Existence of weak solutions to doubly degenerate diffusion equations

Aleš Matas; Jochen Merker

Applications of Mathematics (2012)

  • Volume: 57, Issue: 1, page 43-69
  • ISSN: 0862-7940

Abstract

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We prove existence of weak solutions to doubly degenerate diffusion equations u ˙ = Δ p u m - 1 + f ( m , p 2 ) by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains Ω n with Dirichlet or Neumann boundary conditions. The function f can be an inhomogeneity or a nonlinearity involving terms of the form f ( u ) or div ( F ( u ) ) . In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.

How to cite

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Matas, Aleš, and Merker, Jochen. "Existence of weak solutions to doubly degenerate diffusion equations." Applications of Mathematics 57.1 (2012): 43-69. <http://eudml.org/doc/246610>.

@article{Matas2012,
abstract = {We prove existence of weak solutions to doubly degenerate diffusion equations \begin\{equation*\} \dot\{u\} = \Delta \_p u^\{m-1\} + f \quad (m,p \ge 2) \end\{equation*\} by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains $\Omega \subset \mathbb \{R\}^n$ with Dirichlet or Neumann boundary conditions. The function $f$ can be an inhomogeneity or a nonlinearity involving terms of the form $f(u)$ or $\operatorname\{div\}(F(u))$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.},
author = {Matas, Aleš, Merker, Jochen},
journal = {Applications of Mathematics},
keywords = {$p$-Laplacian; doubly nonlinear evolution equation; weak solution; -Laplacian; doubly nonlinear evolution equation; weak solution},
language = {eng},
number = {1},
pages = {43-69},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of weak solutions to doubly degenerate diffusion equations},
url = {http://eudml.org/doc/246610},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Matas, Aleš
AU - Merker, Jochen
TI - Existence of weak solutions to doubly degenerate diffusion equations
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 43
EP - 69
AB - We prove existence of weak solutions to doubly degenerate diffusion equations \begin{equation*} \dot{u} = \Delta _p u^{m-1} + f \quad (m,p \ge 2) \end{equation*} by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains $\Omega \subset \mathbb {R}^n$ with Dirichlet or Neumann boundary conditions. The function $f$ can be an inhomogeneity or a nonlinearity involving terms of the form $f(u)$ or $\operatorname{div}(F(u))$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.
LA - eng
KW - $p$-Laplacian; doubly nonlinear evolution equation; weak solution; -Laplacian; doubly nonlinear evolution equation; weak solution
UR - http://eudml.org/doc/246610
ER -

References

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  1. Adams, R. A., Sobolev Spaces, Academic Press New Yorek-San Francisco-London (1975). (1975) Zbl0314.46030MR0450957
  2. Alt, H. W., Luckhaus, S., 10.1007/BF01176474, Math. Z. 183 (1983), 311-341. (1983) Zbl0497.35049MR0706391DOI10.1007/BF01176474
  3. Benilan, Ph., Equations d'évolution dans un espace de Banach quelconque et applications, PhD. Thèse Université de Paris XI Orsay (1972). (1972) 
  4. DiBenedetto, E., Degenerate Parabolic Equations, Springer New York (1993). (1993) Zbl0794.35090MR1230384
  5. Bonforte, M., Grillo, G., Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Math. Iberoam. 22 (2006), 111-129. (2006) Zbl1103.35021MR2268115
  6. Caisheng, Chen, 10.1006/jmaa.1999.6695, J. Math. Anal. Appl. 244 (2000), 133-146. (2000) MR1746793DOI10.1006/jmaa.1999.6695
  7. Cipriano, F., Grillo, G., 10.1006/jdeq.2000.3985, J. Differ. Equations 177 (2001), 209-234. (2001) Zbl1036.35043MR1867617DOI10.1006/jdeq.2000.3985
  8. Diaz, J. I., Padial, J. F., Uniqueness and existence of solutions in the B V t ( Q ) space to a doubly nonlinear parabolic problem, Publ. Math. Barcelona 40 (1996), 527-560. (1996) MR1425634
  9. Igbida, N., Urbano, J. M., 10.1007/s00030-003-1030-0, NoDEA, Nonlinear Differ. Equ. Appl. 10 (2003), 287-307. (2003) Zbl1024.35054MR1994812DOI10.1007/s00030-003-1030-0
  10. Ivanov, A. V., 10.1007/BF02398459, J. Math. Sci. 83 (1997), 22-37. (1997) MR1328634DOI10.1007/BF02398459
  11. Maitre, E., 10.1155/S0161171203106175, IJMMS, Int. J. Math, Math. Sci. 27 (2003), 1725-1730. (2003) Zbl1032.46032MR1981027DOI10.1155/S0161171203106175
  12. Merker, J., Generalizations of logarithmic Sobolev inequalities, Discrete Contin. Dyn. Syst., Ser. S 1 (2008), 329-338. (2008) Zbl1152.35326MR2379911
  13. Simon, J., Compact sets in the space L p ( 0 , T ; B ) , Ann. Mat. Pura. Appl., IV. Ser. 146 (1987), 65-96. (1987) MR0916688
  14. Wu, Z., Zhao, J., Yin, J., Li, H., Nonlinear Evolution Equations, World Scientific Singapore (2001). (2001) 
  15. Zheng, S., Nonlinear Evolution Equations, CRC-Press/Chapman & Hall Boca Raton (2004). (2004) Zbl1085.47058MR2088362

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