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$p$-Laplacian; doubly nonlinear evolution equation; weak solution
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 $\div (F(u))$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.
[1] Adams, R. A.: Sobolev Spaces. Academic Press New Yorek-San Francisco-London (1975). MR 0450957 | Zbl 0314.46030
[2] Alt, H. W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311-341. DOI 10.1007/BF01176474 | MR 0706391 | Zbl 0497.35049
[3] Benilan, Ph.: Equations d'évolution dans un espace de Banach quelconque et applications. PhD. Thèse Université de Paris XI Orsay (1972).
[4] DiBenedetto, E.: Degenerate Parabolic Equations. Springer New York (1993). MR 1230384 | Zbl 0794.35090
[5] Bonforte, M., Grillo, G.: Super and ultracontractive bounds for doubly nonlinear evolution equations. Rev. Math. Iberoam. 22 (2006), 111-129. MR 2268115 | Zbl 1103.35021
[6] Caisheng, Chen: Global existence and $L^\infty$ estimates of solution for doubly nonlinear parabolic equation. J. Math. Anal. Appl. 244 (2000), 133-146. DOI 10.1006/jmaa.1999.6695 | MR 1746793
[7] Cipriano, F., Grillo, G.: Uniform bounds for solutions to quasilinear parabolic equations. J. Differ. Equations 177 (2001), 209-234. DOI 10.1006/jdeq.2000.3985 | MR 1867617 | Zbl 1036.35043
[8] Diaz, J. I., Padial, J. F.: Uniqueness and existence of solutions in the $BV_t(Q)$ space to a doubly nonlinear parabolic problem. Publ. Math. Barcelona 40 (1996), 527-560. MR 1425634
[9] Igbida, N., Urbano, J. M.: Uniqueness for nonlinear degenerate problems. NoDEA, Nonlinear Differ. Equ. Appl. 10 (2003), 287-307. DOI 10.1007/s00030-003-1030-0 | MR 1994812 | Zbl 1024.35054
[10] Ivanov, A. V.: Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83 (1997), 22-37. DOI 10.1007/BF02398459 | MR 1328634
[11] Maitre, E.: On a nonlinear compactness lemma in $L^p(0,T;B)$. IJMMS, Int. J. Math, Math. Sci. 27 (2003), 1725-1730. DOI 10.1155/S0161171203106175 | MR 1981027 | Zbl 1032.46032
[12] Merker, J.: Generalizations of logarithmic Sobolev inequalities. Discrete Contin. Dyn. Syst., Ser. S 1 (2008), 329-338. MR 2379911 | Zbl 1152.35326
[13] Simon, J.: Compact sets in the space $L^p(0,T;B)$. Ann. Mat. Pura. Appl., IV. Ser. 146 (1987), 65-96. MR 0916688
[14] Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Evolution Equations. World Scientific Singapore (2001).
[15] Zheng, S.: Nonlinear Evolution Equations. CRC-Press/Chapman & Hall Boca Raton (2004). MR 2088362 | Zbl 1085.47058
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