# Article

 Title: Vectorial quasilinear diffusion equation with dynamic boundary condition (English) Author: Nakayashiki, Ryota Language: English Journal: Proceedings of Equadiff 14 Volume: Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 Issue: 2017 Year: Pages: 211-220 . Category: math . Summary: In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S)$_\varepsilon$ with a nonnegative constant $\varepsilon$, and for any $\varepsilon\ge0$, (S)$_\varepsilon$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $\Omega$, and the parabolic equation on the boundary $\Gamma:=\partial \Omega$, having a sufficient smoothness. The objective of this study is to establish a mathematical method, which can enable us to handle the transmission systems of various vectorial mathematical models, such as the Bingham type flow equations, the Ginzburg– Landau type equations, and so on. On this basis, we set the goal of this paper to prove two Main Theorems, concerned with the well-posedness of (S)$_\varepsilon$(S) with the precise representation of solution, and $\varepsilon$-dependence of (S)$_\varepsilon$, for $\varepsilon \ge0$. (English) Keyword: Vectorial parabolic equation, quasilinear diffusion, dynamic boundary condition MSC: 35K40 MSC: 35K59 MSC: 35R35 . Date available: 2019-09-27T07:58:41Z Last updated: 2019-09-27 Stable URL: http://hdl.handle.net/10338.dmlcz/703016 . Reference: [1] Attouch, H.: Variational Convergence for Functions and Operators.. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 0773850 Reference: [2] Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces.. Springer Monographs in Mathematics. Springer, New York, 2010. MR 2582280 Reference: [3] Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert.. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562 Reference: [4] Colli, P., Gilardi, G., Nakayashiki, R., Shirakawa, K.: A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions.. Nonlinear Anal., 158:32–59, 2017. MR 3661429, 10.1016/j.na.2017.03.020 Reference: [5] Giga, Y., Kashima, Y., Yamazaki, N.: Local solvability of a constrained gradient system of total variation.. Abstr. Appl. Anal., (8):651–682, 2004. MR 2096945 Reference: [6] Ito, A., Yamazaki, N., Kenmochi, N.: Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces.. Discrete Contin. Dynam. Systems, (Added Volume I):327–349, 1998. Dynamical systems and differential equations, Vol. I (Springfield, MO, 1996). MR 1720614 Reference: [7] Kenmochi, N.: Solvability of nonlinear evolution equations with time-dependent constraints and applications.. Bull. Fac. Education, Chiba Univ. http://ci.nii.ac.jp/naid/110004715232,30:1–87, 1981. Reference: [8] Kenmochi, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems.. J. Math. Soc. Japan, 27:121–149, 1975. MR 0372419, 10.2969/jmsj/02710121 Reference: [9] Mosco, U.: Convergence of convex sets and of solutions of variational inequalities.. Advances in Math., 3:510–585, 1969. MR 0298508, 10.1016/0001-8708(69)90009-7 Reference: [10] Nakayashiki, R., Shirakawa, K.: Weak formulation for singular diffusion equations with dynamic boundary condition.. Springer INdAM Series. to appear, 2017. MR 3751650 Reference: [11] Savaré, G., Visintin, A.: Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase.. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8(1):49–89, 1997. MR 1484545 .

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