Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Amy Poh Ai Ling; Masahiko Shimojō

Mathematica Bohemica (2019)

  • Volume: 144, Issue: 3, page 287-297
  • ISSN: 0862-7959

Abstract

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We consider solutions of quasilinear equations in with the initial data satisfying and for some constant . It is known that if with , the blow-up set is empty. We find solutions that blow up throughout when .

How to cite

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Ling, Amy Poh Ai, and Shimojō, Masahiko. "Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data." Mathematica Bohemica 144.3 (2019): 287-297. <http://eudml.org/doc/294294>.

@article{Ling2019,
abstract = {We consider solutions of quasilinear equations $u_\{t\}=\Delta u^\{m\} + u^\{p\}$ in $\mathbb \{R\}^\{N\}$ with the initial data $u_\{0\}$ satisfying $0 < u_\{0\}< M$ and $\lim _\{|x|\rightarrow \infty \}u_\{0\}(x)=M$ for some constant $M>0$. It is known that if $0<m<p$ with $p>1$, the blow-up set is empty. We find solutions $u$ that blow up throughout $\mathbb \{R\}^\{N\}$ when $m>p>1$.},
author = {Ling, Amy Poh Ai, Shimojō, Masahiko},
journal = {Mathematica Bohemica},
keywords = {quasilinear heat equation; total blow-up; blow-up only at space infinity},
language = {eng},
number = {3},
pages = {287-297},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data},
url = {http://eudml.org/doc/294294},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Ling, Amy Poh Ai
AU - Shimojō, Masahiko
TI - Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 3
SP - 287
EP - 297
AB - We consider solutions of quasilinear equations $u_{t}=\Delta u^{m} + u^{p}$ in $\mathbb {R}^{N}$ with the initial data $u_{0}$ satisfying $0 < u_{0}< M$ and $\lim _{|x|\rightarrow \infty }u_{0}(x)=M$ for some constant $M>0$. It is known that if $0<m<p$ with $p>1$, the blow-up set is empty. We find solutions $u$ that blow up throughout $\mathbb {R}^{N}$ when $m>p>1$.
LA - eng
KW - quasilinear heat equation; total blow-up; blow-up only at space infinity
UR - http://eudml.org/doc/294294
ER -

References

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  1. Galaktionov, V. A., Asymptotic behavior of unbounded solutions of the nonlinear equation near a “singular” point, Sov. Math., Dokl. 33 (1986), 840-844 translated from Dokl. Akad. Nauk SSSR 288 1986 1293-1297. (1986) Zbl0629.35061MR0852454
  2. Galaktionov, V. A., Kurdyumov, S. P., Mikhailov, A. P., Samarskii, A. A., Unbounded solutions of the Cauchy problem for the parabolic equation , Sov. Phys., Dokl. 25 (1980), 458-459 translated from Dokl. Akad. Nauk SSSR 252 1980 1362-1364. (1980) MR0581597
  3. Giga, Y., Umeda, N., 10.5269/bspm.v23i1-2.7450, Bol. Soc. Parana. Mat. (3) 23 (2005), 9-28 correction ibid. 24 2006 19-24. (2005) Zbl1173.35531MR2242285DOI10.5269/bspm.v23i1-2.7450
  4. Giga, Y., Umeda, N., 10.1016/j.jmaa.2005.05.007, J. Math. Anal. Appl. 316 (2006), 538-555. (2006) Zbl1106.35029MR2206688DOI10.1016/j.jmaa.2005.05.007
  5. Lacey, A. A., 10.1017/S0308210500025609, Proc. R. Soc. Edinb., Sect. A 98 (1984), 183-202. (1984) Zbl0556.35077MR0765494DOI10.1017/S0308210500025609
  6. Ladyženskaja, O. A., Solonnikov, V. A., Ural'ceva, N. N., 10.1090/mmono/023, Translations of Mathematical Monographs 23. AMS, Providence (1968). (1968) Zbl0174.15403MR0241822DOI10.1090/mmono/023
  7. Mochizuki, K., Suzuki, R., 10.2969/jmsj/04430485, J. Math. Soc. Japan 44 (1992), 485-504. (1992) Zbl0805.35065MR1167379DOI10.2969/jmsj/04430485
  8. Oleinik, O. A., Kalašinkov, A. S., Chou, Y.-L., The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR, Ser. Mat. 22 (1958), 667-704 Russian. (1958) Zbl0093.10302MR0099834
  9. Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P., Mikhailov, A. P., 10.1515/9783110889864.535, De Gruyter Expositions in Mathematics 19. Walter de Gruyter, Berlin (1995). (1995) Zbl1020.35001MR1330922DOI10.1515/9783110889864.535
  10. Seki, Y., 10.1016/j.jmaa.2007.05.033, J. Math. Anal. Appl. 338 (2008), 572-587. (2008) Zbl1144.35030MR2386440DOI10.1016/j.jmaa.2007.05.033
  11. Seki, Y., Umeda, N., Suzuki, R., 10.1017/S0308210506000801, Proc. R. Soc. Edinb., Sect. A, Math. 138 (2008), 379-405. (2008) Zbl1167.35393MR2406697DOI10.1017/S0308210506000801
  12. Shimojō, M., 10.1215/kjm/1250271415, J. Math. Kyoto Univ. 48 (2008), 339-361. (2008) Zbl1184.35078MR2436740DOI10.1215/kjm/1250271415
  13. Suzuki, R., 10.2977/prims/1195169661, Publ. Res. Inst. Math. Sci. 27 (1991), 375-398. (1991) Zbl0789.35024MR1121244DOI10.2977/prims/1195169661

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