar

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M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic problems,, Publ. RIMS Kyoto Univ., 8 (): 211. Google Scholar

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J.L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations,, Oxford lecture series in mathematics and its applications 33, 33 (2006). Google Scholar

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J.L. Vázquez, The porous medium equation,, Oxford mathematical monographs, (2007). Google Scholar

show all references

References:
[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. Google Scholar

[2]

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

[3]

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

[4]

P. Drábek and J. Milota, Methods of Nonlinear Analysis,, Birkhäuser Advanced Texts, (2007). Google Scholar

[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. Google Scholar

[6]

V.A. Galaktionov, Conditions for global nonexistence and localization for a class of nonlinear parabolic equations,, USSR Computational Mathematics and Mathematical Physics, 23 (1983), 35. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[13]

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

[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, 19 (1995). Google Scholar

[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. Google Scholar

[16]

M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic problems,, Publ. RIMS Kyoto Univ., 8 (): 211. Google Scholar

[17]

J.L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations,, Oxford lecture series in mathematics and its applications 33, 33 (2006). Google Scholar

[18]

J.L. Vázquez, The porous medium equation,, Oxford mathematical monographs, (2007). Google Scholar

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