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July  2011, 10(4): 1225-1237. doi: 10.3934/cpaa.2011.10.1225

## Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation

 1 Mathematical Institute, Tohoku University, Sendai 980-8578

Received  July 2010 Revised  October 2010 Published  April 2011

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It was shown in our previous paper that in some parameter range, the problem has a solution with a moving singularity that becomes anomalous in finite time. Our concern is a blow-up solution with a moving singularity. In this paper, we show that there exists a solution with a moving singularity such that it blows up at space infinity.
Citation: Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
##### References:
 [1] C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations,, J. Geometric Analysis, 9 (1999), 221.   Google Scholar [2] 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. IA Math., 16 (1966), 105.   Google Scholar [3] Y. Giga and N. Umeda, Blow-up directions at space infinity for solutions of semilinear heat equations,, Bol. Soc. Parana. Mat., 23 (2005), 9.  doi: 10.5269/bspm.v23i1-2.7450.  Google Scholar [4] Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heat equations,, J. Math. Anal. Appl., 316 (2006), 538.  doi: 10.1016/j.jmaa.2005.05.007.  Google Scholar [5] A. A. Lacey, The form of blow-up for nonlinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 183.   Google Scholar [6] O. A. Ladyž zenskaja, V. A. Solonnikov and N. M. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type,", Amer. Math. Soc., 23 (1968).   Google Scholar [7] N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion,, J. Math. Anal. Appl., 261 (2001), 350.  doi: 10.1006/jmaa.2001.7530.  Google Scholar [8] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts,, Birkh\, (2007).   Google Scholar [9] S. Sato and E. Yanagida, Solutions with Moving Singularities for a Semilinear Parabolic Equation,, J. Differential Equations, 246 (2009), 724.  doi: 10.1016/j.jde.2008.09.004.  Google Scholar [10] S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation,, Discrete and Continuous Dynamical Systems-Series A, 26 (2010), 313.  doi: 10.3934/dcds.2010.26.313.  Google Scholar [11] S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation,, preprint., ().   Google Scholar [12] Y. Seki, On directional blow-up for quasilinear parabolic equations with fast diffusion,, J. Math. Anal. Appl., 338 (2008), 572.  doi: 10.1016/j.jmaa.2007.05.033.  Google Scholar [13] Y. Seki, R. Suzuki and N. Umeda, Blow-up directions for quasilinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 379.  doi: 10.1017/S0308210506000801.  Google Scholar [14] M. Shimojō, The global profile of blow-up at space infinity in semilinear heat equations,, J. Math. Kyoto Univ., 48 (2008), 339.   Google Scholar [15] L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,", Pitman Research Notes in Mathematics Series, 353 (1996).   Google Scholar

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##### References:
 [1] C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations,, J. Geometric Analysis, 9 (1999), 221.   Google Scholar [2] 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. IA Math., 16 (1966), 105.   Google Scholar [3] Y. Giga and N. Umeda, Blow-up directions at space infinity for solutions of semilinear heat equations,, Bol. Soc. Parana. Mat., 23 (2005), 9.  doi: 10.5269/bspm.v23i1-2.7450.  Google Scholar [4] Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heat equations,, J. Math. Anal. Appl., 316 (2006), 538.  doi: 10.1016/j.jmaa.2005.05.007.  Google Scholar [5] A. A. Lacey, The form of blow-up for nonlinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 183.   Google Scholar [6] O. A. Ladyž zenskaja, V. A. Solonnikov and N. M. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type,", Amer. Math. Soc., 23 (1968).   Google Scholar [7] N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion,, J. Math. Anal. Appl., 261 (2001), 350.  doi: 10.1006/jmaa.2001.7530.  Google Scholar [8] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts,, Birkh\, (2007).   Google Scholar [9] S. Sato and E. Yanagida, Solutions with Moving Singularities for a Semilinear Parabolic Equation,, J. Differential Equations, 246 (2009), 724.  doi: 10.1016/j.jde.2008.09.004.  Google Scholar [10] S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation,, Discrete and Continuous Dynamical Systems-Series A, 26 (2010), 313.  doi: 10.3934/dcds.2010.26.313.  Google Scholar [11] S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation,, preprint., ().   Google Scholar [12] Y. Seki, On directional blow-up for quasilinear parabolic equations with fast diffusion,, J. Math. Anal. Appl., 338 (2008), 572.  doi: 10.1016/j.jmaa.2007.05.033.  Google Scholar [13] Y. Seki, R. Suzuki and N. Umeda, Blow-up directions for quasilinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 379.  doi: 10.1017/S0308210506000801.  Google Scholar [14] M. Shimojō, The global profile of blow-up at space infinity in semilinear heat equations,, J. Math. Kyoto Univ., 48 (2008), 339.   Google Scholar [15] L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,", Pitman Research Notes in Mathematics Series, 353 (1996).   Google Scholar
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